-- Mathlib/RingTheory/HahnSeries/Summable.lean
theorem smul_hsum {R} {V} [Semiring R] [AddCommMonoid V] [Module R V]
    (s : SummableFamily Γ R α) (t : SummableFamily Γ' V β) :
    (smul s t).hsum = (of R).symm (s.hsum • (of R) (t.hsum))

Proof state:
case coeff.h
Γ : Type u_1
Γ' : Type u_2
α : Type u_5
β : Type u_6
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
R : Type u_7
V : Type u_8
inst✝² : Semiring R
inst✝¹ : AddCommMonoid V
inst✝ : Module R V
s : SummableFamily Γ R α
t : SummableFamily Γ' V β
g : Γ'
⊢ (s.smul t).hsum.coeff g = ((of R).symm (s.hsum • (of R) t.hsum)).coeff g

Tactic:
rw [coeff_smul s t g, HahnModule.coeff_smul, Equiv.symm_apply_apply]

Result:
case coeff.h
Γ : Type u_1
Γ' : Type u_2
α : Type u_5
β : Type u_6
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
R : Type u_7
V : Type u_8
inst✝² : Semiring R
inst✝¹ : AddCommMonoid V
inst✝ : Module R V
s : SummableFamily Γ R α
t : SummableFamily Γ' V β
g : Γ'
⊢ ∑ gh ∈ VAddAntidiagonal ⋯ ⋯ g, s.hsum.coeff gh.1 • t.hsum.coeff gh.2 =
    ∑ ij ∈ VAddAntidiagonal ⋯ ⋯ g, s.hsum.coeff ij.1 • t.hsum.coeff ij.2

Proof state:
case neg
Γ : Type u_1
Γ' : Type u_2
α : Type u_5
β : Type u_6
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
R : Type u_7
V : Type u_8
inst✝² : Semiring R
inst✝¹ : AddCommMonoid V
inst✝ : Module R V
s : SummableFamily Γ R α
t : SummableFamily Γ' V β
g : Γ'
gh : Γ × Γ'
x✝ : gh ∈ VAddAntidiagonal ⋯ ⋯ g
hgh : ¬s.hsum.coeff gh.1 = 0 → ¬t.hsum.coeff gh.2 = 0 → ¬gh.1 +ᵥ gh.2 = g
h : ¬s.hsum.coeff gh.1 = 0
⊢ s.hsum.coeff gh.1 • t.hsum.coeff gh.2 = 0

Tactic:
simp_all

Result:
no goals

Proof state:
case coeff.h.refine_1
Γ : Type u_1
Γ' : Type u_2
α : Type u_5
β : Type u_6
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
R : Type u_7
V : Type u_8
inst✝² : Semiring R
inst✝¹ : AddCommMonoid V
inst✝ : Module R V
s : SummableFamily Γ R α
t : SummableFamily Γ' V β
g : Γ'
x✝ : Γ × Γ'
hgh : ¬s.hsum.coeff x✝.1 = 0 ∧ ¬t.hsum.coeff x✝.2 = 0 ∧ x✝.1 +ᵥ x✝.2 = g
⊢ (∃ i, ¬(s i).coeff x✝.1 = 0) ∧ ∃ i, ¬(t i).coeff x✝.2 = 0

Tactic:
constructor

Result:
case coeff.h.refine_1.left
Γ : Type u_1
Γ' : Type u_2
α : Type u_5
β : Type u_6
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
R : Type u_7
V : Type u_8
inst✝² : Semiring R
inst✝¹ : AddCommMonoid V
inst✝ : Module R V
s : SummableFamily Γ R α
t : SummableFamily Γ' V β
g : Γ'
x✝ : Γ × Γ'
hgh : ¬s.hsum.coeff x✝.1 = 0 ∧ ¬t.hsum.coeff x✝.2 = 0 ∧ x✝.1 +ᵥ x✝.2 = g
⊢ ∃ i, ¬(s i).coeff x✝.1 = 0

case coeff.h.refine_1.right
Γ : Type u_1
Γ' : Type u_2
α : Type u_5
β : Type u_6
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
R : Type u_7
V : Type u_8
inst✝² : Semiring R
inst✝¹ : AddCommMonoid V
inst✝ : Module R V
s : SummableFamily Γ R α
t : SummableFamily Γ' V β
g : Γ'
x✝ : Γ × Γ'
hgh : ¬s.hsum.coeff x✝.1 = 0 ∧ ¬t.hsum.coeff x✝.2 = 0 ∧ x✝.1 +ᵥ x✝.2 = g
⊢ ∃ i, ¬(t i).coeff x✝.2 = 0

