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

tactic: tfae_have 1 ↔ 2
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 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))
⊢ [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 ↔ 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))
⊢ [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 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
⊢ [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 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
⊢ [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 → 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
⊢ [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 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
⊢ [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_finish
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: exact iff_rTensor_injective' R M
state_before: 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))
state_after: no goals

tactic: exact forall_vanishesTrivially_iff_forall_rTensor_injective R
state_before: 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))
state_after: no goals

tactic: simp [(TensorProduct.lid R M).injective.eq_iff.symm, isTrivialRelation_iff_vanishesTrivially]
state_before: 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
state_after: no goals

tactic: intro h₄ ι hι f x hfx
state_before: 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
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
⊢ ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0

tactic: let f' : ι → R := f
state_before: 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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
⊢ ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
⊢ ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0

tactic: let x' : ι → M := fun i ↦ x (single i 1)
state_before: 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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
⊢ ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
⊢ ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0

tactic: have := calc
  ∑ i, f' i • x' i
  _ = ∑ i, f i • x (single i 1)         := rfl
  _ = x (∑ i, f i • Finsupp.single i 1) := by simp_rw [map_sum, map_smul]
  _ = x f                               := by
    simp_rw [smul_single, smul_eq_mul, mul_one, univ_sum_single]
  _ = 0                                 := hfx
state_before: 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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
⊢ ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
⊢ ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0

tactic: obtain ⟨κ, hκ, a', y', ⟨ha'y', ha'⟩⟩ := h₄ this
state_before: 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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
⊢ ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
state_after: case tfae_4_to_5.intro.intro.intro.intro.intro
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
⊢ ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0

tactic: use κ, hκ
state_before: case tfae_4_to_5.intro.intro.intro.intro.intro
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
⊢ ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
state_after: case h
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
⊢ ∃ a y, x = y ∘ₗ a ∧ a f = 0

tactic: use Finsupp.total ι (κ →₀ R) R (fun i ↦ equivFunOnFinite.symm (a' i))
state_before: case h
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
⊢ ∃ a y, x = y ∘ₗ a ∧ a f = 0
state_after: case h
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
⊢ ∃ y,
    (x = y ∘ₗ Finsupp.total ι (κ →₀ R) R fun i => equivFunOnFinite.symm (a' i)) ∧
      (Finsupp.total ι (κ →₀ R) R fun i => equivFunOnFinite.symm (a' i)) f = 0

tactic: use Finsupp.total κ M R y'
state_before: case h
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
⊢ ∃ y,
    (x = y ∘ₗ Finsupp.total ι (κ →₀ R) R fun i => equivFunOnFinite.symm (a' i)) ∧
      (Finsupp.total ι (κ →₀ R) R fun i => equivFunOnFinite.symm (a' i)) f = 0
state_after: case h
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
⊢ (x = Finsupp.total κ M R y' ∘ₗ Finsupp.total ι (κ →₀ R) R fun i => equivFunOnFinite.symm (a' i)) ∧
    (Finsupp.total ι (κ →₀ R) R fun i => equivFunOnFinite.symm (a' i)) f = 0

tactic: constructor
state_before: case h
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
⊢ (x = Finsupp.total κ M R y' ∘ₗ Finsupp.total ι (κ →₀ R) R fun i => equivFunOnFinite.symm (a' i)) ∧
    (Finsupp.total ι (κ →₀ R) R fun i => equivFunOnFinite.symm (a' i)) f = 0
state_after: case h.left
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
⊢ x = Finsupp.total κ M R y' ∘ₗ Finsupp.total ι (κ →₀ R) R fun i => equivFunOnFinite.symm (a' i)

case h.right
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
⊢ (Finsupp.total ι (κ →₀ R) R fun i => equivFunOnFinite.symm (a' i)) f = 0

tactic: simp_rw [map_sum, map_smul]
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
⊢ ∑ i : ι, f i • x (Finsupp.single i 1) = x (∑ i : ι, f i • Finsupp.single i 1)
state_after: no goals

tactic: simp_rw [smul_single, smul_eq_mul, mul_one, univ_sum_single]
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
⊢ x (∑ i : ι, f i • Finsupp.single i 1) = x f
state_after: no goals

