mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹³ : Monoid G
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : TopologicalSpace V
inst✝⁹ : IsTopologicalAddGroup V
inst✝⁸ : Module R V
inst✝⁷ : AddCommGroup W
inst✝⁶ : TopologicalSpace W
inst✝⁵ : IsTopologicalAddGroup W
inst✝⁴ : Module R W
inst✝³ : AddCommGroup U
inst✝² : Module R U
inst✝¹ : TopologicalSpace U
inst✝ : IsTopologicalAddGroup U
π₁ : ContRepresentation R G V
π₂ : ContRepresentation R G W
π₃ : ContRepresentation R G U
f g : π₁ →ⁱL π₂
⊢ ∀ (g_1 : G),
    (f.toContinuousLinearMap + g.toContinuousLinearMap) ∘SL π₁ g_1 =
      π₂ g_1 ∘SL (f.toContinuousLinearMap + g.toContinuousLinearMap)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹³ : Monoid G
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : TopologicalSpace V
inst✝⁹ : IsTopologicalAddGroup V
inst✝⁸ : Module R V
inst✝⁷ : AddCommGroup W
inst✝⁶ : TopologicalSpace W
inst✝⁵ : IsTopologicalAddGroup W
inst✝⁴ : Module R W
inst✝³ : AddCommGroup U
inst✝² : Module R U
inst✝¹ : TopologicalSpace U
inst✝ : IsTopologicalAddGroup U
π₁ : ContRepresentation R G V
π₂ : ContRepresentation R G W
π₃ : ContRepresentation R G U
f : π₂ →ⁱL π₃
g h : π₁ →ⁱL π₂
⊢ f.comp (g + h) = f.comp g + f.comp h

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹³ : Monoid G
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : TopologicalSpace V
inst✝⁹ : IsTopologicalAddGroup V
inst✝⁸ : Module R V
inst✝⁷ : AddCommGroup W
inst✝⁶ : TopologicalSpace W
inst✝⁵ : IsTopologicalAddGroup W
inst✝⁴ : Module R W
inst✝³ : AddCommGroup U
inst✝² : Module R U
inst✝¹ : TopologicalSpace U
inst✝ : IsTopologicalAddGroup U
π₁ : ContRepresentation R G V
π₂ : ContRepresentation R G W
π₃ : ContRepresentation R G U
f g : π₂ →ⁱL π₃
h : π₁ →ⁱL π₂
⊢ (f + g).comp h = f.comp h + g.comp h

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹³ : Monoid G
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : TopologicalSpace V
inst✝⁹ : IsTopologicalAddGroup V
inst✝⁸ : Module R V
inst✝⁷ : AddCommGroup W
inst✝⁶ : TopologicalSpace W
inst✝⁵ : IsTopologicalAddGroup W
inst✝⁴ : Module R W
inst✝³ : AddCommGroup U
inst✝² : Module R U
inst✝¹ : TopologicalSpace U
inst✝ : IsTopologicalAddGroup U
π₁ : ContRepresentation R G V
π₂ : ContRepresentation R G W
π₃ : ContRepresentation R G U
⊢ ∀ (g : G), 0 ∘SL π₁ g = π₂ g ∘SL 0

