-- Mathlib/LinearAlgebra/LinearIndependent/Basic.lean
theorem surjective_of_linearIndependent_of_span [Nontrivial R] (hv : LinearIndependent R v)
    (f : ι' ↪ ι) (hss : range v ⊆ span R (range (v ∘ f))) : Surjective f

Proof state:
ι : Type u'
ι' : Type u_1
R : Type u_2
M : Type u_4
v : ι → M
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
hv : LinearIndependent R v
f : ι' ↪ ι
hss : range v ⊆ ↑(span R (range (v ∘ ⇑f)))
i : ι
repr : ↥(span R (range (v ∘ ⇑f))) → ι' →₀ R := ⇑⋯.repr
l : ι →₀ R := Finsupp.mapDomain (⇑f) (repr ⟨v i, ⋯⟩)
h_total_l : (Finsupp.linearCombination R v) l = v i
h_total_eq : (Finsupp.linearCombination R v) l = (Finsupp.linearCombination R v) (Finsupp.single i 1)
l_eq : Finsupp.embDomain f (repr ⟨v i, ⋯⟩) = Finsupp.single i 1
⊢ ∃ a, f a = i

Tactic:
rcases Finsupp.single_of_embDomain_single (repr ⟨v i, _⟩) f i (1 : R) zero_ne_one.symm l_eq with
  ⟨i', hi'⟩

Result:
case intro
ι : Type u'
ι' : Type u_1
R : Type u_2
M : Type u_4
v : ι → M
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
hv : LinearIndependent R v
f : ι' ↪ ι
hss : range v ⊆ ↑(span R (range (v ∘ ⇑f)))
i : ι
repr : ↥(span R (range (v ∘ ⇑f))) → ι' →₀ R := ⇑⋯.repr
l : ι →₀ R := Finsupp.mapDomain (⇑f) (repr ⟨v i, ⋯⟩)
h_total_l : (Finsupp.linearCombination R v) l = v i
h_total_eq : (Finsupp.linearCombination R v) l = (Finsupp.linearCombination R v) (Finsupp.single i 1)
l_eq : Finsupp.embDomain f (repr ⟨v i, ⋯⟩) = Finsupp.single i 1
i' : ι'
hi' : repr ⟨v i, ⋯⟩ = Finsupp.single i' 1 ∧ f i' = i
⊢ ∃ a, f a = i

Proof state:
ι : Type u'
ι' : Type u_1
R : Type u_2
M : Type u_4
v : ι → M
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
hv : LinearIndependent R v
f : ι' ↪ ι
hss : range v ⊆ ↑(span R (range (v ∘ ⇑f)))
i : ι
repr : ↥(span R (range (v ∘ ⇑f))) → ι' →₀ R := ⇑⋯.repr
l : ι →₀ R := Finsupp.mapDomain (⇑f) (repr ⟨v i, ⋯⟩)
⊢ (Finsupp.linearCombination R (v ∘ ⇑f)) (repr ⟨v i, ⋯⟩) = v i

Tactic:
rw [(hv.comp f f.injective).linearCombination_repr]

Result:
no goals

Proof state:
ι : Type u'
ι' : Type u_1
R : Type u_2
M : Type u_4
v : ι → M
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
hv : LinearIndependent R v
f : ι' ↪ ι
hss : range v ⊆ ↑(span R (range (v ∘ ⇑f)))
⊢ Surjective ⇑f

Tactic:
intro i

Result:
ι : Type u'
ι' : Type u_1
R : Type u_2
M : Type u_4
v : ι → M
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
hv : LinearIndependent R v
f : ι' ↪ ι
hss : range v ⊆ ↑(span R (range (v ∘ ⇑f)))
i : ι
⊢ ∃ a, f a = i

Proof state:
ι : Type u'
ι' : Type u_1
R : Type u_2
M : Type u_4
v : ι → M
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
hv : LinearIndependent R v
f : ι' ↪ ι
hss : range v ⊆ ↑(span R (range (v ∘ ⇑f)))
i : ι
repr : ↥(span R (range (v ∘ ⇑f))) → ι' →₀ R := ⇑⋯.repr
l : ι →₀ R := Finsupp.mapDomain (⇑f) (repr ⟨v i, ⋯⟩)
h_total_l : (Finsupp.linearCombination R v) l = v i
h_total_eq : (Finsupp.linearCombination R v) l = (Finsupp.linearCombination R v) (Finsupp.single i 1)
⊢ ∃ a, f a = i

Tactic:
have l_eq : l = _ := hv h_total_eq

Result:
ι : Type u'
ι' : Type u_1
R : Type u_2
M : Type u_4
v : ι → M
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
hv : LinearIndependent R v
f : ι' ↪ ι
hss : range v ⊆ ↑(span R (range (v ∘ ⇑f)))
i : ι
repr : ↥(span R (range (v ∘ ⇑f))) → ι' →₀ R := ⇑⋯.repr
l : ι →₀ R := Finsupp.mapDomain (⇑f) (repr ⟨v i, ⋯⟩)
h_total_l : (Finsupp.linearCombination R v) l = v i
h_total_eq : (Finsupp.linearCombination R v) l = (Finsupp.linearCombination R v) (Finsupp.single i 1)
l_eq : l = Finsupp.single i 1
⊢ ∃ a, f a = i

Proof state:
ι : Type u'
ι' : Type u_1
R : Type u_2
M : Type u_4
v : ι → M
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
hv : LinearIndependent R v
f : ι' ↪ ι
hss : range v ⊆ ↑(span R (range (v ∘ ⇑f)))
i : ι
⊢ ∃ a, f a = i

Tactic:
let repr : (span R (range (v ∘ f)) : Type _) → ι' →₀ R := (hv.comp f f.injective).repr

Result:
ι : Type u'
ι' : Type u_1
R : Type u_2
M : Type u_4
v : ι → M
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
hv : LinearIndependent R v
f : ι' ↪ ι
hss : range v ⊆ ↑(span R (range (v ∘ ⇑f)))
i : ι
repr : ↥(span R (range (v ∘ ⇑f))) → ι' →₀ R := ⇑⋯.repr
⊢ ∃ a, f a = i

Proof state:
ι : Type u'
ι' : Type u_1
R : Type u_2
M : Type u_4
v : ι → M
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
hv : LinearIndependent R v
f : ι' ↪ ι
hss : range v ⊆ ↑(span R (range (v ∘ ⇑f)))
i : ι
repr : ↥(span R (range (v ∘ ⇑f))) → ι' →₀ R := ⇑⋯.repr
l : ι →₀ R := Finsupp.mapDomain (⇑f) (repr ⟨v i, ⋯⟩)
h_total_l : (Finsupp.linearCombination R v) l = v i
⊢ (Finsupp.linearCombination R v) l = (Finsupp.linearCombination R v) (Finsupp.single i 1)

Tactic:
rw [h_total_l, Finsupp.linearCombination_single, one_smul]

Result:
no goals

Proof state:
case intro
ι : Type u'
ι' : Type u_1
R : Type u_2
M : Type u_4
v : ι → M
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
hv : LinearIndependent R v
f : ι' ↪ ι
hss : range v ⊆ ↑(span R (range (v ∘ ⇑f)))
i : ι
repr : ↥(span R (range (v ∘ ⇑f))) → ι' →₀ R := ⇑⋯.repr
l : ι →₀ R := Finsupp.mapDomain (⇑f) (repr ⟨v i, ⋯⟩)
h_total_l : (Finsupp.linearCombination R v) l = v i
h_total_eq : (Finsupp.linearCombination R v) l = (Finsupp.linearCombination R v) (Finsupp.single i 1)
l_eq : Finsupp.embDomain f (repr ⟨v i, ⋯⟩) = Finsupp.single i 1
i' : ι'
hi' : repr ⟨v i, ⋯⟩ = Finsupp.single i' 1 ∧ f i' = i
⊢ ∃ a, f a = i

Tactic:
use i'

Result:
case h
ι : Type u'
ι' : Type u_1
R : Type u_2
M : Type u_4
v : ι → M
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
hv : LinearIndependent R v
f : ι' ↪ ι
hss : range v ⊆ ↑(span R (range (v ∘ ⇑f)))
i : ι
repr : ↥(span R (range (v ∘ ⇑f))) → ι' →₀ R := ⇑⋯.repr
l : ι →₀ R := Finsupp.mapDomain (⇑f) (repr ⟨v i, ⋯⟩)
h_total_l : (Finsupp.linearCombination R v) l = v i
h_total_eq : (Finsupp.linearCombination R v) l = (Finsupp.linearCombination R v) (Finsupp.single i 1)
l_eq : Finsupp.embDomain f (repr ⟨v i, ⋯⟩) = Finsupp.single i 1
i' : ι'
hi' : repr ⟨v i, ⋯⟩ = Finsupp.single i' 1 ∧ f i' = i
⊢ f i' = i

Proof state:
case h
ι : Type u'
ι' : Type u_1
R : Type u_2
M : Type u_4
v : ι → M
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
hv : LinearIndependent R v
f : ι' ↪ ι
hss : range v ⊆ ↑(span R (range (v ∘ ⇑f)))
i : ι
repr : ↥(span R (range (v ∘ ⇑f))) → ι' →₀ R := ⇑⋯.repr
l : ι →₀ R := Finsupp.mapDomain (⇑f) (repr ⟨v i, ⋯⟩)
h_total_l : (Finsupp.linearCombination R v) l = v i
h_total_eq : (Finsupp.linearCombination R v) l = (Finsupp.linearCombination R v) (Finsupp.single i 1)
l_eq : Finsupp.embDomain f (repr ⟨v i, ⋯⟩) = Finsupp.single i 1
i' : ι'
hi' : repr ⟨v i, ⋯⟩ = Finsupp.single i' 1 ∧ f i' = i
⊢ f i' = i

Tactic:
exact hi'.2

Result:
no goals