mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ A.ρ g ≫ 𝟙 A.V = 𝟙 A.V ≫ ((RepToAction k G).obj ((ActionToRep k G).obj A)).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
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
⊢ ↑φ.symm ∘SL σ g = ρ g ∘SL ↑φ.symm

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 g : X ⟶ Y
⊢ (forget₂ (Rep k G) (ModuleCat k)).map (f + g) =
    (forget₂ (Rep k G) (ModuleCat k)).map f + (forget₂ (Rep k G) (ModuleCat k)).map 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✝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 : ρ.Equiv σ
ψ : σ.Equiv τ
g : G
⊢ ↑(φ.trans ψ.toContinuousLinearEquiv) ∘SL ρ g = τ g ∘SL ↑(φ.trans ψ.toContinuousLinearEquiv)

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 A B : Rep k G
⊢ ofHom (Representation.IntertwiningMap.inl k A.ρ B.ρ) ≫ ofHom (Representation.IntertwiningMap.fst k A.ρ B.ρ) = 𝟙 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
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 : ρ.Equiv σ
⊢ φ.trans φ.symm = refl ρ

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 A B : Rep k G
⊢ ofHom (Representation.IntertwiningMap.inl k A.ρ B.ρ) ≫ ofHom (Representation.IntertwiningMap.snd k A.ρ B.ρ) = 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
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 : ρ.Equiv σ
⊢ φ.symm.trans φ = refl σ

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 A B : Rep k G
⊢ ofHom (Representation.IntertwiningMap.inr k A.ρ B.ρ) ≫ ofHom (Representation.IntertwiningMap.fst k A.ρ B.ρ) = 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✝¹⁴ : Monoid G
inst✝¹³ : Ring R
inst✝¹² : AddCommGroup V
inst✝¹¹ : TopologicalSpace V
inst✝¹⁰ : IsTopologicalAddGroup V
inst✝⁹ : Module R V
inst✝⁸ : AddCommGroup W
inst✝⁷ : TopologicalSpace W
inst✝⁶ : IsTopologicalAddGroup W
inst✝⁵ : Module R W
inst✝⁴ : AddCommGroup U
inst✝³ : Module R U
inst✝² : TopologicalSpace U
inst✝¹ : IsTopologicalAddGroup U
H : Type u_6
inst✝ : Monoid H
π : ContRepresentation R G V
π' : ContRepresentation R G W
φ : H →* G
f : π →ⁱL π'
h : H
⊢ f.toContinuousLinearMap ∘SL (π.restrict φ) h = (π'.restrict φ) h ∘SL f.toContinuousLinearMap

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.FinGroupCharZero

Goal state before:
pp: k : Type u
inst✝³ : Field k
G : Type u
inst✝² : Finite G
inst✝¹ : Group G
inst✝ : NeZero ↑(Nat.card G)
V : Rep k G
⊢ Injective 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 A B : Rep k G
⊢ ofHom (Representation.IntertwiningMap.inr k A.ρ B.ρ) ≫ ofHom (Representation.IntertwiningMap.snd k A.ρ B.ρ) = 𝟙 B

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
π : ContRepresentation R G V
a✝ b✝ : V
x✝¹ : a✝ ∈ {v | ∀ (g : G), (π g) v = v}
x✝ : b✝ ∈ {v | ∀ (g : G), (π g) v = v}
⊢ a✝ + b✝ ∈ {v | ∀ (g : G), (π g) v = v}

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.FinGroupCharZero

Goal state before:
pp: k : Type u
inst✝³ : Field k
G : Type u
inst✝² : Finite G
inst✝¹ : Group G
inst✝ : NeZero ↑(Nat.card G)
V : Rep k G
⊢ Projective V

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
π : ContRepresentation R G V
⊢ 0 ∈ {v | ∀ (g : G), (π g) v = v}

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
π : ContRepresentation R G V
x✝¹ : R
x✝ : V
hv : x✝ ∈ {v | ∀ (g : G), (π g) v = v}
g : G
⊢ (π g) (x✝¹ • x✝) = x✝¹ • x✝

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.FinGroupCharZero

Goal state before:
pp: k : Type u
inst✝⁴ : Field k
G : Type u
inst✝³ : Finite G
inst✝² : Group G
inst✝¹ : IsAlgClosed k
inst✝ : NeZero ↑(Nat.card G)
V : FDRep k G
h : Module.finrank k (V ⟶ V) = 1
W : FDRep k G
f : W ⟶ V
x✝ : Mono f
hf : IsZero W ∧ IsZero V
habs : f = 0
⊢ 0 < Module.finrank k (V ⟶ V)

