Module: Mathlib.Algebra.Star.Subalgebra

Goal state before:
pp: F : Type u_1
R : Type u_2
A : Type u_3
B : Type u_4
C : Type u_5
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : Semiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra R A
inst✝⁸ : StarModule R A
inst✝⁷ : Semiring B
inst✝⁶ : StarRing B
inst✝⁵ : Algebra R B
inst✝⁴ : StarModule R B
inst✝³ : Semiring C
inst✝² : StarRing C
inst✝¹ : Algebra R C
inst✝ : StarModule R C
q : StarSubalgebra R A
carrier✝ : Set A
mul_mem'✝ : ∀ {a b : A}, a ∈ carrier✝ → b ∈ carrier✝ → a * b ∈ carrier✝
one_mem'✝ : 1 ∈ { carrier := carrier✝, mul_mem' := mul_mem'✝ }.carrier
add_mem'✝ :
  ∀ {a b : A},
    a ∈ { carrier := carrier✝, mul_mem' := mul_mem'✝, one_mem' := one_mem'✝ }.carrier →
      b ∈ { carrier := carrier✝, mul_mem' := mul_mem'✝, one_mem' := one_mem'✝ }.carrier →
        a + b ∈ { carrier := carrier✝, mul_mem' := mul_mem'✝, one_mem' := one_mem'✝ }.carrier
zero_mem'✝ : 0 ∈ { carrier := carrier✝, mul_mem' := mul_mem'✝, one_mem' := one_mem'✝ }.carrier
algebraMap_mem'✝ :
  ∀ (r : R),
    (algebraMap R A) r ∈
      { carrier := carrier✝, mul_mem' := mul_mem'✝, one_mem' := one_mem'✝, add_mem' := add_mem'✝,
          zero_mem' := zero_mem'✝ }.carrier
star_mem'✝ :
  ∀ {a : A},
    a ∈
        { carrier := carrier✝, mul_mem' := mul_mem'✝, one_mem' := one_mem'✝, add_mem' := add_mem'✝,
            zero_mem' := zero_mem'✝, algebraMap_mem' := algebraMap_mem'✝ }.carrier →
      star a ∈
        { carrier := carrier✝, mul_mem' := mul_mem'✝, one_mem' := one_mem'✝, add_mem' := add_mem'✝,
            zero_mem' := zero_mem'✝, algebraMap_mem' := algebraMap_mem'✝ }.carrier
h :
  (fun S ↦ S.carrier)
      { carrier := carrier✝, mul_mem' := mul_mem'✝, one_mem' := one_mem'✝, add_mem' := add_mem'✝,
        zero_mem' := zero_mem'✝, algebraMap_mem' := algebraMap_mem'✝, star_mem' := star_mem'✝ } =
    (fun S ↦ S.carrier) q
⊢ { carrier := carrier✝, mul_mem' := mul_mem'✝, one_mem' := one_mem'✝, add_mem' := add_mem'✝, zero_mem' := zero_mem'✝,
      algebraMap_mem' := algebraMap_mem'✝, star_mem' := star_mem'✝ } =
    q

Tactic:
cases q

Goal state after:
case mk
F : Type u_1
R : Type u_2
A : Type u_3
B : Type u_4
C : Type u_5
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : Semiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra R A
inst✝⁸ : StarModule R A
inst✝⁷ : Semiring B
inst✝⁶ : StarRing B
inst✝⁵ : Algebra R B
inst✝⁴ : StarModule R B
inst✝³ : Semiring C
inst✝² : StarRing C
inst✝¹ : Algebra R C
inst✝ : StarModule R C
carrier✝ : Set A
mul_mem'✝ : ∀ {a b : A}, a ∈ carrier✝ → b ∈ carrier✝ → a * b ∈ carrier✝
one_mem'✝ : 1 ∈ { carrier := carrier✝, mul_mem' := mul_mem'✝ }.carrier
add_mem'✝ :
  ∀ {a b : A},
    a ∈ { carrier := carrier✝, mul_mem' := mul_mem'✝, one_mem' := one_mem'✝ }.carrier →
      b ∈ { carrier := carrier✝, mul_mem' := mul_mem'✝, one_mem' := one_mem'✝ }.carrier →
        a + b ∈ { carrier := carrier✝, mul_mem' := mul_mem'✝, one_mem' := one_mem'✝ }.carrier
zero_mem'✝ : 0 ∈ { carrier := carrier✝, mul_mem' := mul_mem'✝, one_mem' := one_mem'✝ }.carrier
algebraMap_mem'✝ :
  ∀ (r : R),
    (algebraMap R A) r ∈
      { carrier := carrier✝, mul_mem' := mul_mem'✝, one_mem' := one_mem'✝, add_mem' := add_mem'✝,
          zero_mem' := zero_mem'✝ }.carrier
star_mem'✝¹ :
  ∀ {a : A},
    a ∈
        { carrier := carrier✝, mul_mem' := mul_mem'✝, one_mem' := one_mem'✝, add_mem' := add_mem'✝,
            zero_mem' := zero_mem'✝, algebraMap_mem' := algebraMap_mem'✝ }.carrier →
      star a ∈
        { carrier := carrier✝, mul_mem' := mul_mem'✝, one_mem' := one_mem'✝, add_mem' := add_mem'✝,
            zero_mem' := zero_mem'✝, algebraMap_mem' := algebraMap_mem'✝ }.carrier
toSubalgebra✝ : Subalgebra R A
star_mem'✝ : ∀ {a : A}, a ∈ toSubalgebra✝.carrier → star a ∈ toSubalgebra✝.carrier
h :
  (fun S ↦ S.carrier)
      { carrier := carrier✝, mul_mem' := mul_mem'✝, one_mem' := one_mem'✝, add_mem' := add_mem'✝,
        zero_mem' := zero_mem'✝, algebraMap_mem' := algebraMap_mem'✝, star_mem' := star_mem'✝¹ } =
    (fun S ↦ S.carrier) { toSubalgebra := toSubalgebra✝, star_mem' := star_mem'✝ }
⊢ { carrier := carrier✝, mul_mem' := mul_mem'✝, one_mem' := one_mem'✝, add_mem' := add_mem'✝, zero_mem' := zero_mem'✝,
      algebraMap_mem' := algebraMap_mem'✝, star_mem' := star_mem'✝¹ } =
    { toSubalgebra := toSubalgebra✝, star_mem' := star_mem'✝ }