tactic: apply Finsupp.basisSingleOne.ext
state_before: case h.left
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
⊢ x = Finsupp.total κ M R y' ∘ₗ Finsupp.total ι (κ →₀ R) R fun i => equivFunOnFinite.symm (a' i)
state_after: case h.left
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
⊢ ∀ (i : ι),
    x (Finsupp.basisSingleOne i) =
      (Finsupp.total κ M R y' ∘ₗ Finsupp.total ι (κ →₀ R) R fun i => equivFunOnFinite.symm (a' i))
        (Finsupp.basisSingleOne i)

tactic: intro i
state_before: case h.left
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
⊢ ∀ (i : ι),
    x (Finsupp.basisSingleOne i) =
      (Finsupp.total κ M R y' ∘ₗ Finsupp.total ι (κ →₀ R) R fun i => equivFunOnFinite.symm (a' i))
        (Finsupp.basisSingleOne i)
state_after: case h.left
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
i : ι
⊢ x (Finsupp.basisSingleOne i) =
    (Finsupp.total κ M R y' ∘ₗ Finsupp.total ι (κ →₀ R) R fun i => equivFunOnFinite.symm (a' i))
      (Finsupp.basisSingleOne i)

tactic: simpa [total_apply, sum_fintype, single_apply] using ha'y' i
state_before: case h.left
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
i : ι
⊢ x (Finsupp.basisSingleOne i) =
    (Finsupp.total κ M R y' ∘ₗ Finsupp.total ι (κ →₀ R) R fun i => equivFunOnFinite.symm (a' i))
      (Finsupp.basisSingleOne i)
state_after: no goals

tactic: ext j
state_before: case h.right
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
⊢ (Finsupp.total ι (κ →₀ R) R fun i => equivFunOnFinite.symm (a' i)) f = 0
state_after: case h.right.h
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
j : κ
⊢ ((Finsupp.total ι (κ →₀ R) R fun i => equivFunOnFinite.symm (a' i)) f) j = 0 j

tactic: simp only [total_apply, zero_smul, implies_true, sum_fintype, finset_sum_apply]
state_before: case h.right.h
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
j : κ
⊢ ((Finsupp.total ι (κ →₀ R) R fun i => equivFunOnFinite.symm (a' i)) f) j = 0 j
state_after: case h.right.h
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
j : κ
⊢ ∑ i : ι, (f i • equivFunOnFinite.symm (a' i)) j = 0 j

tactic: exact ha' j
state_before: case h.right.h
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
h₄ : ∀ {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
hι : Fintype ι
f : ι →₀ R
x : (ι →₀ R) →ₗ[R] M
hfx : x f = 0
f' : ι → R := ⇑f
x' : ι → M := fun i => x (Finsupp.single i 1)
this : ∑ i : ι, f' i • x' i = 0
κ : Type u
hκ : Fintype κ
a' : ι → κ → R
y' : κ → M
ha'y' : ∀ (i : ι), x' i = ∑ j : κ, a' i j • y' j
ha' : ∀ (j : κ), ∑ i : ι, f' i * a' i j = 0
j : κ
⊢ ∑ i : ι, (f i • equivFunOnFinite.symm (a' i)) j = 0 j
state_after: no goals

tactic: intro h₅ ι hi f x hfx
state_before: 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
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
h₅ : ∀ {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
hi : Fintype ι
f : ι → R
x : ι → M
hfx : ∑ i : ι, f i • x i = 0
⊢ IsTrivialRelation f x

tactic: let f' : ι →₀ R := equivFunOnFinite.symm f
state_before: 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
h₅ : ∀ {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
hi : Fintype ι
f : ι → R
x : ι → M
hfx : ∑ i : ι, f i • x i = 0
⊢ IsTrivialRelation f x
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
h₅ : ∀ {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
hi : Fintype ι
f : ι → R
x : ι → M
hfx : ∑ i : ι, f i • x i = 0
f' : ι →₀ R := equivFunOnFinite.symm f
⊢ IsTrivialRelation f x

tactic: let x' : (ι →₀ R) →ₗ[R] M := Finsupp.total ι M R x
state_before: 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
h₅ : ∀ {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
hi : Fintype ι
f : ι → R
x : ι → M
hfx : ∑ i : ι, f i • x i = 0
f' : ι →₀ R := equivFunOnFinite.symm f
⊢ IsTrivialRelation f x
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
h₅ : ∀ {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
hi : Fintype ι
f : ι → R
x : ι → M
hfx : ∑ i : ι, f i • x i = 0
f' : ι →₀ R := equivFunOnFinite.symm f
x' : (ι →₀ R) →ₗ[R] M := Finsupp.total ι M R x
⊢ IsTrivialRelation f x