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
π : ContRepresentation R G V
π' : ContRepresentation R G W
f : π →ⁱL π'
⊢ ∀ x ∈ π.invariants, f.toContinuousLinearMap x ∈ π'.invariants

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✝¹⁴ : Monoid G
inst✝¹³ : Ring R
inst✝¹² : AddCommGroup V
inst✝¹¹ : TopologicalSpace V
inst✝¹⁰ : IsTopologicalAddGroup V
inst✝⁹ : Module R V
inst✝⁸ : AddCommGroup W
inst✝⁷ : TopologicalSpace W
inst✝⁶ : IsTopologicalAddGroup W
inst✝⁵ : Module R W
inst✝⁴ : AddCommGroup U
inst✝³ : Module R U
inst✝² : TopologicalSpace U
inst✝¹ : IsTopologicalAddGroup U
H : Type u_6
inst✝ : Monoid H
π : ContRepresentation R G V
π' : ContRepresentation R H W
φ : H →* G
f : π.restrict φ →ⁱL π'
⊢ ∀ x ∈ π.invariants, f.toContinuousLinearMap x ∈ π'.invariants

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.FinGroupCharZero

Goal state before:
pp: k : Type u
inst✝⁴ : Field k
G : Type u
inst✝³ : Finite G
inst✝² : Group G
inst✝¹ : IsAlgClosed k
inst✝ : NeZero ↑(Nat.card G)
V : FDRep k G
h : Module.finrank k (V ⟶ V) = 1
W : FDRep k G
f : W ⟶ V
x✝ : Mono f
hf : f ≠ 0
this : Epi f
⊢ IsIso f

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✝²¹ : Monoid G✝
inst✝²⁰ : Ring R
inst✝¹⁹ : AddCommGroup V
inst✝¹⁸ : TopologicalSpace V
inst✝¹⁷ : IsTopologicalAddGroup V
inst✝¹⁶ : Module R V
inst✝¹⁵ : AddCommGroup W
inst✝¹⁴ : TopologicalSpace W
inst✝¹³ : IsTopologicalAddGroup W
inst✝¹² : Module R W
inst✝¹¹ : AddCommGroup U
inst✝¹⁰ : Module R U
inst✝⁹ : TopologicalSpace U
inst✝⁸ : IsTopologicalAddGroup U
H✝ : Type u_6
inst✝⁷ : Monoid H✝
π✝ : ContRepresentation R G✝ V
π' : ContRepresentation R H✝ W
G : Type u_7
H : Type u_8
inst✝⁶ : Group G
inst✝⁵ : TopologicalSpace G
inst✝⁴ : TopologicalSpace R
inst✝³ : ContinuousSMul R V
inst✝² : ContinuousSMul R W
inst✝¹ : Group H
inst✝ : TopologicalSpace H
φ : G →ₜ* H
π : ContRepresentation R G V
⊢ ∀ {a b : C(H, V)},
    a ∈ {f | ∀ (g : G) (h : H), f (φ g * h) = (π g) (f h)} →
      b ∈ {f | ∀ (g : G) (h : H), f (φ g * h) = (π g) (f h)} →
        a + b ∈ {f | ∀ (g : G) (h : H), f (φ g * h) = (π g) (f h)}

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.FinGroupCharZero

Goal state before:
pp: k : Type u
inst✝⁴ : Field k
G : Type u
inst✝³ : Finite G
inst✝² : Group G
inst✝¹ : IsAlgClosed k
inst✝ : NeZero ↑(Nat.card G)
V : FDRep k G
h : Module.finrank k (V ⟶ V) = 1
W : FDRep k G
f : W ⟶ V
x✝ : Mono f
hf : f ≠ 0
this : Epi (Abelian.image.ι f)
⊢ Epi f

