Alternating Maps

We construct the bundled function AlternatingMap, which extends MultilinearMap with all the arguments of the same type.

Main definitions

• AlternatingMap R M N ι is the space of R-linear alternating maps from ι → M to N.
• f.map_eq_zero_of_eq expresses that f is zero when two inputs are equal.
• f.map_swap expresses that f is negated when two inputs are swapped.
• f.map_perm expresses how f varies by a sign change under a permutation of its inputs.
• An AddCommMonoid, AddCommGroup, and Module structure over AlternatingMaps that matches the definitions over MultilinearMaps.
• MultilinearMap.domDomCongr, for permuting the elements within a family.
• MultilinearMap.alternatization, which makes an alternating map out of a non-alternating one.
• AlternatingMap.curryLeft, for binding the leftmost argument of an alternating map indexed by Fin n.succ.

Implementation notes

AlternatingMap is defined in terms of map_eq_zero_of_eq, as this is easier to work with than using map_swap as a definition, and does not require Neg N.

AlternatingMaps are provided with a coercion to MultilinearMap, along with a set of norm_cast lemmas that act on the algebraic structure:

• AlternatingMap.coe_add
• AlternatingMap.coe_zero
• AlternatingMap.coe_sub
• AlternatingMap.coe_neg
• AlternatingMap.coe_smul

structure AlternatingMap (R : Type u_1) [Semiring R] (M : Type u_2) [AddCommMonoid M] [Module R M] (N : Type u_3) [AddCommMonoid N] [Module R N] (ι : Type u_7) extends MultilinearMap : Type (max (max u_2 u_3) u_7)

An alternating map from ι → M to N, denoted M [⋀^ι]→ₗ[R] N, is a multilinear map that vanishes when two of its arguments are equal.

• toFun : (ι → M) → N
• map_add' : ∀ [inst : DecidableEq ι] (m : ι → M) (i : ι) (x y : M), (↑self).toFun (Function.update m i (x + y)) = (↑self).toFun (Function.update m i x) + (↑self).toFun (Function.update m i y)
• map_smul' : ∀ [inst : DecidableEq ι] (m : ι → M) (i : ι) (c : R) (x : M), (↑self).toFun (Function.update m i (c • x)) = c • (↑self).toFun (Function.update m i x)
• map_eq_zero_of_eq' : ∀ (v : ι → M) (i j : ι), v i = v j → i ≠ j → (↑self).toFun v = 0

  The map is alternating: if v has two equal coordinates, then f v = 0.

theorem AlternatingMap.map_eq_zero_of_eq' {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (self : M [⋀^ι]→ₗ[R] N) (v : ι → M) (i : ι) (j : ι) : v i = v j → i ≠ j → (↑self).toFun v = 0

The map is alternating: if v has two equal coordinates, then f v = 0.

def «term_[⋀^_]→ₗ[_]_» : Lean.TrailingParserDescr

An alternating map from ι → M to N, denoted M [⋀^ι]→ₗ[R] N, is a multilinear map that vanishes when two of its arguments are equal.

Equations

• One or more equations did not get rendered due to their size.

Basic coercion simp lemmas, largely copied from RingHom and MultilinearMap

instance AlternatingMap.instFunLike {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} : FunLike (M [⋀^ι]→ₗ[R] N) (ι → M) N

Equations

• AlternatingMap.instFunLike = { coe := fun (f : M [⋀^ι]→ₗ[R] N) => (↑f).toFun, coe_injective' := ⋯ }

instance AlternatingMap.coeFun {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} : CoeFun (M [⋀^ι]→ₗ[R] N) fun (x : M [⋀^ι]→ₗ[R] N) => (ι → M) → N

Equations

• AlternatingMap.coeFun = { coe := DFunLike.coe }

@[simp]

theorem AlternatingMap.toFun_eq_coe {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) : (↑f).toFun = ⇑f

@[simp]

theorem AlternatingMap.coe_mk {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : MultilinearMap R (fun (x : ι) => M) N) (h : ∀ (v : ι → M) (i j : ι), v i = v j → i ≠ j → f.toFun v = 0) : ⇑{ toMultilinearMap := f, map_eq_zero_of_eq' := h } = ⇑f

theorem AlternatingMap.congr_fun {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {f : M [⋀^ι]→ₗ[R] N} {g : M [⋀^ι]→ₗ[R] N} (h : f = g) (x : ι → M) : f x = g x

theorem AlternatingMap.congr_arg {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) {x : ι → M} {y : ι → M} (h : x = y) : f x = f y

theorem AlternatingMap.coe_injective {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} : Function.Injective DFunLike.coe

theorem AlternatingMap.coe_inj {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {f : M [⋀^ι]→ₗ[R] N} {g : M [⋀^ι]→ₗ[R] N} : ⇑f = ⇑g ↔ f = g

theorem AlternatingMap.ext {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {f : M [⋀^ι]→ₗ[R] N} {f' : M [⋀^ι]→ₗ[R] N} (H : ∀ (x : ι → M), f x = f' x) : f = f'

theorem AlternatingMap.ext_iff {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {f : M [⋀^ι]→ₗ[R] N} {g : M [⋀^ι]→ₗ[R] N} : f = g ↔ ∀ (x : ι → M), f x = g x

instance AlternatingMap.coe {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} : Coe (M [⋀^ι]→ₗ[R] N) (MultilinearMap R (fun (x : ι) => M) N)

Equations

• AlternatingMap.coe = { coe := fun (x : M [⋀^ι]→ₗ[R] N) => ↑x }

@[simp]

theorem AlternatingMap.coe_multilinearMap {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) : ⇑↑f = ⇑f

theorem AlternatingMap.coe_multilinearMap_injective {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} : Function.Injective AlternatingMap.toMultilinearMap

theorem AlternatingMap.coe_multilinearMap_mk {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : (ι → M) → N) (h₁ : ∀ [inst : DecidableEq ι] (m : ι → M) (i : ι) (x y : M), f (Function.update m i (x + y)) = f (Function.update m i x) + f (Function.update m i y)) (h₂ : ∀ [inst : DecidableEq ι] (m : ι → M) (i : ι) (c : R) (x : M), f (Function.update m i (c • x)) = c • f (Function.update m i x)) (h₃ : ∀ (v : ι → M) (i j : ι), v i = v j → i ≠ j → { toFun := f, map_add' := ⋯, map_smul' := ⋯ }.toFun v = 0) : ↑{ toFun := f, map_add' := ⋯, map_smul' := ⋯, map_eq_zero_of_eq' := h₃ } = { toFun := f, map_add' := h₁, map_smul' := h₂ }

Simp-normal forms of the structure fields

These are expressed in terms of ⇑f instead of f.toFun.

@[simp]

theorem AlternatingMap.map_add {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (v : ι → M) [DecidableEq ι] (i : ι) (x : M) (y : M) : f (Function.update v i (x + y)) = f (Function.update v i x) + f (Function.update v i y)

@[simp]

theorem AlternatingMap.map_sub {R : Type u_1} [Semiring R] {M' : Type u_5} [AddCommGroup M'] [Module R M'] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g' : M' [⋀^ι]→ₗ[R] N') (v' : ι → M') [DecidableEq ι] (i : ι) (x : M') (y : M') : g' (Function.update v' i (x - y)) = g' (Function.update v' i x) - g' (Function.update v' i y)

@[simp]

theorem AlternatingMap.map_neg {R : Type u_1} [Semiring R] {M' : Type u_5} [AddCommGroup M'] [Module R M'] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g' : M' [⋀^ι]→ₗ[R] N') (v' : ι → M') [DecidableEq ι] (i : ι) (x : M') : g' (Function.update v' i (-x)) = -g' (Function.update v' i x)

@[simp]

theorem AlternatingMap.map_smul {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (v : ι → M) [DecidableEq ι] (i : ι) (r : R) (x : M) : f (Function.update v i (r • x)) = r • f (Function.update v i x)

@[simp]

theorem AlternatingMap.map_eq_zero_of_eq {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (v : ι → M) {i : ι} {j : ι} (h : v i = v j) (hij : i ≠ j) : f v = 0

theorem AlternatingMap.map_coord_zero {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) {m : ι → M} (i : ι) (h : m i = 0) : f m = 0

@[simp]

theorem AlternatingMap.map_update_zero {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) [DecidableEq ι] (m : ι → M) (i : ι) : f (Function.update m i 0) = 0

@[simp]

theorem AlternatingMap.map_zero {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) [Nonempty ι] : f 0 = 0

theorem AlternatingMap.map_eq_zero_of_not_injective {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (v : ι → M) (hv : ¬Function.Injective v) : f v = 0

Algebraic structure inherited from MultilinearMap

AlternatingMap carries the same AddCommMonoid, AddCommGroup, and Module structure as MultilinearMap

instance AlternatingMap.smul {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {S : Type u_10} [Monoid S] [DistribMulAction S N] [SMulCommClass R S N] : SMul S (M [⋀^ι]→ₗ[R] N)

Equations

• AlternatingMap.smul = { smul := fun (c : S) (f : M [⋀^ι]→ₗ[R] N) => let __src := c • ↑f; { toMultilinearMap := __src, map_eq_zero_of_eq' := ⋯ } }

@[simp]

theorem AlternatingMap.smul_apply {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) {S : Type u_10} [Monoid S] [DistribMulAction S N] [SMulCommClass R S N] (c : S) (m : ι → M) : (c • f) m = c • f m

theorem AlternatingMap.coe_smul {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) {S : Type u_10} [Monoid S] [DistribMulAction S N] [SMulCommClass R S N] (c : S) : ↑(c • f) = c • ↑f

theorem AlternatingMap.coeFn_smul {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {S : Type u_10} [Monoid S] [DistribMulAction S N] [SMulCommClass R S N] (c : S) (f : M [⋀^ι]→ₗ[R] N) : ⇑(c • f) = c • ⇑f

instance AlternatingMap.isCentralScalar {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {S : Type u_10} [Monoid S] [DistribMulAction S N] [SMulCommClass R S N] [DistribMulAction Sᵐᵒᵖ N] [IsCentralScalar S N] : IsCentralScalar S (M [⋀^ι]→ₗ[R] N)

Equations

• ⋯ = ⋯

@[simp]

theorem AlternatingMap.prod_apply {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {P : Type u_4} [AddCommMonoid P] [Module R P] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (g : M [⋀^ι]→ₗ[R] P) (m : (i : ι) → (fun (x : ι) => M) i) : (f.prod g) m = (f m, g m)

def AlternatingMap.prod {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {P : Type u_4} [AddCommMonoid P] [Module R P] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (g : M [⋀^ι]→ₗ[R] P) : M [⋀^ι]→ₗ[R] (N × P)

The cartesian product of two alternating maps, as an alternating map.

Equations

• f.prod g = let __src := (↑f).prod ↑g; { toMultilinearMap := __src, map_eq_zero_of_eq' := ⋯ }

@[simp]

theorem AlternatingMap.coe_prod {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {P : Type u_4} [AddCommMonoid P] [Module R P] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (g : M [⋀^ι]→ₗ[R] P) : ↑(f.prod g) = (↑f).prod ↑g

@[simp]

theorem AlternatingMap.pi_apply {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {ι : Type u_7} {ι' : Type u_10} {N : ι' → Type u_11} [(i : ι') → AddCommMonoid (N i)] [(i : ι') → Module R (N i)] (f : (i : ι') → M [⋀^ι]→ₗ[R] N i) (m : (i : ι) → (fun (x : ι) => M) i) (i : ι') : (AlternatingMap.pi f) m i = (f i) m

def AlternatingMap.pi {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {ι : Type u_7} {ι' : Type u_10} {N : ι' → Type u_11} [(i : ι') → AddCommMonoid (N i)] [(i : ι') → Module R (N i)] (f : (i : ι') → M [⋀^ι]→ₗ[R] N i) : M [⋀^ι]→ₗ[R] ((i : ι') → N i)

Combine a family of alternating maps with the same domain and codomains N i into an alternating map taking values in the space of functions Π i, N i.