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹³ : Monoid G
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : TopologicalSpace V
inst✝⁹ : IsTopologicalAddGroup V
inst✝⁸ : Module R V
inst✝⁷ : AddCommGroup W
inst✝⁶ : TopologicalSpace W
inst✝⁵ : IsTopologicalAddGroup W
inst✝⁴ : Module R W
inst✝³ : AddCommGroup U
inst✝² : Module R U
inst✝¹ : TopologicalSpace U
inst✝ : IsTopologicalAddGroup U
π₁ : ContRepresentation R G V
π₂ : ContRepresentation R G W
π₃ : ContRepresentation R G U
f : π₁ →ⁱL π₂
⊢ ∀ (g : G), (-f.toContinuousLinearMap) ∘SL π₁ g = π₂ g ∘SL (-f.toContinuousLinearMap)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹³ : Monoid G
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : TopologicalSpace V
inst✝⁹ : IsTopologicalAddGroup V
inst✝⁸ : Module R V
inst✝⁷ : AddCommGroup W
inst✝⁶ : TopologicalSpace W
inst✝⁵ : IsTopologicalAddGroup W
inst✝⁴ : Module R W
inst✝³ : AddCommGroup U
inst✝² : Module R U
inst✝¹ : TopologicalSpace U
inst✝ : IsTopologicalAddGroup U
π₁ : ContRepresentation R G V
π₂ : ContRepresentation R G W
π₃ : ContRepresentation R G U
f g : π₁ →ⁱL π₂
⊢ ∀ (g_1 : G),
    (f.toContinuousLinearMap - g.toContinuousLinearMap) ∘SL π₁ g_1 =
      π₂ g_1 ∘SL (f.toContinuousLinearMap - g.toContinuousLinearMap)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹³ : Monoid G
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : TopologicalSpace V
inst✝⁹ : IsTopologicalAddGroup V
inst✝⁸ : Module R V
inst✝⁷ : AddCommGroup W
inst✝⁶ : TopologicalSpace W
inst✝⁵ : IsTopologicalAddGroup W
inst✝⁴ : Module R W
inst✝³ : AddCommGroup U
inst✝² : Module R U
inst✝¹ : TopologicalSpace U
inst✝ : IsTopologicalAddGroup U
π₁ : ContRepresentation R G V
π₂ : ContRepresentation R G W
π₃ : ContRepresentation R G U
f g : π₂ →ⁱL π₃
h : π₁ →ⁱL π₂
⊢ (f - g).comp h = f.comp h - g.comp h

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹³ : Monoid G
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : TopologicalSpace V
inst✝⁹ : IsTopologicalAddGroup V
inst✝⁸ : Module R V
inst✝⁷ : AddCommGroup W
inst✝⁶ : TopologicalSpace W
inst✝⁵ : IsTopologicalAddGroup W
inst✝⁴ : Module R W
inst✝³ : AddCommGroup U
inst✝² : Module R U
inst✝¹ : TopologicalSpace U
inst✝ : IsTopologicalAddGroup U
π₁ : ContRepresentation R G V
π₂ : ContRepresentation R G W
π₃ : ContRepresentation R G U
f : π₂ →ⁱL π₃
g h : π₁ →ⁱL π₂
⊢ f.comp (g - h) = f.comp g - f.comp h

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹⁸ : Monoid G
inst✝¹⁷ : Ring R
inst✝¹⁶ : AddCommGroup V
inst✝¹⁵ : TopologicalSpace V
inst✝¹⁴ : IsTopologicalAddGroup V
inst✝¹³ : Module R V
inst✝¹² : AddCommGroup W
inst✝¹¹ : TopologicalSpace W
inst✝¹⁰ : IsTopologicalAddGroup W
inst✝⁹ : Module R W
inst✝⁸ : AddCommGroup U
inst✝⁷ : Module R U
inst✝⁶ : TopologicalSpace U
inst✝⁵ : IsTopologicalAddGroup U
π₁ : ContRepresentation R G V
π₂ : ContRepresentation R G W
π₃ : ContRepresentation R G U
S : Type u_6
inst✝⁴ : Monoid S
inst✝³ : DistribMulAction S W
inst✝² : SMulCommClass R S W
inst✝¹ : ContinuousConstSMul S W
inst✝ : LinearMap.CompatibleSMul W W S R
s : S
f : π₁ →ⁱL π₂
g : G
⊢ (s • f.toContinuousLinearMap) ∘SL π₁ g = π₂ g ∘SL (s • f.toContinuousLinearMap)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹⁸ : Monoid G
inst✝¹⁷ : Ring R
inst✝¹⁶ : AddCommGroup V
inst✝¹⁵ : TopologicalSpace V
inst✝¹⁴ : IsTopologicalAddGroup V
inst✝¹³ : Module R V
inst✝¹² : AddCommGroup W
inst✝¹¹ : TopologicalSpace W
inst✝¹⁰ : IsTopologicalAddGroup W
inst✝⁹ : Module R W
inst✝⁸ : AddCommGroup U
inst✝⁷ : Module R U
inst✝⁶ : TopologicalSpace U
inst✝⁵ : IsTopologicalAddGroup U
π₁ : ContRepresentation R G V
π₂ : ContRepresentation R G W
π₃ : ContRepresentation R G U
S : Type u_7
inst✝⁴ : Monoid S
inst✝³ : DistribMulAction S U
inst✝² : SMulCommClass R S U
inst✝¹ : ContinuousConstSMul S U
inst✝ : LinearMap.CompatibleSMul U U S R
s : S
f : π₂ →ⁱL π₃
g : π₁ →ⁱL π₂
⊢ (s • f).comp g = s • f.comp g