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✝²¹ : Monoid G✝
inst✝²⁰ : Ring R
inst✝¹⁹ : AddCommGroup V
inst✝¹⁸ : TopologicalSpace V
inst✝¹⁷ : IsTopologicalAddGroup V
inst✝¹⁶ : Module R V
inst✝¹⁵ : AddCommGroup W
inst✝¹⁴ : TopologicalSpace W
inst✝¹³ : IsTopologicalAddGroup W
inst✝¹² : Module R W
inst✝¹¹ : AddCommGroup U
inst✝¹⁰ : Module R U
inst✝⁹ : TopologicalSpace U
inst✝⁸ : IsTopologicalAddGroup U
H✝ : Type u_6
inst✝⁷ : Monoid H✝
π✝ : ContRepresentation R G✝ V
π' : ContRepresentation R H✝ W
G : Type u_7
H : Type u_8
inst✝⁶ : Group G
inst✝⁵ : TopologicalSpace G
inst✝⁴ : TopologicalSpace R
inst✝³ : ContinuousSMul R V
inst✝² : ContinuousSMul R W
inst✝¹ : Group H
inst✝ : TopologicalSpace H
φ : G →ₜ* H
π : ContRepresentation R G V
⊢ 0 ∈ {f | ∀ (g : G) (h : H), f (φ g * h) = (π g) (f 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✝²¹ : Monoid G✝
inst✝²⁰ : Ring R
inst✝¹⁹ : AddCommGroup V
inst✝¹⁸ : TopologicalSpace V
inst✝¹⁷ : IsTopologicalAddGroup V
inst✝¹⁶ : Module R V
inst✝¹⁵ : AddCommGroup W
inst✝¹⁴ : TopologicalSpace W
inst✝¹³ : IsTopologicalAddGroup W
inst✝¹² : Module R W
inst✝¹¹ : AddCommGroup U
inst✝¹⁰ : Module R U
inst✝⁹ : TopologicalSpace U
inst✝⁸ : IsTopologicalAddGroup U
H✝ : Type u_6
inst✝⁷ : Monoid H✝
π✝ : ContRepresentation R G✝ V
π' : ContRepresentation R H✝ W
G : Type u_7
H : Type u_8
inst✝⁶ : Group G
inst✝⁵ : TopologicalSpace G
inst✝⁴ : TopologicalSpace R
inst✝³ : ContinuousSMul R V
inst✝² : ContinuousSMul R W
inst✝¹ : Group H
inst✝ : TopologicalSpace H
φ : G →ₜ* H
π : ContRepresentation R G V
⊢ ∀ (c : R) {x : C(H, V)},
    x ∈ {f | ∀ (g : G) (h : H), f (φ g * h) = (π g) (f h)} → c • x ∈ {f | ∀ (g : G) (h : H), f (φ g * h) = (π g) (f h)}

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 A B : Rep k G
⊢ { pt := of (A.ρ.prod B.ρ), fst := ofHom (Representation.IntertwiningMap.fst k A.ρ B.ρ),
            snd := ofHom (Representation.IntertwiningMap.snd k A.ρ B.ρ),
            inl := ofHom (Representation.IntertwiningMap.inl k A.ρ B.ρ),
            inr := ofHom (Representation.IntertwiningMap.inr k A.ρ B.ρ), inl_fst := ⋯, inl_snd := ⋯, inr_fst := ⋯,
            inr_snd := ⋯ }.fst ≫
        { pt := of (A.ρ.prod B.ρ), fst := ofHom (Representation.IntertwiningMap.fst k A.ρ B.ρ),
            snd := ofHom (Representation.IntertwiningMap.snd k A.ρ B.ρ),
            inl := ofHom (Representation.IntertwiningMap.inl k A.ρ B.ρ),
            inr := ofHom (Representation.IntertwiningMap.inr k A.ρ B.ρ), inl_fst := ⋯, inl_snd := ⋯, inr_fst := ⋯,
            inr_snd := ⋯ }.inl +
      { pt := of (A.ρ.prod B.ρ), fst := ofHom (Representation.IntertwiningMap.fst k A.ρ B.ρ),
            snd := ofHom (Representation.IntertwiningMap.snd k A.ρ B.ρ),
            inl := ofHom (Representation.IntertwiningMap.inl k A.ρ B.ρ),
            inr := ofHom (Representation.IntertwiningMap.inr k A.ρ B.ρ), inl_fst := ⋯, inl_snd := ⋯, inr_fst := ⋯,
            inr_snd := ⋯ }.snd ≫
        { pt := of (A.ρ.prod B.ρ), fst := ofHom (Representation.IntertwiningMap.fst k A.ρ B.ρ),
            snd := ofHom (Representation.IntertwiningMap.snd k A.ρ B.ρ),
            inl := ofHom (Representation.IntertwiningMap.inl k A.ρ B.ρ),
            inr := ofHom (Representation.IntertwiningMap.inr k A.ρ B.ρ), inl_fst := ⋯, inl_snd := ⋯, inr_fst := ⋯,
            inr_snd := ⋯ }.inr =
    𝟙
      { pt := of (A.ρ.prod B.ρ), fst := ofHom (Representation.IntertwiningMap.fst k A.ρ B.ρ),
          snd := ofHom (Representation.IntertwiningMap.snd k A.ρ B.ρ),
          inl := ofHom (Representation.IntertwiningMap.inl k A.ρ B.ρ),
          inr := ofHom (Representation.IntertwiningMap.inr k A.ρ B.ρ), inl_fst := ⋯, inl_snd := ⋯, inr_fst := ⋯,
          inr_snd := ⋯ }.pt