Equations

• AlternatingMap.pi f = let __src := MultilinearMap.pi fun (a : ι') => ↑(f a); { toMultilinearMap := __src, map_eq_zero_of_eq' := ⋯ }

@[simp]

theorem AlternatingMap.coe_pi {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {ι : Type u_7} {ι' : Type u_10} {N : ι' → Type u_11} [(i : ι') → AddCommMonoid (N i)] [(i : ι') → Module R (N i)] (f : (i : ι') → M [⋀^ι]→ₗ[R] N i) : ↑(AlternatingMap.pi f) = MultilinearMap.pi fun (a : ι') => ↑(f a)

@[simp]

theorem AlternatingMap.smulRight_apply {R : Type u_10} {M₁ : Type u_11} {M₂ : Type u_12} {ι : Type u_13} [CommSemiring R] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module R M₂] (f : M₁ [⋀^ι]→ₗ[R] R) (z : M₂) : ∀ (a : (i : ι) → (fun (x : ι) => M₁) i), (f.smulRight z) a = f a • z

def AlternatingMap.smulRight {R : Type u_10} {M₁ : Type u_11} {M₂ : Type u_12} {ι : Type u_13} [CommSemiring R] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module R M₂] (f : M₁ [⋀^ι]→ₗ[R] R) (z : M₂) : M₁ [⋀^ι]→ₗ[R] M₂

Given an alternating R-multilinear map f taking values in R, f.smul_right z is the map sending m to f m • z.

Equations

• f.smulRight z = let __src := (↑f).smulRight z; { toMultilinearMap := __src, map_eq_zero_of_eq' := ⋯ }

@[simp]

theorem AlternatingMap.coe_smulRight {R : Type u_10} {M₁ : Type u_11} {M₂ : Type u_12} {ι : Type u_13} [CommSemiring R] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module R M₂] (f : M₁ [⋀^ι]→ₗ[R] R) (z : M₂) : ↑(f.smulRight z) = (↑f).smulRight z

instance AlternatingMap.add {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} : Add (M [⋀^ι]→ₗ[R] N)

Equations

• AlternatingMap.add = { add := fun (a b : M [⋀^ι]→ₗ[R] N) => let __src := ↑a + ↑b; { toMultilinearMap := __src, map_eq_zero_of_eq' := ⋯ } }

@[simp]

theorem AlternatingMap.add_apply {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (f' : M [⋀^ι]→ₗ[R] N) (v : ι → M) : (f + f') v = f v + f' v

theorem AlternatingMap.coe_add {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (f' : M [⋀^ι]→ₗ[R] N) : ↑(f + f') = ↑f + ↑f'

instance AlternatingMap.zero {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} : Zero (M [⋀^ι]→ₗ[R] N)

Equations

• AlternatingMap.zero = { zero := let __src := 0; { toMultilinearMap := __src, map_eq_zero_of_eq' := ⋯ } }

@[simp]

theorem AlternatingMap.zero_apply {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (v : ι → M) : 0 v = 0

theorem AlternatingMap.coe_zero {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} : ↑0 = 0

@[simp]

theorem AlternatingMap.mk_zero {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} : { toMultilinearMap := 0, map_eq_zero_of_eq' := ⋯ } = 0

instance AlternatingMap.inhabited {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} : Inhabited (M [⋀^ι]→ₗ[R] N)

Equations

• AlternatingMap.inhabited = { default := 0 }

instance AlternatingMap.addCommMonoid {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} : AddCommMonoid (M [⋀^ι]→ₗ[R] N)

Equations

• AlternatingMap.addCommMonoid = Function.Injective.addCommMonoid DFunLike.coe ⋯ ⋯ ⋯ ⋯

instance AlternatingMap.neg {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} : Neg (M [⋀^ι]→ₗ[R] N')

Equations

• AlternatingMap.neg = { neg := fun (f : M [⋀^ι]→ₗ[R] N') => let __src := -↑f; { toMultilinearMap := __src, map_eq_zero_of_eq' := ⋯ } }

@[simp]

theorem AlternatingMap.neg_apply {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g : M [⋀^ι]→ₗ[R] N') (m : ι → M) : (-g) m = -g m

theorem AlternatingMap.coe_neg {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g : M [⋀^ι]→ₗ[R] N') : ↑(-g) = -↑g

instance AlternatingMap.sub {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} : Sub (M [⋀^ι]→ₗ[R] N')

Equations

• AlternatingMap.sub = { sub := fun (f g : M [⋀^ι]→ₗ[R] N') => let __src := ↑f - ↑g; { toMultilinearMap := __src, map_eq_zero_of_eq' := ⋯ } }

@[simp]

theorem AlternatingMap.sub_apply {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g : M [⋀^ι]→ₗ[R] N') (g₂ : M [⋀^ι]→ₗ[R] N') (m : ι → M) : (g - g₂) m = g m - g₂ m

theorem AlternatingMap.coe_sub {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g : M [⋀^ι]→ₗ[R] N') (g₂ : M [⋀^ι]→ₗ[R] N') : ↑(g - g₂) = ↑g - ↑g₂

instance AlternatingMap.addCommGroup {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} : AddCommGroup (M [⋀^ι]→ₗ[R] N')

Equations

• AlternatingMap.addCommGroup = Function.Injective.addCommGroup DFunLike.coe ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance AlternatingMap.distribMulAction {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {S : Type u_10} [Monoid S] [DistribMulAction S N] [SMulCommClass R S N] : DistribMulAction S (M [⋀^ι]→ₗ[R] N)

Equations

• AlternatingMap.distribMulAction = DistribMulAction.mk ⋯ ⋯

instance AlternatingMap.module {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {S : Type u_10} [Semiring S] [Module S N] [SMulCommClass R S N] : Module S (M [⋀^ι]→ₗ[R] N)

The space of multilinear maps over an algebra over R is a module over R, for the pointwise addition and scalar multiplication.

