-- Mathlib/RingTheory/LocalRing/Module.lean
/-- If `M` is of finite presentation over a local ring `(R, 𝔪, k)` such that
`𝔪 ⊗ M → M` is injective, then every family of elements that is a `k`-basis of
`k ⊗ M` is an `R`-basis of `M`. -/
lemma exists_basis_of_basis_baseChange [Module.FinitePresentation R M]
    {ι : Type u} (v : ι → M) (hli : LinearIndependent k (TensorProduct.mk R k M 1 ∘ v))
    (hsp : Submodule.span k (Set.range (TensorProduct.mk R k M 1 ∘ v)) = ⊤)
    (H : Function.Injective ((𝔪).subtype.rTensor M)) :
    ∃ (b : Basis ι R M), ∀ i, b i = v i

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsLocalRing R
inst✝ : FinitePresentation R M
ι : Type u
v : ι → M
hli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)
hsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1) ∘ v)) = ⊤
H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪))
bk : Basis ι k (k ⊗[R] M) := Basis.mk hli ⋯
this✝² : Finite ι
this✝¹ : Fintype ι := Fintype.ofFinite ι
this✝ : IsNoetherian k (k ⊗[R] (ι →₀ R)) := isNoetherian_of_isNoetherianRing_of_finite k (k ⊗[R] (ι →₀ R))
i : (ι →₀ R) →ₗ[R] M := Finsupp.linearCombination R v
hi : Surjective ⇑i
this : Module.Finite R ↥(LinearMap.ker i)
⊢ Function.Injective ⇑i

Tactic:
rw [← LinearMap.ker_eq_bot, ← Submodule.subsingleton_iff_eq_bot,
  ← IsLocalRing.subsingleton_tensorProduct (R := R), subsingleton_iff_forall_eq 0]

Result:
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsLocalRing R
inst✝ : FinitePresentation R M
ι : Type u
v : ι → M
hli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)
hsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1) ∘ v)) = ⊤
H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪))
bk : Basis ι k (k ⊗[R] M) := Basis.mk hli ⋯
this✝² : Finite ι
this✝¹ : Fintype ι := Fintype.ofFinite ι
this✝ : IsNoetherian k (k ⊗[R] (ι →₀ R)) := isNoetherian_of_isNoetherianRing_of_finite k (k ⊗[R] (ι →₀ R))
i : (ι →₀ R) →ₗ[R] M := Finsupp.linearCombination R v
hi : Surjective ⇑i
this : Module.Finite R ↥(LinearMap.ker i)
⊢ ∀ (y : k ⊗[R] ↥(LinearMap.ker i)), y = 0

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsLocalRing R
inst✝ : FinitePresentation R M
ι : Type u
v : ι → M
hli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)
hsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1) ∘ v)) = ⊤
H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪))
bk : Basis ι k (k ⊗[R] M) := Basis.mk hli ⋯
this✝¹ : Finite ι
this✝ : Fintype ι := Fintype.ofFinite ι
this : IsNoetherian k (k ⊗[R] (ι →₀ R)) := isNoetherian_of_isNoetherianRing_of_finite k (k ⊗[R] (ι →₀ R))
i : (ι →₀ R) →ₗ[R] M := Finsupp.linearCombination R v
⊢ Submodule.span R (Set.range v) = ⊤

Tactic:
refine IsLocalRing.span_eq_top_of_tmul_eq_basis (R := R) (f := v) bk
  (fun _ ↦ by simp [bk])

Result:
no goals

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsLocalRing R
inst✝ : FinitePresentation R M
ι : Type u
v : ι → M
hli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)
hsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1) ∘ v)) = ⊤
H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪))
bk : Basis ι k (k ⊗[R] M) := Basis.mk hli ⋯
this✝³ : Finite ι
this✝² : Fintype ι := Fintype.ofFinite ι
this✝¹ : IsNoetherian k (k ⊗[R] (ι →₀ R)) := isNoetherian_of_isNoetherianRing_of_finite k (k ⊗[R] (ι →₀ R))
i : (ι →₀ R) →ₗ[R] M := Finsupp.linearCombination R v
hi : Surjective ⇑i
this✝ : Module.Finite R ↥(LinearMap.ker i)
this : Surjective ⇑(LinearMap.baseChange k i)
⊢ Fintype.card ι = finrank k (k ⊗[R] M)