tactic: have : x' f' = 0 := by simpa [x', f', total_apply, sum_fintype] using hfx
state_before: 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
h₅ : ∀ {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
hi : Fintype ι
f : ι → R
x : ι → M
hfx : ∑ i : ι, f i • x i = 0
f' : ι →₀ R := equivFunOnFinite.symm f
x' : (ι →₀ R) →ₗ[R] M := Finsupp.total ι M R x
⊢ IsTrivialRelation f x
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
h₅ : ∀ {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
hi : Fintype ι
f : ι → R
x : ι → M
hfx : ∑ i : ι, f i • x i = 0
f' : ι →₀ R := equivFunOnFinite.symm f
x' : (ι →₀ R) →ₗ[R] M := Finsupp.total ι M R x
this : x' f' = 0
⊢ IsTrivialRelation f x

tactic: obtain ⟨κ, hκ, a', y', ha'y', ha'⟩ := h₅ this
state_before: 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
h₅ : ∀ {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
hi : Fintype ι
f : ι → R
x : ι → M
hfx : ∑ i : ι, f i • x i = 0
f' : ι →₀ R := equivFunOnFinite.symm f
x' : (ι →₀ R) →ₗ[R] M := Finsupp.total ι M R x
this : x' f' = 0
⊢ IsTrivialRelation f x
state_after: case tfae_5_to_4.intro.intro.intro.intro.intro
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
h₅ : ∀ {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
hi : Fintype ι
f : ι → R
x : ι → M
hfx : ∑ i : ι, f i • x i = 0
f' : ι →₀ R := equivFunOnFinite.symm f
x' : (ι →₀ R) →ₗ[R] M := Finsupp.total ι M R x
this : x' f' = 0
κ : Type u
hκ : Fintype κ
a' : (ι →₀ R) →ₗ[R] κ →₀ R
y' : (κ →₀ R) →ₗ[R] M
ha'y' : x' = y' ∘ₗ a'
ha' : a' f' = 0
⊢ IsTrivialRelation f x

tactic: refine ⟨κ, hκ, fun i ↦ a' (single i 1), fun j ↦ y' (single j 1), fun i ↦ ?_, fun j ↦ ?_⟩
state_before: case tfae_5_to_4.intro.intro.intro.intro.intro
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
h₅ : ∀ {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
hi : Fintype ι
f : ι → R
x : ι → M
hfx : ∑ i : ι, f i • x i = 0
f' : ι →₀ R := equivFunOnFinite.symm f
x' : (ι →₀ R) →ₗ[R] M := Finsupp.total ι M R x
this : x' f' = 0
κ : Type u
hκ : Fintype κ
a' : (ι →₀ R) →ₗ[R] κ →₀ R
y' : (κ →₀ R) →ₗ[R] M
ha'y' : x' = y' ∘ₗ a'
ha' : a' f' = 0
⊢ IsTrivialRelation f x
state_after: case tfae_5_to_4.intro.intro.intro.intro.intro.refine_1
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
h₅ : ∀ {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
hi : Fintype ι
f : ι → R
x : ι → M
hfx : ∑ i : ι, f i • x i = 0
f' : ι →₀ R := equivFunOnFinite.symm f
x' : (ι →₀ R) →ₗ[R] M := Finsupp.total ι M R x
this : x' f' = 0
κ : Type u
hκ : Fintype κ
a' : (ι →₀ R) →ₗ[R] κ →₀ R
y' : (κ →₀ R) →ₗ[R] M
ha'y' : x' = y' ∘ₗ a'
ha' : a' f' = 0
i : ι
⊢ x i = ∑ j : κ, (fun i => ⇑(a' (Finsupp.single i 1))) i j • (fun j => y' (Finsupp.single j 1)) j

case tfae_5_to_4.intro.intro.intro.intro.intro.refine_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))
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
h₅ : ∀ {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
hi : Fintype ι
f : ι → R
x : ι → M
hfx : ∑ i : ι, f i • x i = 0
f' : ι →₀ R := equivFunOnFinite.symm f
x' : (ι →₀ R) →ₗ[R] M := Finsupp.total ι M R x
this : x' f' = 0
κ : Type u
hκ : Fintype κ
a' : (ι →₀ R) →ₗ[R] κ →₀ R
y' : (κ →₀ R) →ₗ[R] M
ha'y' : x' = y' ∘ₗ a'
ha' : a' f' = 0
j : κ
⊢ ∑ i : ι, f i * (fun i => ⇑(a' (Finsupp.single i 1))) i j = 0

tactic: simpa [x', f', total_apply, sum_fintype] using hfx
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
h₅ : ∀ {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
hi : Fintype ι
f : ι → R
x : ι → M
hfx : ∑ i : ι, f i • x i = 0
f' : ι →₀ R := equivFunOnFinite.symm f
x' : (ι →₀ R) →ₗ[R] M := Finsupp.total ι M R x
⊢ x' f' = 0
state_after: no goals

tactic: simpa [x', ← map_smul, ← map_sum, smul_single] using
  LinearMap.congr_fun ha'y' (Finsupp.single i 1)
state_before: case tfae_5_to_4.intro.intro.intro.intro.intro.refine_1
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
h₅ : ∀ {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
hi : Fintype ι
f : ι → R
x : ι → M
hfx : ∑ i : ι, f i • x i = 0
f' : ι →₀ R := equivFunOnFinite.symm f
x' : (ι →₀ R) →ₗ[R] M := Finsupp.total ι M R x
this : x' f' = 0
κ : Type u
hκ : Fintype κ
a' : (ι →₀ R) →ₗ[R] κ →₀ R
y' : (κ →₀ R) →ₗ[R] M
ha'y' : x' = y' ∘ₗ a'
ha' : a' f' = 0
i : ι
⊢ x i = ∑ j : κ, (fun i => ⇑(a' (Finsupp.single i 1))) i j • (fun j => y' (Finsupp.single j 1)) j
state_after: no goals

tactic: simp_rw [← smul_eq_mul, ← Finsupp.smul_apply, ← map_smul, ← finset_sum_apply, ← map_sum,
  smul_single, smul_eq_mul, mul_one,
  ← (fun _ ↦ equivFunOnFinite_symm_apply_toFun _ _ : ∀ x, f' x = f x), univ_sum_single]
state_before: case tfae_5_to_4.intro.intro.intro.intro.intro.refine_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))
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
h₅ : ∀ {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
hi : Fintype ι
f : ι → R
x : ι → M
hfx : ∑ i : ι, f i • x i = 0
f' : ι →₀ R := equivFunOnFinite.symm f
x' : (ι →₀ R) →ₗ[R] M := Finsupp.total ι M R x
this : x' f' = 0
κ : Type u
hκ : Fintype κ
a' : (ι →₀ R) →ₗ[R] κ →₀ R
y' : (κ →₀ R) →ₗ[R] M
ha'y' : x' = y' ∘ₗ a'
ha' : a' f' = 0
j : κ
⊢ ∑ i : ι, f i * (fun i => ⇑(a' (Finsupp.single i 1))) i j = 0
state_after: case tfae_5_to_4.intro.intro.intro.intro.intro.refine_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))
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
h₅ : ∀ {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
hi : Fintype ι
f : ι → R
x : ι → M
hfx : ∑ i : ι, f i • x i = 0
f' : ι →₀ R := equivFunOnFinite.symm f
x' : (ι →₀ R) →ₗ[R] M := Finsupp.total ι M R x
this : x' f' = 0
κ : Type u
hκ : Fintype κ
a' : (ι →₀ R) →ₗ[R] κ →₀ R
y' : (κ →₀ R) →ₗ[R] M
ha'y' : x' = y' ∘ₗ a'
ha' : a' f' = 0
j : κ
⊢ (a' (equivFunOnFinite.symm f)) j = 0

tactic: simpa using DFunLike.congr_fun ha' j
state_before: case tfae_5_to_4.intro.intro.intro.intro.intro.refine_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))
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
h₅ : ∀ {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
hi : Fintype ι
f : ι → R
x : ι → M
hfx : ∑ i : ι, f i • x i = 0
f' : ι →₀ R := equivFunOnFinite.symm f
x' : (ι →₀ R) →ₗ[R] M := Finsupp.total ι M R x
this : x' f' = 0
κ : Type u
hκ : Fintype κ
a' : (ι →₀ R) →ₗ[R] κ →₀ R
y' : (κ →₀ R) →ₗ[R] M
ha'y' : x' = y' ∘ₗ a'
ha' : a' f' = 0
j : κ
⊢ (a' (equivFunOnFinite.symm f)) j = 0
state_after: no goals

tactic: intro h₅ N _ _ _ _ f x hfx
state_before: 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
state_after: case tfae_5_to_6
R M : Type u
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : 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
h₅ : ∀ {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✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Finite R N
f : N
x : N →ₗ[R] M
hfx : x f = 0
⊢ ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0

tactic: have ϕ := Module.Free.repr R N
state_before: case tfae_5_to_6
R M : Type u
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : 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
h₅ : ∀ {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✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Finite R N
f : N
x : N →ₗ[R] M
hfx : x f = 0
⊢ ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
state_after: case tfae_5_to_6
R M : Type u
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : 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
h₅ : ∀ {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✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Finite R N
f : N
x : N →ₗ[R] M
hfx : x f = 0
ϕ : N ≃ₗ[R] Free.ChooseBasisIndex R N →₀ R
⊢ ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0

tactic: have : (x ∘ₗ ϕ.symm) (ϕ f) = 0 := by simpa
state_before: case tfae_5_to_6
R M : Type u
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : 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
h₅ : ∀ {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✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Finite R N
f : N
x : N →ₗ[R] M
hfx : x f = 0
ϕ : N ≃ₗ[R] Free.ChooseBasisIndex R N →₀ R
⊢ ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
state_after: case tfae_5_to_6
R M : Type u
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : 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
h₅ : ∀ {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✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Finite R N
f : N
x : N →ₗ[R] M
hfx : x f = 0
ϕ : N ≃ₗ[R] Free.ChooseBasisIndex R N →₀ R
this : (x ∘ₗ ↑ϕ.symm) (ϕ f) = 0
⊢ ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0

tactic: obtain ⟨κ, hκ, a', y, ha'y, ha'⟩ := h₅ this
state_before: case tfae_5_to_6
R M : Type u
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : 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
h₅ : ∀ {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✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Finite R N
f : N
x : N →ₗ[R] M
hfx : x f = 0
ϕ : N ≃ₗ[R] Free.ChooseBasisIndex R N →₀ R
this : (x ∘ₗ ↑ϕ.symm) (ϕ f) = 0
⊢ ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
state_after: case tfae_5_to_6.intro.intro.intro.intro.intro
R M : Type u
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : 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
h₅ : ∀ {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✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Finite R N
f : N
x : N →ₗ[R] M
hfx : x f = 0
ϕ : N ≃ₗ[R] Free.ChooseBasisIndex R N →₀ R
this : (x ∘ₗ ↑ϕ.symm) (ϕ f) = 0
κ : Type u
hκ : Fintype κ
a' : (Free.ChooseBasisIndex R N →₀ R) →ₗ[R] κ →₀ R
y : (κ →₀ R) →ₗ[R] M
ha'y : x ∘ₗ ↑ϕ.symm = y ∘ₗ a'
ha' : a' (ϕ f) = 0
⊢ ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0

tactic: refine ⟨κ, hκ, a' ∘ₗ ϕ, y, ?_, ?_⟩
state_before: case tfae_5_to_6.intro.intro.intro.intro.intro
R M : Type u
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : 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
h₅ : ∀ {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✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Finite R N
f : N
x : N →ₗ[R] M
hfx : x f = 0
ϕ : N ≃ₗ[R] Free.ChooseBasisIndex R N →₀ R
this : (x ∘ₗ ↑ϕ.symm) (ϕ f) = 0
κ : Type u
hκ : Fintype κ
a' : (Free.ChooseBasisIndex R N →₀ R) →ₗ[R] κ →₀ R
y : (κ →₀ R) →ₗ[R] M
ha'y : x ∘ₗ ↑ϕ.symm = y ∘ₗ a'
ha' : a' (ϕ f) = 0
⊢ ∃ κ x_1 a y, x = y ∘ₗ a ∧ a f = 0
state_after: case tfae_5_to_6.intro.intro.intro.intro.intro.refine_1
R M : Type u
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : 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
h₅ : ∀ {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✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Finite R N
f : N
x : N →ₗ[R] M
hfx : x f = 0
ϕ : N ≃ₗ[R] Free.ChooseBasisIndex R N →₀ R
this : (x ∘ₗ ↑ϕ.symm) (ϕ f) = 0
κ : Type u
hκ : Fintype κ
a' : (Free.ChooseBasisIndex R N →₀ R) →ₗ[R] κ →₀ R
y : (κ →₀ R) →ₗ[R] M
ha'y : x ∘ₗ ↑ϕ.symm = y ∘ₗ a'
ha' : a' (ϕ f) = 0
⊢ x = y ∘ₗ a' ∘ₗ ↑ϕ

