Theorem: AdicCompletion.ofTensorProduct_surjective_of_finite
File: Mathlib/RingTheory/AdicCompletion/AsTensorProduct.lean

Proof steps:
state_before: R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : Module.Finite R M
⊢ Function.Surjective ⇑(ofTensorProduct I M)
state_after: case intro.intro
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : Module.Finite R M
n : ℕ
p : (Fin n → R) →ₗ[R] M
hp : Function.Surjective ⇑p
⊢ Function.Surjective ⇑(ofTensorProduct I M)
tactic: obtain ⟨n, p, hp⟩ := Module.Finite.exists_fin' R M
state_before: case intro.intro
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : Module.Finite R M
n : ℕ
p : (Fin n → R) →ₗ[R] M
hp : Function.Surjective ⇑p
⊢ Function.Surjective ⇑(ofTensorProduct I M)
state_after: case intro.intro
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : Module.Finite R M
n : ℕ
p : (Fin n → R) →ₗ[R] M
hp : Function.Surjective ⇑p
f : AdicCompletion I R ⊗[R] (Fin n → R) →ₗ[AdicCompletion I R] AdicCompletion I M :=
  ofTensorProduct I M ∘ₗ LinearMap.baseChange (AdicCompletion I R) p
⊢ Function.Surjective ⇑(ofTensorProduct I M)
tactic: let f := ofTensorProduct I M ∘ₗ p.baseChange (AdicCompletion I R)
state_before: case intro.intro
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : Module.Finite R M
n : ℕ
p : (Fin n → R) →ₗ[R] M
hp : Function.Surjective ⇑p
f : AdicCompletion I R ⊗[R] (Fin n → R) →ₗ[AdicCompletion I R] AdicCompletion I M :=
  ofTensorProduct I M ∘ₗ LinearMap.baseChange (AdicCompletion I R) p
⊢ Function.Surjective ⇑(ofTensorProduct I M)
state_after: case intro.intro
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : Module.Finite R M
n : ℕ
p : (Fin n → R) →ₗ[R] M
hp : Function.Surjective ⇑p
f : AdicCompletion I R ⊗[R] (Fin n → R) →ₗ[AdicCompletion I R] AdicCompletion I M :=
  ofTensorProduct I M ∘ₗ LinearMap.baseChange (AdicCompletion I R) p
g : AdicCompletion I R ⊗[R] (Fin n → R) →ₗ[AdicCompletion I R] AdicCompletion I M :=
  map I p ∘ₗ ofTensorProduct I (Fin n → R)
⊢ Function.Surjective ⇑(ofTensorProduct I M)
tactic: let g := map I p ∘ₗ ofTensorProduct I (Fin n → R)
state_before: case intro.intro
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : Module.Finite R M
n : ℕ
p : (Fin n → R) →ₗ[R] M
hp : Function.Surjective ⇑p
f : AdicCompletion I R ⊗[R] (Fin n → R) →ₗ[AdicCompletion I R] AdicCompletion I M :=
  ofTensorProduct I M ∘ₗ LinearMap.baseChange (AdicCompletion I R) p
g : AdicCompletion I R ⊗[R] (Fin n → R) →ₗ[AdicCompletion I R] AdicCompletion I M :=
  map I p ∘ₗ ofTensorProduct I (Fin n → R)
⊢ Function.Surjective ⇑(ofTensorProduct I M)
state_after: case intro.intro
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : Module.Finite R M
n : ℕ
p : (Fin n → R) →ₗ[R] M
hp : Function.Surjective ⇑p
f : AdicCompletion I R ⊗[R] (Fin n → R) →ₗ[AdicCompletion I R] AdicCompletion I M :=
  ofTensorProduct I M ∘ₗ LinearMap.baseChange (AdicCompletion I R) p
g : AdicCompletion I R ⊗[R] (Fin n → R) →ₗ[AdicCompletion I R] AdicCompletion I M :=
  map I p ∘ₗ ofTensorProduct I (Fin n → R)
hfg : f = g
⊢ Function.Surjective ⇑(ofTensorProduct I M)
tactic: have hfg : f = g := by
    ext
    simp [f, g]
state_before: case intro.intro
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : Module.Finite R M
n : ℕ
p : (Fin n → R) →ₗ[R] M
hp : Function.Surjective ⇑p
f : AdicCompletion I R ⊗[R] (Fin n → R) →ₗ[AdicCompletion I R] AdicCompletion I M :=
  ofTensorProduct I M ∘ₗ LinearMap.baseChange (AdicCompletion I R) p
g : AdicCompletion I R ⊗[R] (Fin n → R) →ₗ[AdicCompletion I R] AdicCompletion I M :=
  map I p ∘ₗ ofTensorProduct I (Fin n → R)
hfg : f = g
hf : Function.Surjective ⇑f
⊢ Function.Surjective ⇑(ofTensorProduct I M)
state_after: no goals
tactic: exact Function.Surjective.of_comp hf