Theorem: ContinuousAlgHom.coe_inj
File: Mathlib/Topology/Algebra/Algebra.lean

Proof steps:
state_before: R : Type u_1
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : TopologicalSpace R
inst✝⁹ : TopologicalSemiring R
A : Type u_2
inst✝⁸ : Semiring A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSemiring A
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSemiring R
B : Type u_3
inst✝³ : Semiring B
inst✝² : TopologicalSpace B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
f g : A →A[R] B
⊢ ↑f = ↑g ↔ f = g
state_after: case mk
R : Type u_1
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : TopologicalSpace R
inst✝⁹ : TopologicalSemiring R
A : Type u_2
inst✝⁸ : Semiring A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSemiring A
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSemiring R
B : Type u_3
inst✝³ : Semiring B
inst✝² : TopologicalSpace B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
g : A →A[R] B
toAlgHom✝ : A →ₐ[R] B
cont✝ : Continuous (↑↑toAlgHom✝.toRingHom).toFun
⊢ ↑{ toAlgHom := toAlgHom✝, cont := cont✝ } = ↑g ↔ { toAlgHom := toAlgHom✝, cont := cont✝ } = g
tactic: cases f
state_before: case mk
R : Type u_1
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : TopologicalSpace R
inst✝⁹ : TopologicalSemiring R
A : Type u_2
inst✝⁸ : Semiring A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSemiring A
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSemiring R
B : Type u_3
inst✝³ : Semiring B
inst✝² : TopologicalSpace B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
g : A →A[R] B
toAlgHom✝ : A →ₐ[R] B
cont✝ : Continuous (↑↑toAlgHom✝.toRingHom).toFun
⊢ ↑{ toAlgHom := toAlgHom✝, cont := cont✝ } = ↑g ↔ { toAlgHom := toAlgHom✝, cont := cont✝ } = g
state_after: case mk.mk
R : Type u_1
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : TopologicalSpace R
inst✝⁹ : TopologicalSemiring R
A : Type u_2
inst✝⁸ : Semiring A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSemiring A
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSemiring R
B : Type u_3
inst✝³ : Semiring B
inst✝² : TopologicalSpace B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
toAlgHom✝¹ : A →ₐ[R] B
cont✝¹ : Continuous (↑↑toAlgHom✝¹.toRingHom).toFun
toAlgHom✝ : A →ₐ[R] B
cont✝ : Continuous (↑↑toAlgHom✝.toRingHom).toFun
⊢ ↑{ toAlgHom := toAlgHom✝¹, cont := cont✝¹ } = ↑{ toAlgHom := toAlgHom✝, cont := cont✝ } ↔
    { toAlgHom := toAlgHom✝¹, cont := cont✝¹ } = { toAlgHom := toAlgHom✝, cont := cont✝ }
tactic: cases g
state_before: case mk.mk
R : Type u_1
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : TopologicalSpace R
inst✝⁹ : TopologicalSemiring R
A : Type u_2
inst✝⁸ : Semiring A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSemiring A
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSemiring R
B : Type u_3
inst✝³ : Semiring B
inst✝² : TopologicalSpace B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
toAlgHom✝¹ : A →ₐ[R] B
cont✝¹ : Continuous (↑↑toAlgHom✝¹.toRingHom).toFun
toAlgHom✝ : A →ₐ[R] B
cont✝ : Continuous (↑↑toAlgHom✝.toRingHom).toFun
⊢ ↑{ toAlgHom := toAlgHom✝¹, cont := cont✝¹ } = ↑{ toAlgHom := toAlgHom✝, cont := cont✝ } ↔
    { toAlgHom := toAlgHom✝¹, cont := cont✝¹ } = { toAlgHom := toAlgHom✝, cont := cont✝ }
state_after: case mk.mk
R : Type u_1
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : TopologicalSpace R
inst✝⁹ : TopologicalSemiring R
A : Type u_2
inst✝⁸ : Semiring A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSemiring A
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSemiring R
B : Type u_3
inst✝³ : Semiring B
inst✝² : TopologicalSpace B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
toAlgHom✝¹ : A →ₐ[R] B
cont✝¹ : Continuous (↑↑toAlgHom✝¹.toRingHom).toFun
toAlgHom✝ : A →ₐ[R] B
cont✝ : Continuous (↑↑toAlgHom✝.toRingHom).toFun
⊢ ↑{ toAlgHom := toAlgHom✝¹, cont := cont✝¹ } = ↑{ toAlgHom := toAlgHom✝, cont := cont✝ } ↔ toAlgHom✝¹ = toAlgHom✝
tactic: simp only [mk.injEq]
state_before: case mk.mk
R : Type u_1
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : TopologicalSpace R
inst✝⁹ : TopologicalSemiring R
A : Type u_2
inst✝⁸ : Semiring A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSemiring A
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSemiring R
B : Type u_3
inst✝³ : Semiring B
inst✝² : TopologicalSpace B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
toAlgHom✝¹ : A →ₐ[R] B
cont✝¹ : Continuous (↑↑toAlgHom✝¹.toRingHom).toFun
toAlgHom✝ : A →ₐ[R] B
cont✝ : Continuous (↑↑toAlgHom✝.toRingHom).toFun
⊢ ↑{ toAlgHom := toAlgHom✝¹, cont := cont✝¹ } = ↑{ toAlgHom := toAlgHom✝, cont := cont✝ } ↔ toAlgHom✝¹ = toAlgHom✝
state_after: no goals
tactic: exact Eq.congr_right rfl