Tactic:
rw [Module.finrank_eq_card_basis bk]

Result:
no goals

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsLocalRing R
inst✝ : FinitePresentation R M
ι : Type u
v : ι → M
hli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)
hsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1) ∘ v)) = ⊤
H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪))
bk : Basis ι k (k ⊗[R] M) := Basis.mk hli ⋯
this✝¹ : Finite ι
this✝ : Fintype ι := Fintype.ofFinite ι
this : IsNoetherian k (k ⊗[R] (ι →₀ R)) := isNoetherian_of_isNoetherianRing_of_finite k (k ⊗[R] (ι →₀ R))
i : (ι →₀ R) →ₗ[R] M := Finsupp.linearCombination R v
⊢ ∃ b, ∀ (i : ι), b i = v i

Tactic:
have hi : Surjective i := by
  rw [← LinearMap.range_eq_top, Finsupp.range_linearCombination]
  refine IsLocalRing.span_eq_top_of_tmul_eq_basis (R := R) (f := v) bk
    (fun _ ↦ by simp [bk])

Result:
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsLocalRing R
inst✝ : FinitePresentation R M
ι : Type u
v : ι → M
hli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)
hsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1) ∘ v)) = ⊤
H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪))
bk : Basis ι k (k ⊗[R] M) := Basis.mk hli ⋯
this✝¹ : Finite ι
this✝ : Fintype ι := Fintype.ofFinite ι
this : IsNoetherian k (k ⊗[R] (ι →₀ R)) := isNoetherian_of_isNoetherianRing_of_finite k (k ⊗[R] (ι →₀ R))
i : (ι →₀ R) →ₗ[R] M := Finsupp.linearCombination R v
hi : Surjective ⇑i
⊢ ∃ b, ∀ (i : ι), b i = v i

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsLocalRing R
inst✝ : FinitePresentation R M
ι : Type u
v : ι → M
hli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)
hsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1) ∘ v)) = ⊤
H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪))
bk : Basis ι k (k ⊗[R] M) := Basis.mk hli ⋯
this✝ : Finite ι
this : Fintype ι := Fintype.ofFinite ι
⊢ ∃ b, ∀ (i : ι), b i = v i

Tactic:
letI : IsNoetherian k (k ⊗[R] (ι →₀ R)) :=
  isNoetherian_of_isNoetherianRing_of_finite k (k ⊗[R] (ι →₀ R))

Result:
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsLocalRing R
inst✝ : FinitePresentation R M
ι : Type u
v : ι → M
hli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)
hsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1) ∘ v)) = ⊤
H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪))
bk : Basis ι k (k ⊗[R] M) := Basis.mk hli ⋯
this✝¹ : Finite ι
this✝ : Fintype ι := Fintype.ofFinite ι
this : IsNoetherian k (k ⊗[R] (ι →₀ R)) := isNoetherian_of_isNoetherianRing_of_finite k (k ⊗[R] (ι →₀ R))
⊢ ∃ b, ∀ (i : ι), b i = v i

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsLocalRing R
inst✝ : FinitePresentation R M
ι : Type u
v : ι → M
hli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)
hsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1) ∘ v)) = ⊤
H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪))
bk : Basis ι k (k ⊗[R] M) := Basis.mk hli ⋯
⊢ ∃ b, ∀ (i : ι), b i = v i

Tactic:
haveI : Finite ι := Module.Finite.finite_basis bk

Result:
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsLocalRing R
inst✝ : FinitePresentation R M
ι : Type u
v : ι → M
hli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)
hsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1) ∘ v)) = ⊤
H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪))
bk : Basis ι k (k ⊗[R] M) := Basis.mk hli ⋯
this : Finite ι
⊢ ∃ b, ∀ (i : ι), b i = v i