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝²³ : Monoid G
inst✝²² : Ring R
inst✝²¹ : AddCommGroup V
inst✝²⁰ : TopologicalSpace V
inst✝¹⁹ : IsTopologicalAddGroup V
inst✝¹⁸ : Module R V
inst✝¹⁷ : AddCommGroup W
inst✝¹⁶ : TopologicalSpace W
inst✝¹⁵ : IsTopologicalAddGroup W
inst✝¹⁴ : Module R W
inst✝¹³ : AddCommGroup U
inst✝¹² : Module R U
inst✝¹¹ : TopologicalSpace U
inst✝¹⁰ : IsTopologicalAddGroup U
π₁ : ContRepresentation R G V
π₂ : ContRepresentation R G W
π₃ : ContRepresentation R G U
S : Type u_7
inst✝⁹ : Monoid S
inst✝⁸ : DistribMulAction S U
inst✝⁷ : SMulCommClass R S U
inst✝⁶ : ContinuousConstSMul S U
inst✝⁵ : LinearMap.CompatibleSMul U U S R
inst✝⁴ : DistribMulAction S W
inst✝³ : SMulCommClass R S W
inst✝² : ContinuousConstSMul S W
inst✝¹ : LinearMap.CompatibleSMul W W S R
inst✝ : LinearMap.CompatibleSMul W U S R
s : S
f : π₂ →ⁱL π₃
g : π₁ →ⁱL π₂
⊢ f.comp (s • g) = s • f.comp g

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹⁸ : Monoid G
inst✝¹⁷ : Ring R
inst✝¹⁶ : AddCommGroup V
inst✝¹⁵ : TopologicalSpace V
inst✝¹⁴ : IsTopologicalAddGroup V
inst✝¹³ : Module R V
inst✝¹² : AddCommGroup W
inst✝¹¹ : TopologicalSpace W
inst✝¹⁰ : IsTopologicalAddGroup W
inst✝⁹ : Module R W
inst✝⁸ : AddCommGroup U
inst✝⁷ : Module R U
inst✝⁶ : TopologicalSpace U
inst✝⁵ : IsTopologicalAddGroup U
π₁ : ContRepresentation R G V
π₂ : ContRepresentation R G W
π₃ : ContRepresentation R G U
S : Type u_6
inst✝⁴ : Monoid S
inst✝³ : DistribMulAction S W
inst✝² : SMulCommClass R S W
inst✝¹ : ContinuousConstSMul S W
inst✝ : LinearMap.CompatibleSMul W W S R
x✝² x✝¹ : S
x✝ : π₁ →ⁱL π₂
⊢ (x✝² * x✝¹) • x✝ = x✝² • x✝¹ • x✝

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝³ : Semiring k
inst✝² : Monoid G
X : Type w
inst✝¹ : AddCommGroup X
inst✝ : Module k X
ρ : Representation k G X
⊢ ↑(of ρ) = X

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹⁸ : Monoid G
inst✝¹⁷ : Ring R
inst✝¹⁶ : AddCommGroup V
inst✝¹⁵ : TopologicalSpace V
inst✝¹⁴ : IsTopologicalAddGroup V
inst✝¹³ : Module R V
inst✝¹² : AddCommGroup W
inst✝¹¹ : TopologicalSpace W
inst✝¹⁰ : IsTopologicalAddGroup W
inst✝⁹ : Module R W
inst✝⁸ : AddCommGroup U
inst✝⁷ : Module R U
inst✝⁶ : TopologicalSpace U
inst✝⁵ : IsTopologicalAddGroup U
π₁ : ContRepresentation R G V
π₂ : ContRepresentation R G W
π₃ : ContRepresentation R G U
S : Type u_6
inst✝⁴ : Monoid S
inst✝³ : DistribMulAction S W
inst✝² : SMulCommClass R S W
inst✝¹ : ContinuousConstSMul S W
inst✝ : LinearMap.CompatibleSMul W W S R
x✝ : π₁ →ⁱL π₂
⊢ 1 • x✝ = x✝

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝³ : Semiring k
inst✝² : Monoid G
X : Type w
inst✝¹ : AddCommGroup X
inst✝ : Module k X
ρ : Representation k G X
⊢ (of ρ).ρ = ρ

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Semiring k
inst✝ : Monoid G
A : Rep k G
a : ↑A
⊢ (ConcreteCategory.hom (𝟙 A)) a = a

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹⁸ : Monoid G
inst✝¹⁷ : Ring R
inst✝¹⁶ : AddCommGroup V
inst✝¹⁵ : TopologicalSpace V
inst✝¹⁴ : IsTopologicalAddGroup V
inst✝¹³ : Module R V
inst✝¹² : AddCommGroup W
inst✝¹¹ : TopologicalSpace W
inst✝¹⁰ : IsTopologicalAddGroup W
inst✝⁹ : Module R W
inst✝⁸ : AddCommGroup U
inst✝⁷ : Module R U
inst✝⁶ : TopologicalSpace U
inst✝⁵ : IsTopologicalAddGroup U
π₁ : ContRepresentation R G V
π₂ : ContRepresentation R G W
π₃ : ContRepresentation R G U
S : Type u_6
inst✝⁴ : Monoid S
inst✝³ : DistribMulAction S W
inst✝² : SMulCommClass R S W
inst✝¹ : ContinuousConstSMul S W
inst✝ : LinearMap.CompatibleSMul W W S R
x✝ : S
⊢ x✝ • 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Semiring k
inst✝ : Monoid G
A B C : Rep k G
f : A ⟶ B
g : B ⟶ C
a : ↑A
⊢ (ConcreteCategory.hom (f ≫ g)) a = (ConcreteCategory.hom g) ((ConcreteCategory.hom f) a)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹⁸ : Monoid G
inst✝¹⁷ : Ring R
inst✝¹⁶ : AddCommGroup V
inst✝¹⁵ : TopologicalSpace V
inst✝¹⁴ : IsTopologicalAddGroup V
inst✝¹³ : Module R V
inst✝¹² : AddCommGroup W
inst✝¹¹ : TopologicalSpace W
inst✝¹⁰ : IsTopologicalAddGroup W
inst✝⁹ : Module R W
inst✝⁸ : AddCommGroup U
inst✝⁷ : Module R U
inst✝⁶ : TopologicalSpace U
inst✝⁵ : IsTopologicalAddGroup U
π₁ : ContRepresentation R G V
π₂ : ContRepresentation R G W
π₃ : ContRepresentation R G U
S : Type u_6
inst✝⁴ : Monoid S
inst✝³ : DistribMulAction S W
inst✝² : SMulCommClass R S W
inst✝¹ : ContinuousConstSMul S W
inst✝ : LinearMap.CompatibleSMul W W S R
x✝² : S
x✝¹ x✝ : π₁ →ⁱL π₂
⊢ x✝² • (x✝¹ + x✝) = x✝² • x✝¹ + x✝² • x✝

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Semiring k
inst✝ : Monoid G
A B : Rep k G
f : A ⟶ B
g : G
a : ↑A
⊢ (Hom.hom f) ((A.ρ g) a) = (B.ρ g) ((Hom.hom f) a)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Semiring k
inst✝ : Monoid G
A B : Rep k G
e : A ≅ B
x : ↑A
⊢ (Hom.hom e.inv) ((Hom.hom e.hom) x) = x

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Semiring k
inst✝ : Monoid G
A B : Rep k G
e : A ≅ B
x : ↑B
⊢ (Hom.hom e.hom) ((Hom.hom e.inv) x) = x

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝²³ : Monoid G
inst✝²² : Ring R
inst✝²¹ : AddCommGroup V
inst✝²⁰ : TopologicalSpace V
inst✝¹⁹ : IsTopologicalAddGroup V
inst✝¹⁸ : Module R V
inst✝¹⁷ : AddCommGroup W
inst✝¹⁶ : TopologicalSpace W
inst✝¹⁵ : IsTopologicalAddGroup W
inst✝¹⁴ : Module R W
inst✝¹³ : AddCommGroup U
inst✝¹² : Module R U
inst✝¹¹ : TopologicalSpace U
inst✝¹⁰ : IsTopologicalAddGroup U
π₁ : ContRepresentation R G V
π₂ : ContRepresentation R G W
π₃ : ContRepresentation R G U
S✝ : Type u_6
inst✝⁹ : Monoid S✝
inst✝⁸ : DistribMulAction S✝ W
inst✝⁷ : SMulCommClass R S✝ W
inst✝⁶ : ContinuousConstSMul S✝ W
inst✝⁵ : LinearMap.CompatibleSMul W W S✝ R
S : Type u_7
inst✝⁴ : Ring S
inst✝³ : Module S W
inst✝² : SMulCommClass R S W
inst✝¹ : ContinuousConstSMul S W
inst✝ : LinearMap.CompatibleSMul W W S R
x✝² x✝¹ : S
x✝ : π₁ →ⁱL π₂
⊢ (x✝² + x✝¹) • x✝ = x✝² • x✝ + x✝¹ • x✝

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝⁷ : Semiring k
inst✝⁶ : Monoid G
X Y : Type w
inst✝⁵ : AddCommGroup X
inst✝⁴ : AddCommGroup Y
inst✝³ : Module k X
inst✝² : Module k Y
ρ : Representation k G X
σ : Representation k G Y
A B C : Rep k G
Z : Type w
inst✝¹ : AddCommGroup Z
inst✝ : Module k Z
τ : Representation k G Z
i : A ≅ B
x : ↑A
⊢ (ConcreteCategory.hom i.inv) ((ConcreteCategory.hom i.hom) x) = x

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝⁷ : Semiring k
inst✝⁶ : Monoid G
X Y : Type w
inst✝⁵ : AddCommGroup X
inst✝⁴ : AddCommGroup Y
inst✝³ : Module k X
inst✝² : Module k Y
ρ : Representation k G X
σ : Representation k G Y
A B C : Rep k G
Z : Type w
inst✝¹ : AddCommGroup Z
inst✝ : Module k Z
τ : Representation k G Z
i : A ≅ B
x : ↑B
⊢ (ConcreteCategory.hom i.hom) ((ConcreteCategory.hom i.inv) x) = x

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝²³ : Monoid G
inst✝²² : Ring R
inst✝²¹ : AddCommGroup V
inst✝²⁰ : TopologicalSpace V
inst✝¹⁹ : IsTopologicalAddGroup V
inst✝¹⁸ : Module R V
inst✝¹⁷ : AddCommGroup W
inst✝¹⁶ : TopologicalSpace W
inst✝¹⁵ : IsTopologicalAddGroup W
inst✝¹⁴ : Module R W
inst✝¹³ : AddCommGroup U
inst✝¹² : Module R U
inst✝¹¹ : TopologicalSpace U
inst✝¹⁰ : IsTopologicalAddGroup U
π₁ : ContRepresentation R G V
π₂ : ContRepresentation R G W
π₃ : ContRepresentation R G U
S✝ : Type u_6
inst✝⁹ : Monoid S✝
inst✝⁸ : DistribMulAction S✝ W
inst✝⁷ : SMulCommClass R S✝ W
inst✝⁶ : ContinuousConstSMul S✝ W
inst✝⁵ : LinearMap.CompatibleSMul W W S✝ R
S : Type u_7
inst✝⁴ : Ring S
inst✝³ : Module S W
inst✝² : SMulCommClass R S W
inst✝¹ : ContinuousConstSMul S W
inst✝ : LinearMap.CompatibleSMul W W S R
x✝ : π₁ →ⁱL π₂
⊢ 0 • x✝ = 0

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝⁷ : Semiring k
inst✝⁶ : Monoid G
X✝ Y✝ : Type w
inst✝⁵ : AddCommGroup X✝
inst✝⁴ : AddCommGroup Y✝
inst✝³ : Module k X✝
inst✝² : Module k Y✝
ρ : Representation k G X✝
σ : Representation k G Y✝
A B C : Rep k G
Z : Type w
inst✝¹ : AddCommGroup Z
inst✝ : Module k Z
τ : Representation k G Z
X Y : Rep k G
f : X ⟶ Y
x✝ : IsIso ((forget (Rep k G)).map f)
⊢ IsIso f

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Semiring k
inst✝ : Monoid G
A B C : Rep k G
f₁ f₂ : A ⟶ B
g : B ⟶ C
⊢ (f₁ + f₂) ≫ g = f₁ ≫ g + f₂ ≫ g

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Semiring k
inst✝ : Monoid G
A B C : Rep k G
f : A ⟶ B
g₁ g₂ : B ⟶ C
⊢ Hom.hom (f ≫ (g₁ + g₂)) = Hom.hom (f ≫ g₁ + f ≫ g₂)

Tactic:
simp [add_hom, Representation.IntertwiningMap.comp_add]

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Semiring k
inst✝ : Monoid G
A B C : Rep k G
f : A ⟶ B
g₁ g₂ : B ⟶ C
⊢ f ≫ (g₁ + g₂) = f ≫ g₁ + f ≫ g₂

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
inst✝⁹ : Monoid G
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup V
inst✝⁶ : TopologicalSpace V
inst✝⁵ : IsTopologicalAddGroup V
inst✝⁴ : Module R V
inst✝³ : AddCommGroup W
inst✝² : TopologicalSpace W
inst✝¹ : IsTopologicalAddGroup W
inst✝ : Module R W
ρ : ContRepresentation R G V
σ : ContRepresentation R G W
φ ψ : σ.Equiv ρ
h : φ.toContinuousLinearEquiv = ψ.toContinuousLinearEquiv
⊢ φ = ψ

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹³ : Monoid G
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : TopologicalSpace V
inst✝⁹ : IsTopologicalAddGroup V
inst✝⁸ : Module R V
inst✝⁷ : AddCommGroup W
inst✝⁶ : TopologicalSpace W
inst✝⁵ : IsTopologicalAddGroup W
inst✝⁴ : Module R W
inst✝³ : AddCommGroup U
inst✝² : Module R U
inst✝¹ : TopologicalSpace U
inst✝ : IsTopologicalAddGroup U
ρ : ContRepresentation R G V
σ : ContRepresentation R G W
τ : ContRepresentation R G U
φ✝ φ ψ : ρ.Equiv σ
h1 : ⇑φ.toContinuousLinearEquiv = ⇑ψ.toContinuousLinearEquiv
h2 : (↑φ.toContinuousLinearEquiv).invFun = (↑ψ.toContinuousLinearEquiv).invFun
⊢ φ = ψ

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Semiring k
inst✝ : Monoid G
A B : Rep k G
ι : Type u'
f : ι → (A ⟶ B)
s : Finset ι
⊢ Hom.hom (∑ i ∈ s, f i) = ∑ i ∈ s, Hom.hom (f i)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Semiring k
inst✝ : Monoid G
A B : Rep k G
ι : Type u'
f : ι → (A ⟶ B)
s : Finset ι
⊢ Hom.hom (∑ i ∈ s, f i) = ∑ i ∈ s, Hom.hom (f i)

Tactic:
by
  classical induction s using Finset.induction with
  | empty => simp
  | insert a s ha h => simp [Finset.sum_insert ha, add_hom, h]

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝⁵ : Semiring k
inst✝⁴ : Monoid G
ι : Type u'
M N : Type v'
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module k M
inst✝ : Module k N
σ : Representation k G M
ρ : Representation k G N
f : ι → σ.IntertwiningMap ρ
s : Finset ι
⊢ ofHom (∑ i ∈ s, f i) = ∑ i ∈ s, ofHom (f i)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Semiring k
inst✝ : Monoid G
A : Rep k G
g1 g2 : G
⊢ A.ρ (g1 * g2) = A.ρ g1 ∘ₗ A.ρ g2

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G✝ : Type v
inst✝⁸ : Semiring k
inst✝⁷ : Monoid G✝
X Y : Type w
inst✝⁶ : AddCommGroup X
inst✝⁵ : AddCommGroup Y
inst✝⁴ : Module k X
inst✝³ : Module k Y
ρ : Representation k G✝ X
σ : Representation k G✝ Y
A✝ B C : Rep k G✝
Z : Type w
inst✝² : AddCommGroup Z
inst✝¹ : Module k Z
τ : Representation k G✝ Z
G : Type v
inst✝ : CommMonoid G
A : Rep k G
g : G
⊢ ∀ (g_1 : G), A.ρ g ∘ₗ A.ρ g_1 = A.ρ g_1 ∘ₗ A.ρ g