case tfae_5_to_6.intro.intro.intro.intro.intro.refine_2
R M : Type u
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : 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
h₅ : ∀ {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✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Finite R N
f : N
x : N →ₗ[R] M
hfx : x f = 0
ϕ : N ≃ₗ[R] Free.ChooseBasisIndex R N →₀ R
this : (x ∘ₗ ↑ϕ.symm) (ϕ f) = 0
κ : Type u
hκ : Fintype κ
a' : (Free.ChooseBasisIndex R N →₀ R) →ₗ[R] κ →₀ R
y : (κ →₀ R) →ₗ[R] M
ha'y : x ∘ₗ ↑ϕ.symm = y ∘ₗ a'
ha' : a' (ϕ f) = 0
⊢ (a' ∘ₗ ↑ϕ) f = 0

tactic: simpa
state_before: R M : Type u
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : 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
h₅ : ∀ {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✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Finite R N
f : N
x : N →ₗ[R] M
hfx : x f = 0
ϕ : N ≃ₗ[R] Free.ChooseBasisIndex R N →₀ R
⊢ (x ∘ₗ ↑ϕ.symm) (ϕ f) = 0
state_after: no goals

tactic: simpa [LinearMap.comp_assoc] using congrArg (fun g ↦ (g ∘ₗ ϕ : N →ₗ[R] M)) ha'y
state_before: case tfae_5_to_6.intro.intro.intro.intro.intro.refine_1
R M : Type u
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : 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
h₅ : ∀ {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✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Finite R N
f : N
x : N →ₗ[R] M
hfx : x f = 0
ϕ : N ≃ₗ[R] Free.ChooseBasisIndex R N →₀ R
this : (x ∘ₗ ↑ϕ.symm) (ϕ f) = 0
κ : Type u
hκ : Fintype κ
a' : (Free.ChooseBasisIndex R N →₀ R) →ₗ[R] κ →₀ R
y : (κ →₀ R) →ₗ[R] M
ha'y : x ∘ₗ ↑ϕ.symm = y ∘ₗ a'
ha' : a' (ϕ f) = 0
⊢ x = y ∘ₗ a' ∘ₗ ↑ϕ
state_after: no goals

tactic: simpa using ha'
state_before: case tfae_5_to_6.intro.intro.intro.intro.intro.refine_2
R M : Type u
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : 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
h₅ : ∀ {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✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Finite R N
f : N
x : N →ₗ[R] M
hfx : x f = 0
ϕ : N ≃ₗ[R] Free.ChooseBasisIndex R N →₀ R
this : (x ∘ₗ ↑ϕ.symm) (ϕ f) = 0
κ : Type u
hκ : Fintype κ
a' : (Free.ChooseBasisIndex R N →₀ R) →ₗ[R] κ →₀ R
y : (κ →₀ R) →ₗ[R] M
ha'y : x ∘ₗ ↑ϕ.symm = y ∘ₗ a'
ha' : a' (ϕ f) = 0
⊢ (a' ∘ₗ ↑ϕ) f = 0
state_after: no goals

tactic: intro h₆ _ _ _ _ hfx
state_before: 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
state_after: case tfae_6_to_5
R M : Type u
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : 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
h₆ :
  ∀ {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
hfx : x✝ f✝ = 0
⊢ ∃ κ x a y, x✝ = y ∘ₗ a ∧ a f✝ = 0

tactic: exact h₆ hfx
state_before: case tfae_6_to_5
R M : Type u
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : 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
h₆ :
  ∀ {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
hfx : x✝ f✝ = 0
⊢ ∃ κ x a y, x✝ = y ∘ₗ a ∧ a f✝ = 0
state_after: no goals