Equations

• AlternatingMap.module = Module.mk ⋯ ⋯

instance AlternatingMap.noZeroSMulDivisors {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {S : Type u_10} [Semiring S] [Module S N] [SMulCommClass R S N] [NoZeroSMulDivisors S N] : NoZeroSMulDivisors S (M [⋀^ι]→ₗ[R] N)

Equations

• ⋯ = ⋯

@[simp]

theorem AlternatingMap.ofSubsingleton_symm_apply_apply (R : Type u_1) [Semiring R] (M : Type u_2) [AddCommMonoid M] [Module R M] (N : Type u_3) [AddCommMonoid N] [Module R N] {ι : Type u_7} [Subsingleton ι] (i : ι) (f : M [⋀^ι]→ₗ[R] N) (x : M) : ((AlternatingMap.ofSubsingleton R M N i).symm f) x = f fun (x_1 : ι) => x

@[simp]

theorem AlternatingMap.ofSubsingleton_apply_apply (R : Type u_1) [Semiring R] (M : Type u_2) [AddCommMonoid M] [Module R M] (N : Type u_3) [AddCommMonoid N] [Module R N] {ι : Type u_7} [Subsingleton ι] (i : ι) (f : M →ₗ[R] N) (x : ι → M) : ((AlternatingMap.ofSubsingleton R M N i) f) x = f (x i)

def AlternatingMap.ofSubsingleton (R : Type u_1) [Semiring R] (M : Type u_2) [AddCommMonoid M] [Module R M] (N : Type u_3) [AddCommMonoid N] [Module R N] {ι : Type u_7} [Subsingleton ι] (i : ι) : (M →ₗ[R] N) ≃ M [⋀^ι]→ₗ[R] N

The natural equivalence between linear maps from M to N and 1-multilinear alternating maps from M to N.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem AlternatingMap.constOfIsEmpty_apply (R : Type u_1) [Semiring R] (M : Type u_2) [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] (ι : Type u_7) [IsEmpty ι] (m : N) : ⇑(AlternatingMap.constOfIsEmpty R M ι m) = Function.const (ι → M) m

def AlternatingMap.constOfIsEmpty (R : Type u_1) [Semiring R] (M : Type u_2) [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] (ι : Type u_7) [IsEmpty ι] (m : N) : M [⋀^ι]→ₗ[R] N

The constant map is alternating when ι is empty.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem AlternatingMap.codRestrict_apply_coe {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (p : Submodule R N) (h : ∀ (v : ι → M), f v ∈ p) (v : ι → M) : ↑((f.codRestrict p h) v) = f v

def AlternatingMap.codRestrict {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (p : Submodule R N) (h : ∀ (v : ι → M), f v ∈ p) : M [⋀^ι]→ₗ[R] ↥p

Restrict the codomain of an alternating map to a submodule.

Equations

• f.codRestrict p h = let __src := (↑f).codRestrict p h; { toFun := fun (v : ι → M) => ⟨f v, ⋯⟩, map_add' := ⋯, map_smul' := ⋯, map_eq_zero_of_eq' := ⋯ }

Composition with linear maps

def LinearMap.compAlternatingMap {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {N₂ : Type u_10} [AddCommMonoid N₂] [Module R N₂] (g : N →ₗ[R] N₂) : M [⋀^ι]→ₗ[R] N →+ M [⋀^ι]→ₗ[R] N₂

Composing an alternating map with a linear map on the left gives again an alternating map.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem LinearMap.coe_compAlternatingMap {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {N₂ : Type u_10} [AddCommMonoid N₂] [Module R N₂] (g : N →ₗ[R] N₂) (f : M [⋀^ι]→ₗ[R] N) : ⇑(g.compAlternatingMap f) = ⇑g ∘ ⇑f

@[simp]

theorem LinearMap.compAlternatingMap_apply {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {N₂ : Type u_10} [AddCommMonoid N₂] [Module R N₂] (g : N →ₗ[R] N₂) (f : M [⋀^ι]→ₗ[R] N) (m : ι → M) : (g.compAlternatingMap f) m = g (f m)

theorem LinearMap.smulRight_eq_comp {R : Type u_11} {M₁ : Type u_12} {M₂ : Type u_13} {ι : Type u_14} [CommSemiring R] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module R M₂] (f : M₁ [⋀^ι]→ₗ[R] R) (z : M₂) : f.smulRight z = (LinearMap.id.smulRight z).compAlternatingMap f

@[simp]

theorem LinearMap.subtype_compAlternatingMap_codRestrict {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (p : Submodule R N) (h : ∀ (v : ι → M), f v ∈ p) : p.subtype.compAlternatingMap (f.codRestrict p h) = f

@[simp]

theorem LinearMap.compAlternatingMap_codRestrict {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {N₂ : Type u_10} [AddCommMonoid N₂] [Module R N₂] (g : N →ₗ[R] N₂) (f : M [⋀^ι]→ₗ[R] N) (p : Submodule R N₂) (h : ∀ (c : N), g c ∈ p) : (LinearMap.codRestrict p g h).compAlternatingMap f = (g.compAlternatingMap f).codRestrict p ⋯

def AlternatingMap.compLinearMap {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (f : M [⋀^ι]→ₗ[R] N) (g : M₂ →ₗ[R] M) : M₂ [⋀^ι]→ₗ[R] N

Composing an alternating map with the same linear map on each argument gives again an alternating map.

Equations

• f.compLinearMap g = let __src := (↑f).compLinearMap fun (x : ι) => g; { toMultilinearMap := __src, map_eq_zero_of_eq' := ⋯ }

theorem AlternatingMap.coe_compLinearMap {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (f : M [⋀^ι]→ₗ[R] N) (g : M₂ →ₗ[R] M) : ⇑(f.compLinearMap g) = ⇑f ∘ fun (x : ι → M₂) => ⇑g ∘ x

@[simp]

theorem AlternatingMap.compLinearMap_apply {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (f : M [⋀^ι]→ₗ[R] N) (g : M₂ →ₗ[R] M) (v : ι → M₂) : (f.compLinearMap g) v = f fun (i : ι) => g (v i)

theorem AlternatingMap.compLinearMap_assoc {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] {M₃ : Type u_11} [AddCommMonoid M₃] [Module R M₃] (f : M [⋀^ι]→ₗ[R] N) (g₁ : M₂ →ₗ[R] M) (g₂ : M₃ →ₗ[R] M₂) : (f.compLinearMap g₁).compLinearMap g₂ = f.compLinearMap (g₁ ∘ₗ g₂)

Composing an alternating map twice with the same linear map in each argument is the same as composing with their composition.

@[simp]

theorem AlternatingMap.zero_compLinearMap {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (g : M₂ →ₗ[R] M) : AlternatingMap.compLinearMap 0 g = 0

@[simp]

theorem AlternatingMap.add_compLinearMap {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (f₁ : M [⋀^ι]→ₗ[R] N) (f₂ : M [⋀^ι]→ₗ[R] N) (g : M₂ →ₗ[R] M) : (f₁ + f₂).compLinearMap g = f₁.compLinearMap g + f₂.compLinearMap g

@[simp]

theorem AlternatingMap.compLinearMap_zero {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] [Nonempty ι] (f : M [⋀^ι]→ₗ[R] N) : f.compLinearMap 0 = 0

@[simp]

theorem AlternatingMap.compLinearMap_id {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) : f.compLinearMap LinearMap.id = f

Composing an alternating map with the identity linear map in each argument.

theorem AlternatingMap.compLinearMap_injective {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (f : M₂ →ₗ[R] M) (hf : Function.Surjective ⇑f) : Function.Injective fun (g : M [⋀^ι]→ₗ[R] N) => g.compLinearMap f

Composing with a surjective linear map is injective.

theorem AlternatingMap.compLinearMap_inj {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (f : M₂ →ₗ[R] M) (hf : Function.Surjective ⇑f) (g₁ : M [⋀^ι]→ₗ[R] N) (g₂ : M [⋀^ι]→ₗ[R] N) : g₁.compLinearMap f = g₂.compLinearMap f ↔ g₁ = g₂

@[simp]

theorem AlternatingMap.domLCongr_apply (R : Type u_1) [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (N : Type u_3) [AddCommMonoid N] [Module R N] (ι : Type u_7) {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (S : Type u_12) [Semiring S] [Module S N] [SMulCommClass R S N] (e : M ≃ₗ[R] M₂) (f : M [⋀^ι]→ₗ[R] N) : (AlternatingMap.domLCongr R N ι S e) f = f.compLinearMap ↑e.symm

def AlternatingMap.domLCongr (R : Type u_1) [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (N : Type u_3) [AddCommMonoid N] [Module R N] (ι : Type u_7) {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (S : Type u_12) [Semiring S] [Module S N] [SMulCommClass R S N] (e : M ≃ₗ[R] M₂) : M [⋀^ι]→ₗ[R] N ≃ₗ[S] M₂ [⋀^ι]→ₗ[R] N

Construct a linear equivalence between maps from a linear equivalence between domains.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem AlternatingMap.domLCongr_refl (R : Type u_1) [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (N : Type u_3) [AddCommMonoid N] [Module R N] (ι : Type u_7) (S : Type u_12) [Semiring S] [Module S N] [SMulCommClass R S N] : AlternatingMap.domLCongr R N ι S (LinearEquiv.refl R M) = LinearEquiv.refl S (M [⋀^ι]→ₗ[R] N)

@[simp]

theorem AlternatingMap.domLCongr_symm (R : Type u_1) [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (N : Type u_3) [AddCommMonoid N] [Module R N] (ι : Type u_7) {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (S : Type u_12) [Semiring S] [Module S N] [SMulCommClass R S N] (e : M ≃ₗ[R] M₂) : (AlternatingMap.domLCongr R N ι S e).symm = AlternatingMap.domLCongr R N ι S e.symm

theorem AlternatingMap.domLCongr_trans (R : Type u_1) [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (N : Type u_3) [AddCommMonoid N] [Module R N] (ι : Type u_7) {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] {M₃ : Type u_11} [AddCommMonoid M₃] [Module R M₃] (S : Type u_12) [Semiring S] [Module S N] [SMulCommClass R S N] (e : M ≃ₗ[R] M₂) (f : M₂ ≃ₗ[R] M₃) : AlternatingMap.domLCongr R N ι S e ≪≫ₗ AlternatingMap.domLCongr R N ι S f = AlternatingMap.domLCongr R N ι S (e ≪≫ₗ f)

@[simp]

theorem AlternatingMap.compLinearEquiv_eq_zero_iff {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (f : M [⋀^ι]→ₗ[R] N) (g : M₂ ≃ₗ[R] M) : f.compLinearMap ↑g = 0 ↔ f = 0

Composing an alternating map with the same linear equiv on each argument gives the zero map if and only if the alternating map is the zero map.

Other lemmas from MultilinearMap

theorem AlternatingMap.map_update_sum {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) {α : Type u_12} [DecidableEq ι] (t : Finset α) (i : ι) (g : α → M) (m : ι → M) : f (Function.update m i (∑ a ∈ t, g a)) = ∑ a ∈ t, f (Function.update m i (g a))

Theorems specific to alternating maps

Various properties of reordered and repeated inputs which follow from AlternatingMap.map_eq_zero_of_eq.

theorem AlternatingMap.map_update_self {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (v : ι → M) [DecidableEq ι] {i : ι} {j : ι} (hij : i ≠ j) : f (Function.update v i (v j)) = 0

theorem AlternatingMap.map_update_update {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (v : ι → M) [DecidableEq ι] {i : ι} {j : ι} (hij : i ≠ j) (m : M) : f (Function.update (Function.update v i m) j m) = 0

theorem AlternatingMap.map_swap_add {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (v : ι → M) [DecidableEq ι] {i : ι} {j : ι} (hij : i ≠ j) : f (v ∘ ⇑(Equiv.swap i j)) + f v = 0

theorem AlternatingMap.map_add_swap {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (v : ι → M) [DecidableEq ι] {i : ι} {j : ι} (hij : i ≠ j) : f v + f (v ∘ ⇑(Equiv.swap i j)) = 0

theorem AlternatingMap.map_swap {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g : M [⋀^ι]→ₗ[R] N') (v : ι → M) [DecidableEq ι] {i : ι} {j : ι} (hij : i ≠ j) : g (v ∘ ⇑(Equiv.swap i j)) = -g v

theorem AlternatingMap.map_perm {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g : M [⋀^ι]→ₗ[R] N') [DecidableEq ι] [Fintype ι] (v : ι → M) (σ : Equiv.Perm ι) : g (v ∘ ⇑σ) = Equiv.Perm.sign σ • g v

theorem AlternatingMap.map_congr_perm {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g : M [⋀^ι]→ₗ[R] N') (v : ι → M) [DecidableEq ι] [Fintype ι] (σ : Equiv.Perm ι) : g v = Equiv.Perm.sign σ • g (v ∘ ⇑σ)

@[simp]

theorem AlternatingMap.domDomCongr_apply {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (σ : ι ≃ ι') (f : M [⋀^ι]→ₗ[R] N) (v : ι' → M) : (AlternatingMap.domDomCongr σ f) v = f (v ∘ ⇑σ)

def AlternatingMap.domDomCongr {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (σ : ι ≃ ι') (f : M [⋀^ι]→ₗ[R] N) : M [⋀^ι']→ₗ[R] N

Transfer the arguments to a map along an equivalence between argument indices.

This is the alternating version of MultilinearMap.domDomCongr.

Equations

• AlternatingMap.domDomCongr σ f = let __src := MultilinearMap.domDomCongr σ ↑f; { toFun := fun (v : ι' → M) => f (v ∘ ⇑σ), map_add' := ⋯, map_smul' := ⋯, map_eq_zero_of_eq' := ⋯ }

@[simp]

theorem AlternatingMap.domDomCongr_refl {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) : AlternatingMap.domDomCongr (Equiv.refl ι) f = f

theorem AlternatingMap.domDomCongr_trans {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} {ι'' : Type u_9} (σ₁ : ι ≃ ι') (σ₂ : ι' ≃ ι'') (f : M [⋀^ι]→ₗ[R] N) : AlternatingMap.domDomCongr (σ₁.trans σ₂) f = AlternatingMap.domDomCongr σ₂ (AlternatingMap.domDomCongr σ₁ f)

@[simp]

theorem AlternatingMap.domDomCongr_zero {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (σ : ι ≃ ι') : AlternatingMap.domDomCongr σ 0 = 0

@[simp]

theorem AlternatingMap.domDomCongr_add {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (σ : ι ≃ ι') (f : M [⋀^ι]→ₗ[R] N) (g : M [⋀^ι]→ₗ[R] N) : AlternatingMap.domDomCongr σ (f + g) = AlternatingMap.domDomCongr σ f + AlternatingMap.domDomCongr σ g

@[simp]

theorem AlternatingMap.domDomCongr_smul {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} {S : Type u_12} [Monoid S] [DistribMulAction S N] [SMulCommClass R S N] (σ : ι ≃ ι') (c : S) (f : M [⋀^ι]→ₗ[R] N) : AlternatingMap.domDomCongr σ (c • f) = c • AlternatingMap.domDomCongr σ f

@[simp]

theorem AlternatingMap.domDomCongrEquiv_apply {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (σ : ι ≃ ι') (f : M [⋀^ι]→ₗ[R] N) : (AlternatingMap.domDomCongrEquiv σ) f = AlternatingMap.domDomCongr σ f

@[simp]

theorem AlternatingMap.domDomCongrEquiv_symm_apply {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (σ : ι ≃ ι') (f : M [⋀^ι']→ₗ[R] N) : (AlternatingMap.domDomCongrEquiv σ).symm f = AlternatingMap.domDomCongr σ.symm f

def AlternatingMap.domDomCongrEquiv {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (σ : ι ≃ ι') : M [⋀^ι]→ₗ[R] N ≃+ M [⋀^ι']→ₗ[R] N

AlternatingMap.domDomCongr as an equivalence.

This is declared separately because it does not work with dot notation.

Equations

• AlternatingMap.domDomCongrEquiv σ = { toFun := AlternatingMap.domDomCongr σ, invFun := AlternatingMap.domDomCongr σ.symm, left_inv := ⋯, right_inv := ⋯, map_add' := ⋯ }

@[simp]

theorem AlternatingMap.domDomCongrₗ_apply {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (S : Type u_12) [Semiring S] [Module S N] [SMulCommClass R S N] (σ : ι ≃ ι') (f : M [⋀^ι]→ₗ[R] N) : (AlternatingMap.domDomCongrₗ S σ) f = AlternatingMap.domDomCongr σ f

@[simp]

theorem AlternatingMap.domDomCongrₗ_symm_apply {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (S : Type u_12) [Semiring S] [Module S N] [SMulCommClass R S N] (σ : ι ≃ ι') (f : M [⋀^ι']→ₗ[R] N) : (AlternatingMap.domDomCongrₗ S σ).symm f = AlternatingMap.domDomCongr σ.symm f

def AlternatingMap.domDomCongrₗ {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (S : Type u_12) [Semiring S] [Module S N] [SMulCommClass R S N] (σ : ι ≃ ι') : M [⋀^ι]→ₗ[R] N ≃ₗ[S] M [⋀^ι']→ₗ[R] N

AlternatingMap.domDomCongr as a linear equivalence.

Equations

• AlternatingMap.domDomCongrₗ S σ = { toFun := AlternatingMap.domDomCongr σ, map_add' := ⋯, map_smul' := ⋯, invFun := AlternatingMap.domDomCongr σ.symm, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem AlternatingMap.domDomCongrₗ_refl {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (S : Type u_12) [Semiring S] [Module S N] [SMulCommClass R S N] : AlternatingMap.domDomCongrₗ S (Equiv.refl ι) = LinearEquiv.refl S (M [⋀^ι]→ₗ[R] N)

@[simp]

theorem AlternatingMap.domDomCongrₗ_toAddEquiv {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (S : Type u_12) [Semiring S] [Module S N] [SMulCommClass R S N] (σ : ι ≃ ι') : ↑(AlternatingMap.domDomCongrₗ S σ) = AlternatingMap.domDomCongrEquiv σ

@[simp]

theorem AlternatingMap.domDomCongr_eq_iff {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (σ : ι ≃ ι') (f : M [⋀^ι]→ₗ[R] N) (g : M [⋀^ι]→ₗ[R] N) : AlternatingMap.domDomCongr σ f = AlternatingMap.domDomCongr σ g ↔ f = g

The results of applying domDomCongr to two maps are equal if and only if those maps are.

@[simp]

theorem AlternatingMap.domDomCongr_eq_zero_iff {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (σ : ι ≃ ι') (f : M [⋀^ι]→ₗ[R] N) : AlternatingMap.domDomCongr σ f = 0 ↔ f = 0

theorem AlternatingMap.domDomCongr_perm {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g : M [⋀^ι]→ₗ[R] N') [Fintype ι] [DecidableEq ι] (σ : Equiv.Perm ι) : AlternatingMap.domDomCongr σ g = Equiv.Perm.sign σ • g

theorem AlternatingMap.coe_domDomCongr {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (f : M [⋀^ι]→ₗ[R] N) (σ : ι ≃ ι') : ↑(AlternatingMap.domDomCongr σ f) = MultilinearMap.domDomCongr σ ↑f

theorem AlternatingMap.map_linearDependent {ι : Type u_7} {K : Type u_12} [Ring K] {M : Type u_13} [AddCommGroup M] [Module K M] {N : Type u_14} [AddCommGroup N] [Module K N] [NoZeroSMulDivisors K N] (f : M [⋀^ι]→ₗ[K] N) (v : ι → M) (h : ¬LinearIndependent K v) : f v = 0

If the arguments are linearly dependent then the result is 0.

theorem AlternatingMap.map_vecCons_add {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {n : ℕ} (f : M [⋀^Fin n.succ]→ₗ[R] N) (m : Fin n → M) (x : M) (y : M) : f (Matrix.vecCons (x + y) m) = f (Matrix.vecCons x m) + f (Matrix.vecCons y m)

A version of MultilinearMap.cons_add for AlternatingMap.

theorem AlternatingMap.map_vecCons_smul {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {n : ℕ} (f : M [⋀^Fin n.succ]→ₗ[R] N) (m : Fin n → M) (c : R) (x : M) : f (Matrix.vecCons (c • x) m) = c • f (Matrix.vecCons x m)

A version of MultilinearMap.cons_smul for AlternatingMap.

def MultilinearMap.alternatization {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} [Fintype ι] [DecidableEq ι] : MultilinearMap R (fun (x : ι) => M) N' →+ M [⋀^ι]→ₗ[R] N'

Produce an AlternatingMap out of a MultilinearMap, by summing over all argument permutations.

Equations

• One or more equations did not get rendered due to their size.

theorem MultilinearMap.alternatization_def {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} [Fintype ι] [DecidableEq ι] (m : MultilinearMap R (fun (x : ι) => M) N') : ⇑(MultilinearMap.alternatization m) = ⇑(∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • MultilinearMap.domDomCongr σ m)

theorem MultilinearMap.alternatization_coe {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} [Fintype ι] [DecidableEq ι] (m : MultilinearMap R (fun (x : ι) => M) N') : ↑(MultilinearMap.alternatization m) = ∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • MultilinearMap.domDomCongr σ m

theorem MultilinearMap.alternatization_apply {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} [Fintype ι] [DecidableEq ι] (m : MultilinearMap R (fun (x : ι) => M) N') (v : ι → M) : (MultilinearMap.alternatization m) v = ∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • (MultilinearMap.domDomCongr σ m) v

theorem AlternatingMap.coe_alternatization {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} [DecidableEq ι] [Fintype ι] (a : M [⋀^ι]→ₗ[R] N') : MultilinearMap.alternatization ↑a = (Fintype.card ι).factorial • a

Alternatizing a multilinear map that is already alternating results in a scale factor of n!, where n is the number of inputs.

theorem LinearMap.compMultilinearMap_alternatization {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} {N'₂ : Type u_10} [AddCommGroup N'₂] [Module R N'₂] [DecidableEq ι] [Fintype ι] (g : N' →ₗ[R] N'₂) (f : MultilinearMap R (fun (x : ι) => M) N') : MultilinearMap.alternatization (g.compMultilinearMap f) = g.compAlternatingMap (MultilinearMap.alternatization f)

Composition with a linear map before and after alternatization are equivalent.

theorem Basis.ext_alternating {ι : Type u_7} {ι₁ : Type u_10} [Finite ι] {R' : Type u_11} {N₁ : Type u_12} {N₂ : Type u_13} [CommSemiring R'] [AddCommMonoid N₁] [AddCommMonoid N₂] [Module R' N₁] [Module R' N₂] {f : N₁ [⋀^ι]→ₗ[R'] N₂} {g : N₁ [⋀^ι]→ₗ[R'] N₂} (e : Basis ι₁ R' N₁) (h : ∀ (v : ι → ι₁), Function.Injective v → (f fun (i : ι) => e (v i)) = g fun (i : ι) => e (v i)) : f = g

Two alternating maps indexed by a Fintype are equal if they are equal when all arguments are distinct basis vectors.

Currying

@[simp]

theorem AlternatingMap.curryLeft_apply_apply {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] {n : ℕ} (f : M'' [⋀^Fin n.succ]→ₗ[R'] N'') (m : M'') (v : Fin n → M'') : (f.curryLeft m) v = f (Matrix.vecCons m v)

def AlternatingMap.curryLeft {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] {n : ℕ} (f : M'' [⋀^Fin n.succ]→ₗ[R'] N'') : M'' →ₗ[R'] M'' [⋀^Fin n]→ₗ[R'] N''

Given an alternating map f in n+1 variables, split the first variable to obtain a linear map into alternating maps in n variables, given by x ↦ (m ↦ f (Matrix.vecCons x m)). It can be thought of as a map $Hom(\bigwedge^{n+1} M, N) \to Hom(M, Hom(\bigwedge^n M, N))$.

This is MultilinearMap.curryLeft for AlternatingMap. See also AlternatingMap.curryLeftLinearMap.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem AlternatingMap.curryLeft_zero {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] {n : ℕ} : AlternatingMap.curryLeft 0 = 0

@[simp]

theorem AlternatingMap.curryLeft_add {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] {n : ℕ} (f : M'' [⋀^Fin n.succ]→ₗ[R'] N'') (g : M'' [⋀^Fin n.succ]→ₗ[R'] N'') : (f + g).curryLeft = f.curryLeft + g.curryLeft

@[simp]

theorem AlternatingMap.curryLeft_smul {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] {n : ℕ} (r : R') (f : M'' [⋀^Fin n.succ]→ₗ[R'] N'') : (r • f).curryLeft = r • f.curryLeft

@[simp]

theorem AlternatingMap.curryLeftLinearMap_apply {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] {n : ℕ} (f : M'' [⋀^Fin n.succ]→ₗ[R'] N'') : AlternatingMap.curryLeftLinearMap f = f.curryLeft

def AlternatingMap.curryLeftLinearMap {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] {n : ℕ} : M'' [⋀^Fin n.succ]→ₗ[R'] N'' →ₗ[R'] M'' →ₗ[R'] M'' [⋀^Fin n]→ₗ[R'] N''

AlternatingMap.curryLeft as a LinearMap. This is a separate definition as dot notation does not work for this version.

Equations

• AlternatingMap.curryLeftLinearMap = { toFun := fun (f : M'' [⋀^Fin n.succ]→ₗ[R'] N'') => f.curryLeft, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem AlternatingMap.curryLeft_same {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] {n : ℕ} (f : M'' [⋀^Fin n.succ.succ]→ₗ[R'] N'') (m : M'') : (f.curryLeft m).curryLeft m = 0

Currying with the same element twice gives the zero map.

@[simp]

theorem AlternatingMap.curryLeft_compAlternatingMap {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} {N₂'' : Type u_14} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [AddCommMonoid N₂''] [Module R' M''] [Module R' N''] [Module R' N₂''] {n : ℕ} (g : N'' →ₗ[R'] N₂'') (f : M'' [⋀^Fin n.succ]→ₗ[R'] N'') (m : M'') : (g.compAlternatingMap f).curryLeft m = g.compAlternatingMap (f.curryLeft m)

@[simp]

theorem AlternatingMap.curryLeft_compLinearMap {R' : Type u_10} {M'' : Type u_11} {M₂'' : Type u_12} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid M₂''] [AddCommMonoid N''] [Module R' M''] [Module R' M₂''] [Module R' N''] {n : ℕ} (g : M₂'' →ₗ[R'] M'') (f : M'' [⋀^Fin n.succ]→ₗ[R'] N'') (m : M₂'') : (f.compLinearMap g).curryLeft m = (f.curryLeft (g m)).compLinearMap g

@[simp]

theorem AlternatingMap.constLinearEquivOfIsEmpty_symm_apply {ι : Type u_7} {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] [IsEmpty ι] (f : M'' [⋀^ι]→ₗ[R'] N'') : AlternatingMap.constLinearEquivOfIsEmpty.symm f = f 0

@[simp]

theorem AlternatingMap.constLinearEquivOfIsEmpty_apply {ι : Type u_7} {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] [IsEmpty ι] (m : N'') : AlternatingMap.constLinearEquivOfIsEmpty m = AlternatingMap.constOfIsEmpty R' M'' ι m

def AlternatingMap.constLinearEquivOfIsEmpty {ι : Type u_7} {R' : Type u_10} {M'' : Type u_11} {N'' : Type u_13} [CommSemiring R'] [AddCommMonoid M''] [AddCommMonoid N''] [Module R' M''] [Module R' N''] [IsEmpty ι] : N'' ≃ₗ[R'] M'' [⋀^ι]→ₗ[R'] N''

The space of constant maps is equivalent to the space of maps that are alternating with respect to an empty family.

Equations

• One or more equations did not get rendered due to their size.