-- Mathlib/Algebra/Category/ModuleCat/Free.lean
theorem linearIndependent_leftExact : LinearIndependent R u

Proof state:
ι : Type u_1
ι' : Type u_2
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
hv : LinearIndependent R v
u : ι ⊕ ι' → ↑S.X₂
hw : LinearIndependent R (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr)
hm : Mono S.f
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
⊢ LinearIndependent R v

Tactic:
exact hv

Result:
no goals

Proof state:
ι : Type u_1
ι' : Type u_2
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
hv : LinearIndependent R v
u : ι ⊕ ι' → ↑S.X₂
hw : LinearIndependent R (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr)
hm : Mono S.f
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
⊢ LinearIndependent R u

Tactic:
rw [linearIndependent_sum]

Result:
ι : Type u_1
ι' : Type u_2
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
hv : LinearIndependent R v
u : ι ⊕ ι' → ↑S.X₂
hw : LinearIndependent R (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr)
hm : Mono S.f
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
⊢ LinearIndependent R (u ∘ Sum.inl) ∧
    LinearIndependent R (u ∘ Sum.inr) ∧ Disjoint (span R (range (u ∘ Sum.inl))) (span R (range (u ∘ Sum.inr)))

Proof state:
ι : Type u_1
ι' : Type u_2
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
hv : LinearIndependent R v
u : ι ⊕ ι' → ↑S.X₂
hw : LinearIndependent R (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr)
hm : Mono S.f
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
⊢ LinearIndependent R (u ∘ Sum.inl) ∧
    LinearIndependent R (u ∘ Sum.inr) ∧ Disjoint (span R (range (u ∘ Sum.inl))) (span R (range (u ∘ Sum.inr)))

Tactic:
refine ⟨?_, LinearIndependent.of_comp S.g.hom hw, disjoint_span_sum hS hw huv⟩

Result:
ι : Type u_1
ι' : Type u_2
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
hv : LinearIndependent R v
u : ι ⊕ ι' → ↑S.X₂
hw : LinearIndependent R (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr)
hm : Mono S.f
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
⊢ LinearIndependent R (u ∘ Sum.inl)

Proof state:
ι : Type u_1
ι' : Type u_2
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
hv : LinearIndependent R v
u : ι ⊕ ι' → ↑S.X₂
hw : LinearIndependent R (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr)
hm : Mono S.f
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
⊢ LinearIndependent R v

ι : Type u_1
ι' : Type u_2
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
hv : LinearIndependent R v
u : ι ⊕ ι' → ↑S.X₂
hw : LinearIndependent R (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr)
hm : Mono S.f
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
⊢ LinearMap.ker (Hom.hom S.f) = ⊥

Tactic:
swap

Result:
ι : Type u_1
ι' : Type u_2
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
hv : LinearIndependent R v
u : ι ⊕ ι' → ↑S.X₂
hw : LinearIndependent R (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr)
hm : Mono S.f
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
⊢ LinearMap.ker (Hom.hom S.f) = ⊥

ι : Type u_1
ι' : Type u_2
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
hv : LinearIndependent R v
u : ι ⊕ ι' → ↑S.X₂
hw : LinearIndependent R (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr)
hm : Mono S.f
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
⊢ LinearIndependent R v

Proof state:
ι : Type u_1
ι' : Type u_2
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
hv : LinearIndependent R v
u : ι ⊕ ι' → ↑S.X₂
hw : LinearIndependent R (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr)
hm : Mono S.f
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
⊢ LinearMap.ker (Hom.hom S.f) = ⊥

Tactic:
rw [LinearMap.ker_eq_bot, ← mono_iff_injective]

Result:
ι : Type u_1
ι' : Type u_2
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
hv : LinearIndependent R v
u : ι ⊕ ι' → ↑S.X₂
hw : LinearIndependent R (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr)
hm : Mono S.f
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
⊢ Mono S.f

Proof state:
ι : Type u_1
ι' : Type u_2
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
hv : LinearIndependent R v
u : ι ⊕ ι' → ↑S.X₂
hw : LinearIndependent R (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr)
hm : Mono S.f
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
⊢ Mono S.f

Tactic:
infer_instance

Result:
no goals

Proof state:
ι : Type u_1
ι' : Type u_2
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
hv : LinearIndependent R v
u : ι ⊕ ι' → ↑S.X₂
hw : LinearIndependent R (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr)
hm : Mono S.f
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
⊢ LinearIndependent R (u ∘ Sum.inl)

Tactic:
rw [huv, LinearMap.linearIndependent_iff S.f.hom]

Result:
ι : Type u_1
ι' : Type u_2
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
hv : LinearIndependent R v
u : ι ⊕ ι' → ↑S.X₂
hw : LinearIndependent R (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr)
hm : Mono S.f
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
⊢ LinearIndependent R v

ι : Type u_1
ι' : Type u_2
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
hv : LinearIndependent R v
u : ι ⊕ ι' → ↑S.X₂
hw : LinearIndependent R (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr)
hm : Mono S.f
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
⊢ LinearMap.ker (Hom.hom S.f) = ⊥