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
inst✝¹ : Semiring k
G : Type v
inst✝ : CommMonoid G
A B : Rep k G
f : A ⟶ B
g : G
⊢ A.applyAsHom g ≫ f = f ≫ B.applyAsHom g

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Ring k
inst✝ : Monoid G
A : Rep k G
x : ↑A
g : G
⊢ (((Finsupp.lift (↑A) k G) fun g ↦ (A.ρ g) x) ∘ₗ ↑(MonoidAlgebra.coeffLinearEquiv k)) ∘ₗ
      (Representation.ofMulAction k G G) g =
    A.ρ g ∘ₗ ((Finsupp.lift (↑A) k G) fun g ↦ (A.ρ g) x) ∘ₗ ↑(MonoidAlgebra.coeffLinearEquiv k)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Ring k
inst✝ : Monoid G
A : Rep k G
g : G
x : ↑A
r : k
⊢ (Hom.hom (A.leftRegularHom x)) (MonoidAlgebra.single g r) = r • (A.ρ g) x

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Ring k
inst✝ : Monoid G
A B C X✝ Y✝ : Rep k G
f : X✝ ⟶ Y✝
g : G
⊢ ModuleCat.Hom.hom
      ({ V := ModuleCat.of k ↑X✝, ρ := (ModuleCat.of k ↑X✝).endRingEquiv.symm.toMonoidHom.comp X✝.ρ }.ρ g ≫
        Hom.toModuleCatHom f) =
    ModuleCat.Hom.hom
      (Hom.toModuleCatHom f ≫
        { V := ModuleCat.of k ↑Y✝, ρ := (ModuleCat.of k ↑Y✝).endRingEquiv.symm.toMonoidHom.comp Y✝.ρ }.ρ g)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
inst✝⁹ : Monoid G
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup V
inst✝⁶ : TopologicalSpace V
inst✝⁵ : IsTopologicalAddGroup V
inst✝⁴ : Module R V
inst✝³ : AddCommGroup W
inst✝² : TopologicalSpace W
inst✝¹ : IsTopologicalAddGroup W
inst✝ : Module R W
ρ : ContRepresentation R G V
σ : ContRepresentation R G W
φ ψ : ρ.Equiv σ
h : ⇑φ = ⇑ψ
⊢ φ = ψ

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹³ : Monoid G
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : TopologicalSpace V
inst✝⁹ : IsTopologicalAddGroup V
inst✝⁸ : Module R V
inst✝⁷ : AddCommGroup W
inst✝⁶ : TopologicalSpace W
inst✝⁵ : IsTopologicalAddGroup W
inst✝⁴ : Module R W
inst✝³ : AddCommGroup U
inst✝² : Module R U
inst✝¹ : TopologicalSpace U
inst✝ : IsTopologicalAddGroup U
ρ : ContRepresentation R G V
σ : ContRepresentation R G W
τ : ContRepresentation R G U
φ : ρ.Equiv σ
⊢ ∀ (g : G), ↑(ContinuousLinearEquiv.refl R V) ∘SL ρ g = ρ g ∘SL ↑(ContinuousLinearEquiv.refl R V)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Ring k
inst✝ : Monoid G
A B C : Rep k G
X✝ Y✝ : Action (ModuleCat k) G
f : X✝ ⟶ Y✝
g : G
⊢ ModuleCat.Hom.hom f.hom ∘ₗ (X✝.V.endRingEquiv.toMonoidHom.comp X✝.ρ) g =
    (Y✝.V.endRingEquiv.toMonoidHom.comp Y✝.ρ) g ∘ₗ ModuleCat.Hom.hom f.hom

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Ring k
inst✝ : Monoid G
A✝ B C : Rep k G
A : Action (ModuleCat k) G
g : G
⊢ ((RepToAction k G).obj ((ActionToRep k G).obj A)).ρ g ≫ 𝟙 ((RepToAction k G).obj ((ActionToRep k G).obj A)).V =
    𝟙 ((RepToAction k G).obj ((ActionToRep k G).obj A)).V ≫ A.ρ g

Tactic:
by

Goal state after: