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Mathlib.GroupTheory.GroupAction.Prod

Prod instances for additive and multiplicative actions #

This file defines instances for binary product of additive and multiplicative actions and provides scalar multiplication as a homomorphism from α × β to β.

Main declarations #

See also #

Porting notes #

The to_additive attribute can be used to generate both the smul and vadd lemmas from the corresponding pow lemmas, as explained on zulip here: https://leanprover.zulipchat.com/#narrow/near/316087838

This was not done as part of the port in order to stay as close as possible to the mathlib3 code.

instance Prod.vadd {M : Type u_1} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] :
VAdd M (α × β)
Equations
  • Prod.vadd = { vadd := fun (a : M) (p : α × β) => (a +ᵥ p.1, a +ᵥ p.2) }
instance Prod.smul {M : Type u_1} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] :
SMul M (α × β)
Equations
  • Prod.smul = { smul := fun (a : M) (p : α × β) => (a p.1, a p.2) }
@[simp]
theorem Prod.vadd_fst {M : Type u_1} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] (a : M) (x : α × β) :
(a +ᵥ x).1 = a +ᵥ x.1
@[simp]
theorem Prod.smul_fst {M : Type u_1} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] (a : M) (x : α × β) :
(a x).1 = a x.1
@[simp]
theorem Prod.vadd_snd {M : Type u_1} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] (a : M) (x : α × β) :
(a +ᵥ x).2 = a +ᵥ x.2
@[simp]
theorem Prod.smul_snd {M : Type u_1} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] (a : M) (x : α × β) :
(a x).2 = a x.2
@[simp]
theorem Prod.vadd_mk {M : Type u_1} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] (a : M) (b : α) (c : β) :
a +ᵥ (b, c) = (a +ᵥ b, a +ᵥ c)
@[simp]
theorem Prod.smul_mk {M : Type u_1} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] (a : M) (b : α) (c : β) :
a (b, c) = (a b, a c)
theorem Prod.vadd_def {M : Type u_1} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] (a : M) (x : α × β) :
a +ᵥ x = (a +ᵥ x.1, a +ᵥ x.2)
theorem Prod.smul_def {M : Type u_1} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] (a : M) (x : α × β) :
a x = (a x.1, a x.2)
@[simp]
theorem Prod.vadd_swap {M : Type u_1} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] (a : M) (x : α × β) :
(a +ᵥ x).swap = a +ᵥ x.swap
@[simp]
theorem Prod.smul_swap {M : Type u_1} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] (a : M) (x : α × β) :
(a x).swap = a x.swap
theorem Prod.smul_zero_mk {M : Type u_1} {β : Type u_6} [SMul M β] {α : Type u_7} [Monoid M] [AddMonoid α] [DistribMulAction M α] (a : M) (c : β) :
a (0, c) = (0, a c)
theorem Prod.smul_mk_zero {M : Type u_1} {α : Type u_5} [SMul M α] {β : Type u_7} [Monoid M] [AddMonoid β] [DistribMulAction M β] (a : M) (b : α) :
a (b, 0) = (a b, 0)
instance Prod.pow {E : Type u_4} {α : Type u_5} {β : Type u_6} [Pow α E] [Pow β E] :
Pow (α × β) E
Equations
  • Prod.pow = { pow := fun (p : α × β) (c : E) => (p.1 ^ c, p.2 ^ c) }
@[simp]
theorem Prod.pow_fst {E : Type u_4} {α : Type u_5} {β : Type u_6} [Pow α E] [Pow β E] (p : α × β) (c : E) :
(p ^ c).1 = p.1 ^ c
@[simp]
theorem Prod.pow_snd {E : Type u_4} {α : Type u_5} {β : Type u_6} [Pow α E] [Pow β E] (p : α × β) (c : E) :
(p ^ c).2 = p.2 ^ c
@[simp]
theorem Prod.pow_mk {E : Type u_4} {α : Type u_5} {β : Type u_6} [Pow α E] [Pow β E] (c : E) (a : α) (b : β) :
(a, b) ^ c = (a ^ c, b ^ c)
theorem Prod.pow_def {E : Type u_4} {α : Type u_5} {β : Type u_6} [Pow α E] [Pow β E] (p : α × β) (c : E) :
p ^ c = (p.1 ^ c, p.2 ^ c)
@[simp]
theorem Prod.pow_swap {E : Type u_4} {α : Type u_5} {β : Type u_6} [Pow α E] [Pow β E] (p : α × β) (c : E) :
(p ^ c).swap = p.swap ^ c
instance Prod.vaddAssocClass {M : Type u_1} {N : Type u_2} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] [VAdd N α] [VAdd N β] [VAdd M N] [VAddAssocClass M N α] [VAddAssocClass M N β] :
VAddAssocClass M N (α × β)
Equations
  • =
instance Prod.isScalarTower {M : Type u_1} {N : Type u_2} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] [SMul N α] [SMul N β] [SMul M N] [IsScalarTower M N α] [IsScalarTower M N β] :
IsScalarTower M N (α × β)
Equations
  • =
instance Prod.vaddCommClass {M : Type u_1} {N : Type u_2} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] [VAdd N α] [VAdd N β] [VAddCommClass M N α] [VAddCommClass M N β] :
VAddCommClass M N (α × β)
Equations
  • =
instance Prod.smulCommClass {M : Type u_1} {N : Type u_2} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] [SMul N α] [SMul N β] [SMulCommClass M N α] [SMulCommClass M N β] :
SMulCommClass M N (α × β)
Equations
  • =
instance Prod.isCentralVAdd {M : Type u_1} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] [VAdd Mᵃᵒᵖ α] [VAdd Mᵃᵒᵖ β] [IsCentralVAdd M α] [IsCentralVAdd M β] :
IsCentralVAdd M (α × β)
Equations
  • =
instance Prod.isCentralScalar {M : Type u_1} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] [SMul Mᵐᵒᵖ α] [SMul Mᵐᵒᵖ β] [IsCentralScalar M α] [IsCentralScalar M β] :
Equations
  • =
abbrev Prod.faithfulVAddLeft.match_1 {β : Type u_1} (motive : Nonempty βProp) :
∀ (x : Nonempty β), (∀ (b : β), motive )motive x
Equations
  • =
Instances For
    instance Prod.faithfulVAddLeft {M : Type u_1} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] [FaithfulVAdd M α] [Nonempty β] :
    FaithfulVAdd M (α × β)
    Equations
    • =
    instance Prod.faithfulSMulLeft {M : Type u_1} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] [FaithfulSMul M α] [Nonempty β] :
    FaithfulSMul M (α × β)
    Equations
    • =
    instance Prod.faithfulVAddRight {M : Type u_1} {α : Type u_5} {β : Type u_6} [VAdd M α] [VAdd M β] [Nonempty α] [FaithfulVAdd M β] :
    FaithfulVAdd M (α × β)
    Equations
    • =
    instance Prod.faithfulSMulRight {M : Type u_1} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] [Nonempty α] [FaithfulSMul M β] :
    FaithfulSMul M (α × β)
    Equations
    • =
    instance Prod.vaddCommClassBoth {M : Type u_1} {N : Type u_2} {P : Type u_3} [Add N] [Add P] [VAdd M N] [VAdd M P] [VAddCommClass M N N] [VAddCommClass M P P] :
    VAddCommClass M (N × P) (N × P)
    Equations
    • =
    instance Prod.smulCommClassBoth {M : Type u_1} {N : Type u_2} {P : Type u_3} [Mul N] [Mul P] [SMul M N] [SMul M P] [SMulCommClass M N N] [SMulCommClass M P P] :
    SMulCommClass M (N × P) (N × P)
    Equations
    • =
    instance Prod.isScalarTowerBoth {M : Type u_1} {N : Type u_2} {P : Type u_3} [Mul N] [Mul P] [SMul M N] [SMul M P] [IsScalarTower M N N] [IsScalarTower M P P] :
    IsScalarTower M (N × P) (N × P)
    Equations
    • =
    instance Prod.addAction {M : Type u_1} {α : Type u_5} {β : Type u_6} [AddMonoid M] [AddAction M α] [AddAction M β] :
    AddAction M (α × β)
    Equations
    theorem Prod.addAction.proof_1 {M : Type u_3} {α : Type u_1} {β : Type u_2} [AddMonoid M] [AddAction M α] [AddAction M β] :
    ∀ (x : α × β), 0 +ᵥ x = x
    abbrev Prod.addAction.match_1 {α : Type u_1} {β : Type u_2} (motive : α × βProp) :
    ∀ (x : α × β), (∀ (fst : α) (snd : β), motive (fst, snd))motive x
    Equations
    • =
    Instances For
      theorem Prod.addAction.proof_2 {M : Type u_3} {α : Type u_1} {β : Type u_2} [AddMonoid M] [AddAction M α] [AddAction M β] :
      ∀ (x x_1 : M) (x_2 : α × β), (x + x_1 +ᵥ x_2.1, x + x_1 +ᵥ x_2.2) = (x +ᵥ (x_1 +ᵥ x_2).1, x +ᵥ (x_1 +ᵥ x_2).2)
      instance Prod.mulAction {M : Type u_1} {α : Type u_5} {β : Type u_6} [Monoid M] [MulAction M α] [MulAction M β] :
      MulAction M (α × β)
      Equations
      instance Prod.smulZeroClass {R : Type u_7} {M : Type u_8} {N : Type u_9} [Zero M] [Zero N] [SMulZeroClass R M] [SMulZeroClass R N] :
      Equations
      instance Prod.distribSMul {R : Type u_7} {M : Type u_8} {N : Type u_9} [AddZeroClass M] [AddZeroClass N] [DistribSMul R M] [DistribSMul R N] :
      DistribSMul R (M × N)
      Equations
      instance Prod.distribMulAction {M : Type u_1} {N : Type u_2} {R : Type u_7} [Monoid R] [AddMonoid M] [AddMonoid N] [DistribMulAction R M] [DistribMulAction R N] :
      Equations
      • Prod.distribMulAction = let __src := Prod.mulAction; let __src_1 := Prod.distribSMul; DistribMulAction.mk
      instance Prod.mulDistribMulAction {M : Type u_1} {N : Type u_2} {R : Type u_7} [Monoid R] [Monoid M] [Monoid N] [MulDistribMulAction R M] [MulDistribMulAction R N] :
      Equations

      Scalar multiplication as a homomorphism #

      @[simp]
      theorem smulMulHom_apply {α : Type u_5} {β : Type u_6} [Monoid α] [Mul β] [MulAction α β] [IsScalarTower α β β] [SMulCommClass α β β] (a : α × β) :
      smulMulHom a = a.1 a.2
      def smulMulHom {α : Type u_5} {β : Type u_6} [Monoid α] [Mul β] [MulAction α β] [IsScalarTower α β β] [SMulCommClass α β β] :
      α × β →ₙ* β

      Scalar multiplication as a multiplicative homomorphism.

      Equations
      • smulMulHom = { toFun := fun (a : α × β) => a.1 a.2, map_mul' := }
      Instances For
        @[simp]
        theorem smulMonoidHom_apply {α : Type u_5} {β : Type u_6} [Monoid α] [MulOneClass β] [MulAction α β] [IsScalarTower α β β] [SMulCommClass α β β] :
        ∀ (a : α × β), smulMonoidHom a = smulMulHom.toFun a
        def smulMonoidHom {α : Type u_5} {β : Type u_6} [Monoid α] [MulOneClass β] [MulAction α β] [IsScalarTower α β β] [SMulCommClass α β β] :
        α × β →* β

        Scalar multiplication as a monoid homomorphism.

        Equations
        • smulMonoidHom = let __src := smulMulHom; { toFun := __src.toFun, map_one' := , map_mul' := }
        Instances For
          abbrev AddAction.prodOfVAddCommClass (M : Type u_1) (N : Type u_2) (α : Type u_5) [AddMonoid M] [AddMonoid N] [AddAction M α] [AddAction N α] [VAddCommClass M N α] :
          AddAction (M × N) α

          Construct an AddAction by a product monoid from AddActions by the factors. This is not an instance to avoid diamonds for example when α := M × N.

          Equations
          Instances For
            theorem AddAction.prodOfVAddCommClass.proof_1 (M : Type u_2) (N : Type u_3) (α : Type u_1) [AddMonoid M] [AddMonoid N] [AddAction M α] [AddAction N α] (a : α) :
            0 +ᵥ (0.2 +ᵥ a) = a
            theorem AddAction.prodOfVAddCommClass.proof_2 (M : Type u_1) (N : Type u_2) (α : Type u_3) [AddMonoid M] [AddMonoid N] [AddAction M α] [AddAction N α] [VAddCommClass M N α] (x : M × N) (y : M × N) (a : α) :
            x.1 + y.1 +ᵥ (x.2 + y.2 +ᵥ a) = x.1 +ᵥ (x.2 +ᵥ (y.1 +ᵥ (y.2 +ᵥ a)))
            @[reducible, inline]
            abbrev MulAction.prodOfSMulCommClass (M : Type u_1) (N : Type u_2) (α : Type u_5) [Monoid M] [Monoid N] [MulAction M α] [MulAction N α] [SMulCommClass M N α] :
            MulAction (M × N) α

            Construct a MulAction by a product monoid from MulActions by the factors. This is not an instance to avoid diamonds for example when α := M × N.

            Equations
            Instances For
              theorem AddAction.prodEquiv.proof_2 (M : Type u_3) (N : Type u_2) (α : Type u_1) [AddMonoid M] [AddMonoid N] (_insts : (x : AddAction M α) ×' (x_1 : AddAction N α) ×' VAddCommClass M N α) :
              theorem AddAction.prodEquiv.proof_3 (M : Type u_3) (N : Type u_2) (α : Type u_1) [AddMonoid M] [AddMonoid N] :
              ∀ (x : AddAction (M × N) α), (fun (_insts : (x : AddAction M α) ×' (x_1 : AddAction N α) ×' VAddCommClass M N α) => let_fun this := ; AddAction.prodOfVAddCommClass M N α) ((fun (x : AddAction (M × N) α) => AddAction.compHom α (AddMonoidHom.inl M N), AddAction.compHom α (AddMonoidHom.inr M N), ) x) = x
              theorem AddAction.prodEquiv.proof_4 (M : Type u_1) (N : Type u_3) (α : Type u_2) [AddMonoid M] [AddMonoid N] :
              ∀ (x : (x : AddAction M α) ×' (x_1 : AddAction N α) ×' VAddCommClass M N α), (fun (x : AddAction (M × N) α) => AddAction.compHom α (AddMonoidHom.inl M N), AddAction.compHom α (AddMonoidHom.inr M N), ) ((fun (_insts : (x : AddAction M α) ×' (x_1 : AddAction N α) ×' VAddCommClass M N α) => let_fun this := ; AddAction.prodOfVAddCommClass M N α) x) = x
              def AddAction.prodEquiv (M : Type u_1) (N : Type u_2) (α : Type u_5) [AddMonoid M] [AddMonoid N] :
              AddAction (M × N) α (x : AddAction M α) ×' (x_1 : AddAction N α) ×' VAddCommClass M N α

              An AddAction by a product monoid is equivalent to commuting AddActions by the factors.

              Equations
              • One or more equations did not get rendered due to their size.
              Instances For
                theorem AddAction.prodEquiv.proof_1 (M : Type u_3) (N : Type u_2) (α : Type u_1) [AddMonoid M] [AddMonoid N] :
                ∀ (x : AddAction (M × N) α), VAddCommClass M N α
                def MulAction.prodEquiv (M : Type u_1) (N : Type u_2) (α : Type u_5) [Monoid M] [Monoid N] :
                MulAction (M × N) α (x : MulAction M α) ×' (x_1 : MulAction N α) ×' SMulCommClass M N α

                A MulAction by a product monoid is equivalent to commuting MulActions by the factors.

                Equations
                • One or more equations did not get rendered due to their size.
                Instances For
                  @[reducible, inline]
                  abbrev DistribMulAction.prodOfSMulCommClass (M : Type u_1) (N : Type u_2) (α : Type u_5) [Monoid M] [Monoid N] [AddMonoid α] [DistribMulAction M α] [DistribMulAction N α] [SMulCommClass M N α] :

                  Construct a DistribMulAction by a product monoid from DistribMulActions by the factors.

                  Equations
                  Instances For
                    def DistribMulAction.prodEquiv (M : Type u_1) (N : Type u_2) (α : Type u_5) [Monoid M] [Monoid N] [AddMonoid α] :
                    DistribMulAction (M × N) α (x : DistribMulAction M α) ×' (x_1 : DistribMulAction N α) ×' SMulCommClass M N α

                    A DistribMulAction by a product monoid is equivalent to commuting DistribMulActions by the factors.

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