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✝²¹ : Monoid G✝
inst✝²⁰ : Ring R
inst✝¹⁹ : AddCommGroup V
inst✝¹⁸ : TopologicalSpace V
inst✝¹⁷ : IsTopologicalAddGroup V
inst✝¹⁶ : Module R V
inst✝¹⁵ : AddCommGroup W
inst✝¹⁴ : TopologicalSpace W
inst✝¹³ : IsTopologicalAddGroup W
inst✝¹² : Module R W
inst✝¹¹ : AddCommGroup U
inst✝¹⁰ : Module R U
inst✝⁹ : TopologicalSpace U
inst✝⁸ : IsTopologicalAddGroup U
H✝ : Type u_6
inst✝⁷ : Monoid H✝
π✝ : ContRepresentation R G✝ V
π' : ContRepresentation R H✝ W
G : Type u_7
H : Type u_8
inst✝⁶ : Group G
inst✝⁵ : TopologicalSpace G
inst✝⁴ : TopologicalSpace R
inst✝³ : ContinuousSMul R V
inst✝² : ContinuousSMul R W
inst✝¹ : Group H
inst✝ : TopologicalSpace H
φ : G →ₜ* H
π : ContRepresentation R G V
⊢ Continuous fun p ↦ p.1 • p.2

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
H✝ : Type u_6
inst✝⁹ : Monoid H✝
π✝¹ : ContRepresentation R G✝ V
π' : ContRepresentation R H✝ W
G : Type u_7
H : Type u_8
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalSpace R
inst✝⁵ : ContinuousSMul R V
inst✝⁴ : ContinuousSMul R W
inst✝³ : Group H
inst✝² : TopologicalSpace H
φ : G →ₜ* H
π✝ : ContRepresentation R G V
inst✝¹ : IsTopologicalGroup G
inst✝ : IsTopologicalGroup H
π : ContRepresentation R G V
h : H
f : C(H, V)
hf : f ∈ coindV φ π
⊢ f.comp (ContinuousMap.mulRight h) ∈ coindV φ π

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.FinGroupCharZero

Goal state before:
pp: k : Type u
inst✝⁴ : Field k
G : Type u
inst✝³ : Finite G
inst✝² : Group G
inst✝¹ : IsAlgClosed k
inst✝ : NeZero ↑(Nat.card G)
V : FDRep k G
h : Module.finrank k (V ⟶ V) = 1
W : FDRep k G
f : W ⟶ V
x✝ : Mono f
ι : Abelian.image f ⟶ V := Abelian.image.ι f
hf : Abelian.factorThruImage f ≫ ι ≠ 0
φ : V ⟶ Abelian.image f := Injective.factorThru (𝟙 (Abelian.image f)) ι
⊢ φ ≫ ι ≠ 0

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 : Rep k G
f : trivial k G PUnit.{w + 1} ⟶ X
⊢ f = default

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 : Rep k G
f : X ⟶ trivial k G PUnit.{w + 1}
⊢ f = default

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
M : Rep k G
⊢ Limits.IsZero M ↔ Subsingleton ↑M

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
H✝ : Type u_6
inst✝⁹ : Monoid H✝
π✝¹ : ContRepresentation R G✝ V
π' : ContRepresentation R H✝ W
G : Type u_7
H : Type u_8
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalSpace R
inst✝⁵ : ContinuousSMul R V
inst✝⁴ : ContinuousSMul R W
inst✝³ : Group H
inst✝² : TopologicalSpace H
φ : G →ₜ* H
π✝ : ContRepresentation R G V
inst✝¹ : IsTopologicalGroup G
inst✝ : IsTopologicalGroup H
π : ContRepresentation R G V
h : H
x✝¹ x✝ : ↥(coindV φ π)
⊢ (match x✝¹ + x✝ with
    | ⟨f, hf⟩ => ⟨f.comp (ContinuousMap.mulRight h), ⋯⟩) =
    (match x✝¹ with
      | ⟨f, hf⟩ => ⟨f.comp (ContinuousMap.mulRight h), ⋯⟩) +
      match x✝ with
      | ⟨f, hf⟩ => ⟨f.comp (ContinuousMap.mulRight h), ⋯⟩

Tactic:
by

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

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : CommSemiring k
inst✝ : Monoid G
M N O : Rep k G
r : k
f : M ⟶ N
g : N ⟶ O
⊢ (r • f) ≫ g = r • f ≫ g

Tactic:
by

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

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : CommSemiring k
inst✝ : Monoid G
M N O : Rep k G
f : M ⟶ N
r : k
g : N ⟶ O
⊢ f ≫ (r • g) = r • f ≫ g

Tactic:
by

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

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : CommSemiring k
inst✝ : Monoid G
M N : Rep k G
⊢ ∀ (c : k) (x : M ⟶ N),
    { toFun := Hom.hom, map_zero' := ⋯, map_add' := ⋯ } (c • x) =
      c • { toFun := Hom.hom, map_zero' := ⋯, map_add' := ⋯ } 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
H✝ : Type u_6
inst✝⁹ : Monoid H✝
π✝¹ : ContRepresentation R G✝ V
π' : ContRepresentation R H✝ W
G : Type u_7
H : Type u_8
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalSpace R
inst✝⁵ : ContinuousSMul R V
inst✝⁴ : ContinuousSMul R W
inst✝³ : Group H
inst✝² : TopologicalSpace H
φ : G →ₜ* H
π✝ : ContRepresentation R G V
inst✝¹ : IsTopologicalGroup G
inst✝ : IsTopologicalGroup H
π : ContRepresentation R G V
h : H
x✝¹ : R
x✝ : ↥(coindV φ π)
⊢ (match x✝¹ • x✝ with
    | ⟨f, hf⟩ => ⟨f.comp (ContinuousMap.mulRight h), ⋯⟩) =
    (RingHom.id R) x✝¹ •
      match x✝ with
      | ⟨f, hf⟩ => ⟨f.comp (ContinuousMap.mulRight h), ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.FinGroupCharZero

Goal state before:
pp: k : Type u
inst✝⁴ : Field k
G : Type u
inst✝³ : Finite G
inst✝² : Group G
inst✝¹ : IsAlgClosed k
inst✝ : NeZero ↑(Nat.card G)
V : FDRep k G
h : Module.finrank k (V ⟶ V) = 1
⊢ Simple V

Tactic:
by

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

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : CommRing k
inst✝ : Monoid G
X Y : Rep k G
f : X ⟶ Y
r : k
⊢ (forget₂ (Rep k G) (ModuleCat k)).map (r • f) = r • (forget₂ (Rep k G) (ModuleCat k)).map f

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
H✝ : Type u_6
inst✝⁹ : Monoid H✝
π✝¹ : ContRepresentation R G✝ V
π' : ContRepresentation R H✝ W
G : Type u_7
H : Type u_8
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalSpace R
inst✝⁵ : ContinuousSMul R V
inst✝⁴ : ContinuousSMul R W
inst✝³ : Group H
inst✝² : TopologicalSpace H
φ : G →ₜ* H
π✝ : ContRepresentation R G V
inst✝¹ : IsTopologicalGroup G
inst✝ : IsTopologicalGroup H
π : ContRepresentation R G V
h : H
⊢ Continuous
    (Subtype.val ∘ fun x ↦
      match x with
      | ⟨f, hf⟩ => ⟨f.comp (ContinuousMap.mulRight 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✝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
H✝ : Type u_6
inst✝⁹ : Monoid H✝
π✝¹ : ContRepresentation R G✝ V
π' : ContRepresentation R H✝ W
G : Type u_7
H : Type u_8
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalSpace R
inst✝⁵ : ContinuousSMul R V
inst✝⁴ : ContinuousSMul R W
inst✝³ : Group H
inst✝² : TopologicalSpace H
φ : G →ₜ* H
π✝ : ContRepresentation R G V
inst✝¹ : IsTopologicalGroup G
inst✝ : IsTopologicalGroup H
π : ContRepresentation R G V
⊢ {
      toFun := fun x ↦
        match x with
        | ⟨f, hf⟩ => ⟨f.comp (ContinuousMap.mulRight 1), ⋯⟩,
      map_add' := ⋯, map_smul' := ⋯, cont := ⋯ } =
    1

Tactic:
by

Goal state after: