Theorem: Unitization.starLift_symm_apply
File: Mathlib/Algebra/Algebra/Unitization.lean

Proof steps:
state_before: R : Type u_1
A : Type u_2
C : Type u_3
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : NonUnitalSemiring A
inst✝⁸ : StarRing A
inst✝⁷ : Module R A
inst✝⁶ : SMulCommClass R A A
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : StarModule R A
inst✝³ : Semiring C
inst✝² : Algebra R C
inst✝¹ : StarRing C
inst✝ : StarModule R C
φ : A →⋆ₙₐ[R] C
x : Unitization R A
⊢ (↑↑(lift φ.toNonUnitalAlgHom).toRingHom).toFun (star x) = star ((↑↑(lift φ.toNonUnitalAlgHom).toRingHom).toFun x)
state_after: case inl_add_inr
R : Type u_1
A : Type u_2
C : Type u_3
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : NonUnitalSemiring A
inst✝⁸ : StarRing A
inst✝⁷ : Module R A
inst✝⁶ : SMulCommClass R A A
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : StarModule R A
inst✝³ : Semiring C
inst✝² : Algebra R C
inst✝¹ : StarRing C
inst✝ : StarModule R C
φ : A →⋆ₙₐ[R] C
r✝ : R
a✝ : A
⊢ (↑↑(lift φ.toNonUnitalAlgHom).toRingHom).toFun (star (inl r✝ + ↑a✝)) =
    star ((↑↑(lift φ.toNonUnitalAlgHom).toRingHom).toFun (inl r✝ + ↑a✝))
tactic: induction x
state_before: case inl_add_inr
R : Type u_1
A : Type u_2
C : Type u_3
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : NonUnitalSemiring A
inst✝⁸ : StarRing A
inst✝⁷ : Module R A
inst✝⁶ : SMulCommClass R A A
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : StarModule R A
inst✝³ : Semiring C
inst✝² : Algebra R C
inst✝¹ : StarRing C
inst✝ : StarModule R C
φ : A →⋆ₙₐ[R] C
r✝ : R
a✝ : A
⊢ (↑↑(lift φ.toNonUnitalAlgHom).toRingHom).toFun (star (inl r✝ + ↑a✝)) =
    star ((↑↑(lift φ.toNonUnitalAlgHom).toRingHom).toFun (inl r✝ + ↑a✝))
state_after: no goals
tactic: simp [map_star]
state_before: R : Type u_1
A : Type u_2
C : Type u_3
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : NonUnitalSemiring A
inst✝⁸ : StarRing A
inst✝⁷ : Module R A
inst✝⁶ : SMulCommClass R A A
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : StarModule R A
inst✝³ : Semiring C
inst✝² : Algebra R C
inst✝¹ : StarRing C
inst✝ : StarModule R C
φ : A →⋆ₙₐ[R] C
⊢ (fun φ => φ.toNonUnitalStarAlgHom.comp (inrNonUnitalStarAlgHom R A))
      ((fun φ => { toAlgHom := lift φ.toNonUnitalAlgHom, map_star' := ⋯ }) φ) =
    φ
state_after: case h
R : Type u_1
A : Type u_2
C : Type u_3
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : NonUnitalSemiring A
inst✝⁸ : StarRing A
inst✝⁷ : Module R A
inst✝⁶ : SMulCommClass R A A
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : StarModule R A
inst✝³ : Semiring C
inst✝² : Algebra R C
inst✝¹ : StarRing C
inst✝ : StarModule R C
φ : A →⋆ₙₐ[R] C
x✝ : A
⊢ ((fun φ => φ.toNonUnitalStarAlgHom.comp (inrNonUnitalStarAlgHom R A))
        ((fun φ => { toAlgHom := lift φ.toNonUnitalAlgHom, map_star' := ⋯ }) φ))
      x✝ =
    φ x✝
tactic: ext
state_before: case h
R : Type u_1
A : Type u_2
C : Type u_3
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : NonUnitalSemiring A
inst✝⁸ : StarRing A
inst✝⁷ : Module R A
inst✝⁶ : SMulCommClass R A A
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : StarModule R A
inst✝³ : Semiring C
inst✝² : Algebra R C
inst✝¹ : StarRing C
inst✝ : StarModule R C
φ : A →⋆ₙₐ[R] C
x✝ : A
⊢ ((fun φ => φ.toNonUnitalStarAlgHom.comp (inrNonUnitalStarAlgHom R A))
        ((fun φ => { toAlgHom := lift φ.toNonUnitalAlgHom, map_star' := ⋯ }) φ))
      x✝ =
    φ x✝
state_after: no goals
tactic: simp
state_before: R : Type u_1
A : Type u_2
C : Type u_3
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : NonUnitalSemiring A
inst✝⁸ : StarRing A
inst✝⁷ : Module R A
inst✝⁶ : SMulCommClass R A A
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : StarModule R A
inst✝³ : Semiring C
inst✝² : Algebra R C
inst✝¹ : StarRing C
inst✝ : StarModule R C
φ : Unitization R A →⋆ₐ[R] C
⊢ ∀ (a : A),
    ((fun φ => { toAlgHom := lift φ.toNonUnitalAlgHom, map_star' := ⋯ })
          ((fun φ => φ.toNonUnitalStarAlgHom.comp (inrNonUnitalStarAlgHom R A)) φ))
        ↑a =
      φ ↑a
state_after: no goals
tactic: simp