F : Type ?u.459241
R : Type u_2
A : Type u_1
B : Type ?u.459250
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : Semiring A
inst✝⁶ : Algebra R A
inst✝⁵ : StarRing A
inst✝⁴ : StarModule R A
inst✝³ : Semiring B
inst✝² : Algebra R B
inst✝¹ : StarRing B
inst✝ : StarModule R B
s : Set A
S : StarSubalgebra R A
⊢ adjoin R s ≤ S ↔ s ≤ ↑S
F : Type ?u.459241
R : Type u_2
A : Type u_1
B : Type ?u.459250
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : Semiring A
inst✝⁶ : Algebra R A
inst✝⁵ : StarRing A
inst✝⁴ : StarModule R A
inst✝³ : Semiring B
inst✝² : Algebra R B
inst✝¹ : StarRing B
inst✝ : StarModule R B
⊢ GaloisConnection (adjoin R) SetLike.coe
intro s S
F : Type ?u.459241
R : Type u_2
A : Type u_1
B : Type ?u.459250
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : Semiring A
inst✝⁶ : Algebra R A
inst✝⁵ : StarRing A
inst✝⁴ : StarModule R A
inst✝³ : Semiring B
inst✝² : Algebra R B
inst✝¹ : StarRing B
inst✝ : StarModule R B
s : Set A
S : StarSubalgebra R A
⊢ s ∪ star s ⊆ ↑S ↔ s ≤ ↑S
F : Type ?u.459241
R : Type u_2
A : Type u_1
B : Type ?u.459250
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : Semiring A
inst✝⁶ : Algebra R A
inst✝⁵ : StarRing A
inst✝⁴ : StarModule R A
inst✝³ : Semiring B
inst✝² : Algebra R B
inst✝¹ : StarRing B
inst✝ : StarModule R B
s : Set A
S : StarSubalgebra R A
⊢ adjoin R s ≤ S ↔ s ≤ ↑S
rw [← toSubalgebra_le_iff, adjoin_toSubalgebra, Algebra.adjoin_le_iff, coe_toSubalgebra]
no goals
F : Type ?u.459241
R : Type u_2
A : Type u_1
B : Type ?u.459250
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : Semiring A
inst✝⁶ : Algebra R A
inst✝⁵ : StarRing A
inst✝⁴ : StarModule R A
inst✝³ : Semiring B
inst✝² : Algebra R B
inst✝¹ : StarRing B
inst✝ : StarModule R B
s : Set A
S : StarSubalgebra R A
⊢ s ∪ star s ⊆ ↑S ↔ s ≤ ↑S
exact
  ⟨fun h => (Set.subset_union_left s _).trans h, fun h =>
    Set.union_subset h fun x hx => star_star x ▸ star_mem (show star x ∈ S from h hx)⟩