Theorem: exists_isTranscendenceBasis
File: Mathlib/RingTheory/AlgebraicIndependent.lean

tactic: cases' exists_maximal_algebraicIndependent (∅ : Set A) Set.univ (Set.subset_univ _)
    ((algebraicIndependent_empty_iff R A).2 h) with
  s hs
state_before: ι : Type ?u.1352562
ι' : Type ?u.1352565
R : Type u_1
K : Type ?u.1352571
A : Type u_2
A' : Type ?u.1352577
A'' : Type ?u.1352580
V : Type u
V' : Type ?u.1352585
x : ι → A
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : CommRing A'
inst✝³ : CommRing A''
inst✝² : Algebra R A
inst✝¹ : Algebra R A'
inst✝ : Algebra R A''
a b : R
h : Injective ↑(algebraMap R A)
⊢ ∃ s, IsTranscendenceBasis R Subtype.val

tactic: use s, hs.1
state_before: case intro
ι : Type ?u.1352562
ι' : Type ?u.1352565
R : Type u_1
K : Type ?u.1352571
A : Type u_2
A' : Type ?u.1352577
A'' : Type ?u.1352580
V : Type u
V' : Type ?u.1352585
x : ι → A
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : CommRing A'
inst✝³ : CommRing A''
inst✝² : Algebra R A
inst✝¹ : Algebra R A'
inst✝ : Algebra R A''
a b : R
h : Injective ↑(algebraMap R A)
s : Set A
hs :
  AlgebraicIndependent R Subtype.val ∧
    ∅ ⊆ s ∧ s ⊆ univ ∧ ∀ (x : Set A), AlgebraicIndependent R Subtype.val → s ⊆ x → x ⊆ univ → x = s
⊢ ∃ s, IsTranscendenceBasis R Subtype.val

tactic: intro t ht hr
state_before: case intro
ι : Type ?u.1352562
ι' : Type ?u.1352565
R : Type u_1
K : Type ?u.1352571
A : Type u_2
A' : Type ?u.1352577
A'' : Type ?u.1352580
V : Type u
V' : Type ?u.1352585
x : ι → A
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : CommRing A'
inst✝³ : CommRing A''
inst✝² : Algebra R A
inst✝¹ : Algebra R A'
inst✝ : Algebra R A''
a b : R
h : Injective ↑(algebraMap R A)
s : Set A
hs :
  AlgebraicIndependent R Subtype.val ∧
    ∅ ⊆ s ∧ s ⊆ univ ∧ ∀ (x : Set A), AlgebraicIndependent R Subtype.val → s ⊆ x → x ⊆ univ → x = s
⊢ ∀ (s_1 : Set A), AlgebraicIndependent R Subtype.val → range Subtype.val ≤ s_1 → range Subtype.val = s_1

tactic: simp only [Subtype.range_coe_subtype, setOf_mem_eq] at *
state_before: case intro
ι : Type ?u.1352562
ι' : Type ?u.1352565
R : Type u_1
K : Type ?u.1352571
A : Type u_2
A' : Type ?u.1352577
A'' : Type ?u.1352580
V : Type u
V' : Type ?u.1352585
x : ι → A
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : CommRing A'
inst✝³ : CommRing A''
inst✝² : Algebra R A
inst✝¹ : Algebra R A'
inst✝ : Algebra R A''
a b : R
h : Injective ↑(algebraMap R A)
s : Set A
hs :
  AlgebraicIndependent R Subtype.val ∧
    ∅ ⊆ s ∧ s ⊆ univ ∧ ∀ (x : Set A), AlgebraicIndependent R Subtype.val → s ⊆ x → x ⊆ univ → x = s
t : Set A
ht : AlgebraicIndependent R Subtype.val
hr : range Subtype.val ≤ t
⊢ range Subtype.val = t

tactic: exact Eq.symm (hs.2.2.2 t ht hr (Set.subset_univ _))
state_before: case intro
ι : Type ?u.1352562
ι' : Type ?u.1352565
R : Type u_1
K : Type ?u.1352571
A : Type u_2
A' : Type ?u.1352577
A'' : Type ?u.1352580
V : Type u
V' : Type ?u.1352585
x : ι → A
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : CommRing A'
inst✝³ : CommRing A''
inst✝² : Algebra R A
inst✝¹ : Algebra R A'
inst✝ : Algebra R A''
a b : R
h : Injective ↑(algebraMap R A)
s : Set A
hs :
  AlgebraicIndependent R Subtype.val ∧
    ∅ ⊆ s ∧ s ⊆ univ ∧ ∀ (x : Set A), AlgebraicIndependent R Subtype.val → s ⊆ x → x ⊆ univ → x = s
t : Set A
ht : AlgebraicIndependent R Subtype.val
hr : s ≤ t
⊢ s = t