-- Mathlib/Topology/Algebra/StarSubalgebra.lean
theorem _root_.StarAlgHomClass.ext_topologicalClosure [T2Space B] {F : Type*}
    {S : StarSubalgebra R A} [FunLike F S.topologicalClosure B]
    [AlgHomClass F R S.topologicalClosure B] [StarHomClass F S.topologicalClosure B] {φ ψ : F}
    (hφ : Continuous φ) (hψ : Continuous ψ) (h : ∀ x : S,
        φ (inclusion (le_topologicalClosure S) x) = ψ ((inclusion (le_topologicalClosure S)) x)) :
    φ = ψ

Proof state:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : Semiring A
inst✝¹² : Algebra R A
inst✝¹¹ : StarRing A
inst✝¹⁰ : StarModule R A
inst✝⁹ : IsTopologicalSemiring A
inst✝⁸ : ContinuousStar A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : Semiring B
inst✝⁵ : Algebra R B
inst✝⁴ : StarRing B
inst✝³ : T2Space B
F : Type u_4
S : StarSubalgebra R A
inst✝² : FunLike F (↥S.topologicalClosure) B
inst✝¹ : AlgHomClass F R (↥S.topologicalClosure) B
inst✝ : StarHomClass F (↥S.topologicalClosure) B
φ ψ : F
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : ∀ (x : ↥S), φ ((inclusion ⋯) x) = ψ ((inclusion ⋯) x)
this : ↑φ = ↑ψ
⊢ ⇑↑φ = ⇑ψ

Tactic:
apply congrArg _ this

Result:
no goals

Proof state:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : Semiring A
inst✝¹² : Algebra R A
inst✝¹¹ : StarRing A
inst✝¹⁰ : StarModule R A
inst✝⁹ : IsTopologicalSemiring A
inst✝⁸ : ContinuousStar A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : Semiring B
inst✝⁵ : Algebra R B
inst✝⁴ : StarRing B
inst✝³ : T2Space B
F : Type u_4
S : StarSubalgebra R A
inst✝² : FunLike F (↥S.topologicalClosure) B
inst✝¹ : AlgHomClass F R (↥S.topologicalClosure) B
inst✝ : StarHomClass F (↥S.topologicalClosure) B
φ ψ : F
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : ∀ (x : ↥S), φ ((inclusion ⋯) x) = ψ ((inclusion ⋯) x)
⊢ ↑φ = ↑ψ

Tactic:
refine StarAlgHom.ext_topologicalClosure (R := R) (A := A) (B := B) hφ hψ (StarAlgHom.ext ?_)

Result:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : Semiring A
inst✝¹² : Algebra R A
inst✝¹¹ : StarRing A
inst✝¹⁰ : StarModule R A
inst✝⁹ : IsTopologicalSemiring A
inst✝⁸ : ContinuousStar A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : Semiring B
inst✝⁵ : Algebra R B
inst✝⁴ : StarRing B
inst✝³ : T2Space B
F : Type u_4
S : StarSubalgebra R A
inst✝² : FunLike F (↥S.topologicalClosure) B
inst✝¹ : AlgHomClass F R (↥S.topologicalClosure) B
inst✝ : StarHomClass F (↥S.topologicalClosure) B
φ ψ : F
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : ∀ (x : ↥S), φ ((inclusion ⋯) x) = ψ ((inclusion ⋯) x)
⊢ ∀ (x : ↥S), ((↑φ).comp (inclusion ⋯)) x = ((↑ψ).comp (inclusion ⋯)) x

Proof state:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : Semiring A
inst✝¹² : Algebra R A
inst✝¹¹ : StarRing A
inst✝¹⁰ : StarModule R A
inst✝⁹ : IsTopologicalSemiring A
inst✝⁸ : ContinuousStar A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : Semiring B
inst✝⁵ : Algebra R B
inst✝⁴ : StarRing B
inst✝³ : T2Space B
F : Type u_4
S : StarSubalgebra R A
inst✝² : FunLike F (↥S.topologicalClosure) B
inst✝¹ : AlgHomClass F R (↥S.topologicalClosure) B
inst✝ : StarHomClass F (↥S.topologicalClosure) B
φ ψ : F
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : ∀ (x : ↥S), φ ((inclusion ⋯) x) = ψ ((inclusion ⋯) x)
⊢ φ = ψ

Tactic:
have : (φ : S.topologicalClosure →⋆ₐ[R] B) = (ψ : S.topologicalClosure →⋆ₐ[R] B) := by
  refine StarAlgHom.ext_topologicalClosure (R := R) (A := A) (B := B) hφ hψ (StarAlgHom.ext ?_)
  simpa only [StarAlgHom.coe_comp, StarAlgHom.coe_coe] using h

Result:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : Semiring A
inst✝¹² : Algebra R A
inst✝¹¹ : StarRing A
inst✝¹⁰ : StarModule R A
inst✝⁹ : IsTopologicalSemiring A
inst✝⁸ : ContinuousStar A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : Semiring B
inst✝⁵ : Algebra R B
inst✝⁴ : StarRing B
inst✝³ : T2Space B
F : Type u_4
S : StarSubalgebra R A
inst✝² : FunLike F (↥S.topologicalClosure) B
inst✝¹ : AlgHomClass F R (↥S.topologicalClosure) B
inst✝ : StarHomClass F (↥S.topologicalClosure) B
φ ψ : F
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : ∀ (x : ↥S), φ ((inclusion ⋯) x) = ψ ((inclusion ⋯) x)
this : ↑φ = ↑ψ
⊢ φ = ψ

Proof state:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : Semiring A
inst✝¹² : Algebra R A
inst✝¹¹ : StarRing A
inst✝¹⁰ : StarModule R A
inst✝⁹ : IsTopologicalSemiring A
inst✝⁸ : ContinuousStar A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : Semiring B
inst✝⁵ : Algebra R B
inst✝⁴ : StarRing B
inst✝³ : T2Space B
F : Type u_4
S : StarSubalgebra R A
inst✝² : FunLike F (↥S.topologicalClosure) B
inst✝¹ : AlgHomClass F R (↥S.topologicalClosure) B
inst✝ : StarHomClass F (↥S.topologicalClosure) B
φ ψ : F
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : ∀ (x : ↥S), φ ((inclusion ⋯) x) = ψ ((inclusion ⋯) x)
⊢ ∀ (x : ↥S), ((↑φ).comp (inclusion ⋯)) x = ((↑ψ).comp (inclusion ⋯)) x

Tactic:
simpa only [StarAlgHom.coe_comp, StarAlgHom.coe_coe] using h

Result:
no goals

Proof state:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : Semiring A
inst✝¹² : Algebra R A
inst✝¹¹ : StarRing A
inst✝¹⁰ : StarModule R A
inst✝⁹ : IsTopologicalSemiring A
inst✝⁸ : ContinuousStar A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : Semiring B
inst✝⁵ : Algebra R B
inst✝⁴ : StarRing B
inst✝³ : T2Space B
F : Type u_4
S : StarSubalgebra R A
inst✝² : FunLike F (↥S.topologicalClosure) B
inst✝¹ : AlgHomClass F R (↥S.topologicalClosure) B
inst✝ : StarHomClass F (↥S.topologicalClosure) B
φ ψ : F
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : ∀ (x : ↥S), φ ((inclusion ⋯) x) = ψ ((inclusion ⋯) x)
this : ↑φ = ↑ψ
⊢ φ = ψ

Tactic:
rw [DFunLike.ext'_iff, ← StarAlgHom.coe_coe]

Result:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : Semiring A
inst✝¹² : Algebra R A
inst✝¹¹ : StarRing A
inst✝¹⁰ : StarModule R A
inst✝⁹ : IsTopologicalSemiring A
inst✝⁸ : ContinuousStar A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : Semiring B
inst✝⁵ : Algebra R B
inst✝⁴ : StarRing B
inst✝³ : T2Space B
F : Type u_4
S : StarSubalgebra R A
inst✝² : FunLike F (↥S.topologicalClosure) B
inst✝¹ : AlgHomClass F R (↥S.topologicalClosure) B
inst✝ : StarHomClass F (↥S.topologicalClosure) B
φ ψ : F
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : ∀ (x : ↥S), φ ((inclusion ⋯) x) = ψ ((inclusion ⋯) x)
this : ↑φ = ↑ψ
⊢ ⇑↑φ = ⇑ψ