Theorem: Module.Flat.tfae_equational_criterion
File: Mathlib/RingTheory/Flat/EquationalCriterion.lean

Proof steps:
state_before: R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
⊢ [Flat R M, ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I)),
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M},
        x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0,
      ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
        {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0].TFAE
state_after: case tfae_1_iff_2
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
⊢ Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))

R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
⊢ [Flat R M, ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I)),
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M},
        x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0,
      ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
        {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0].TFAE
tactic: tfae_have 1 ↔ 2
state_before: R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
⊢ [Flat R M, ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I)),
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M},
        x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0,
      ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
        {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0].TFAE
state_after: case tfae_3_iff_2
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
⊢ (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))

R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_2 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
⊢ [Flat R M, ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I)),
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M},
        x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0,
      ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
        {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0].TFAE
tactic: tfae_have 3 ↔ 2
state_before: R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_2 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
⊢ [Flat R M, ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I)),
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M},
        x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0,
      ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
        {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0].TFAE
state_after: case tfae_3_iff_4
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_2 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
⊢ (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x

R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_2 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
⊢ [Flat R M, ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I)),
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M},
        x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0,
      ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
        {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0].TFAE
tactic: tfae_have 3 ↔ 4
state_before: R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_2 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
⊢ [Flat R M, ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I)),
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M},
        x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0,
      ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
        {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0].TFAE
state_after: case tfae_4_to_5
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_2 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
⊢ (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0

R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_2 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
tfae_4_to_5 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
⊢ [Flat R M, ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I)),
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M},
        x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0,
      ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
        {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0].TFAE
tactic: tfae_have 4 → 5
state_before: R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_2 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
tfae_4_to_5 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
⊢ [Flat R M, ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I)),
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M},
        x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0,
      ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
        {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0].TFAE
state_after: case tfae_5_to_4
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_2 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
tfae_4_to_5 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
⊢ (∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x

R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_2 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
tfae_4_to_5 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
tfae_5_to_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
⊢ [Flat R M, ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I)),
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M},
        x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0,
      ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
        {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0].TFAE
tactic: tfae_have 5 → 4
state_before: R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_2 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
tfae_4_to_5 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
tfae_5_to_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
⊢ [Flat R M, ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I)),
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M},
        x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0,
      ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
        {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0].TFAE
state_after: case tfae_5_to_6
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_2 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
tfae_4_to_5 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
tfae_5_to_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
⊢ (∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0) →
    ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
      {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0

R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_2 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
tfae_4_to_5 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
tfae_5_to_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
tfae_5_to_6 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0) →
    ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
      {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
⊢ [Flat R M, ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I)),
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M},
        x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0,
      ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
        {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0].TFAE
tactic: tfae_have 5 → 6
state_before: R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_2 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
tfae_4_to_5 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
tfae_5_to_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
tfae_5_to_6 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0) →
    ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
      {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
⊢ [Flat R M, ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I)),
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M},
        x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0,
      ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
        {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0].TFAE
state_after: case tfae_6_to_5
R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_2 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
tfae_4_to_5 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
tfae_5_to_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
tfae_5_to_6 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0) →
    ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
      {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
⊢ (∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
      {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0

R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_2 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
tfae_4_to_5 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
tfae_5_to_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
tfae_5_to_6 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0) →
    ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
      {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
tfae_6_to_5 :
  (∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
      {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
⊢ [Flat R M, ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I)),
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M},
        x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0,
      ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
        {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0].TFAE
tactic: tfae_have 6 → 5
state_before: R M : Type u
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
tfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_2 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))
tfae_3_iff_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
tfae_4_to_5 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
tfae_5_to_4 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x
tfae_5_to_6 :
  (∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0) →
    ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
      {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
tfae_6_to_5 :
  (∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
      {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0) →
    ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
⊢ [Flat R M, ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I)),
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι → R} {x : ι → M}, ∑ i : ι, f i • x i = 0 → IsTrivialRelation f x,
      ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M},
        x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0,
      ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Free R N] [inst_3 : Finite R N] {f : N}
        {x : N →ₗ[R] M}, x f = 0 → ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0].TFAE
state_after: no goals
tactic: tfae_finish