Theorem: Algebra.FormallyUnramified.iff_exists_tensorProduct
File: Mathlib/RingTheory/Unramified/Finite.lean

Proof steps:
state_before: R S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
inst✝ : EssFiniteType R S
⊢ FormallyUnramified R S ↔ ∃ t, (∀ (s : S), (1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) * t = 0) ∧ (TensorProduct.lmul' R) t = 1
state_after: R S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
inst✝ : EssFiniteType R S
⊢ (∃ e, IsIdempotentElem e ∧ KaehlerDifferential.ideal R S = Submodule.span (S ⊗[R] S) {e}) ↔
    ∃ t, (∀ (s : S), (1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) * t = 0) ∧ (TensorProduct.lmul' R) t = 1
tactic: rw [iff_subsingleton_kaehlerDifferential, KaehlerDifferential,
    Ideal.cotangent_subsingleton_iff, Ideal.isIdempotentElem_iff_of_fg _
      (KaehlerDifferential.ideal_fg R S)]
state_before: R S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
inst✝ : EssFiniteType R S
⊢ (∃ e, IsIdempotentElem e ∧ KaehlerDifferential.ideal R S = Submodule.span (S ⊗[R] S) {e}) ↔
    ∃ t, (∀ (s : S), (1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) * t = 0) ∧ (TensorProduct.lmul' R) t = 1
state_after: R S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
inst✝ : EssFiniteType R S
this : ∀ (t : S ⊗[R] S), (TensorProduct.lmul' R) t = 1 ↔ 1 - t ∈ KaehlerDifferential.ideal R S
⊢ (∃ e, IsIdempotentElem e ∧ KaehlerDifferential.ideal R S = Submodule.span (S ⊗[R] S) {e}) ↔
    ∃ t, (∀ (s : S), (1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) * t = 0) ∧ (TensorProduct.lmul' R) t = 1
tactic: have : ∀ t : S ⊗[R] S, TensorProduct.lmul' R t = 1 ↔ 1 - t ∈ KaehlerDifferential.ideal R S := by
    intro t
    simp only [KaehlerDifferential.ideal, RingHom.mem_ker, map_sub, map_one,
      sub_eq_zero, @eq_comm S 1]
state_before: R S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
inst✝ : EssFiniteType R S
this : ∀ (t : S ⊗[R] S), (TensorProduct.lmul' R) t = 1 ↔ 1 - t ∈ KaehlerDifferential.ideal R S
⊢ (∃ e, IsIdempotentElem e ∧ KaehlerDifferential.ideal R S = Submodule.span (S ⊗[R] S) {e}) ↔
    ∃ t, (∀ (s : S), (1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) * t = 0) ∧ (TensorProduct.lmul' R) t = 1
state_after: R S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
inst✝ : EssFiniteType R S
this : ∀ (t : S ⊗[R] S), (TensorProduct.lmul' R) t = 1 ↔ 1 - t ∈ KaehlerDifferential.ideal R S
⊢ (∃ e, IsIdempotentElem e ∧ Ideal.span (Set.range fun s => 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) = Submodule.span (S ⊗[R] S) {e}) ↔
    ∃ t, (∀ (s : S), (1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) * t = 0) ∧ 1 - t ∈ Ideal.span (Set.range fun s => 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)
tactic: simp_rw [this, ← KaehlerDifferential.span_range_eq_ideal]
state_before: R S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
inst✝ : EssFiniteType R S
this : ∀ (t : S ⊗[R] S), (TensorProduct.lmul' R) t = 1 ↔ 1 - t ∈ KaehlerDifferential.ideal R S
⊢ (∃ e, IsIdempotentElem e ∧ Ideal.span (Set.range fun s => 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) = Submodule.span (S ⊗[R] S) {e}) ↔
    ∃ t, (∀ (s : S), (1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) * t = 0) ∧ 1 - t ∈ Ideal.span (Set.range fun s => 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)
state_after: case mp
R S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
inst✝ : EssFiniteType R S
this : ∀ (t : S ⊗[R] S), (TensorProduct.lmul' R) t = 1 ↔ 1 - t ∈ KaehlerDifferential.ideal R S
⊢ (∃ e, IsIdempotentElem e ∧ Ideal.span (Set.range fun s => 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) = Submodule.span (S ⊗[R] S) {e}) →
    ∃ t, (∀ (s : S), (1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) * t = 0) ∧ 1 - t ∈ Ideal.span (Set.range fun s => 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)

case mpr
R S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
inst✝ : EssFiniteType R S
this : ∀ (t : S ⊗[R] S), (TensorProduct.lmul' R) t = 1 ↔ 1 - t ∈ KaehlerDifferential.ideal R S
⊢ (∃ t, (∀ (s : S), (1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) * t = 0) ∧ 1 - t ∈ Ideal.span (Set.range fun s => 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)) →
    ∃ e, IsIdempotentElem e ∧ Ideal.span (Set.range fun s => 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) = Submodule.span (S ⊗[R] S) {e}
tactic: constructor