Proof state:
case coeff.h.refine_1.left
Γ : Type u_1
Γ' : Type u_2
α : Type u_5
β : Type u_6
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
R : Type u_7
V : Type u_8
inst✝² : Semiring R
inst✝¹ : AddCommMonoid V
inst✝ : Module R V
s : SummableFamily Γ R α
t : SummableFamily Γ' V β
g : Γ'
x✝ : Γ × Γ'
hgh : ¬∑ i ∈ (s.coeff x✝.1).support, (s i).coeff x✝.1 = 0 ∧ ¬t.hsum.coeff x✝.2 = 0 ∧ x✝.1 +ᵥ x✝.2 = g
⊢ ∃ i, ¬(s i).coeff x✝.1 = 0

Tactic:
have h' := Finset.exists_ne_zero_of_sum_ne_zero hgh.1

Result:
case coeff.h.refine_1.left
Γ : Type u_1
Γ' : Type u_2
α : Type u_5
β : Type u_6
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
R : Type u_7
V : Type u_8
inst✝² : Semiring R
inst✝¹ : AddCommMonoid V
inst✝ : Module R V
s : SummableFamily Γ R α
t : SummableFamily Γ' V β
g : Γ'
x✝ : Γ × Γ'
hgh : ¬∑ i ∈ (s.coeff x✝.1).support, (s i).coeff x✝.1 = 0 ∧ ¬t.hsum.coeff x✝.2 = 0 ∧ x✝.1 +ᵥ x✝.2 = g
h' : ∃ a ∈ (s.coeff x✝.1).support, (s a).coeff x✝.1 ≠ 0
⊢ ∃ i, ¬(s i).coeff x✝.1 = 0

Proof state:
case coeff.h.refine_1.left
Γ : Type u_1
Γ' : Type u_2
α : Type u_5
β : Type u_6
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
R : Type u_7
V : Type u_8
inst✝² : Semiring R
inst✝¹ : AddCommMonoid V
inst✝ : Module R V
s : SummableFamily Γ R α
t : SummableFamily Γ' V β
g : Γ'
x✝ : Γ × Γ'
hgh : ¬s.hsum.coeff x✝.1 = 0 ∧ ¬t.hsum.coeff x✝.2 = 0 ∧ x✝.1 +ᵥ x✝.2 = g
⊢ ∃ i, ¬(s i).coeff x✝.1 = 0

Tactic:
rw [coeff_hsum_eq_sum] at hgh

Result:
case coeff.h.refine_1.left
Γ : Type u_1
Γ' : Type u_2
α : Type u_5
β : Type u_6
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
R : Type u_7
V : Type u_8
inst✝² : Semiring R
inst✝¹ : AddCommMonoid V
inst✝ : Module R V
s : SummableFamily Γ R α
t : SummableFamily Γ' V β
g : Γ'
x✝ : Γ × Γ'
hgh : ¬∑ i ∈ (s.coeff x✝.1).support, (s i).coeff x✝.1 = 0 ∧ ¬t.hsum.coeff x✝.2 = 0 ∧ x✝.1 +ᵥ x✝.2 = g
⊢ ∃ i, ¬(s i).coeff x✝.1 = 0

Proof state:
case coeff.h
Γ : Type u_1
Γ' : Type u_2
α : Type u_5
β : Type u_6
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
R : Type u_7
V : Type u_8
inst✝² : Semiring R
inst✝¹ : AddCommMonoid V
inst✝ : Module R V
s : SummableFamily Γ R α
t : SummableFamily Γ' V β
g : Γ'
⊢ ∑ gh ∈ VAddAntidiagonal ⋯ ⋯ g, s.hsum.coeff gh.1 • t.hsum.coeff gh.2 =
    ∑ ij ∈ VAddAntidiagonal ⋯ ⋯ g, s.hsum.coeff ij.1 • t.hsum.coeff ij.2

Tactic:
refine Eq.symm (sum_of_injOn (fun a ↦ a) (fun _ _ _ _ h ↦ h) (fun _ hgh => ?_)
  (fun gh _ hgh => ?_) fun _ _ => by simp)

Result:
case coeff.h.refine_1
Γ : Type u_1
Γ' : Type u_2
α : Type u_5
β : Type u_6
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
R : Type u_7
V : Type u_8
inst✝² : Semiring R
inst✝¹ : AddCommMonoid V
inst✝ : Module R V
s : SummableFamily Γ R α
t : SummableFamily Γ' V β
g : Γ'
x✝ : Γ × Γ'
hgh : x✝ ∈ ↑(VAddAntidiagonal ⋯ ⋯ g)
⊢ (fun a => a) x✝ ∈ ↑(VAddAntidiagonal ⋯ ⋯ g)

case coeff.h.refine_2
Γ : Type u_1
Γ' : Type u_2
α : Type u_5
β : Type u_6
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
R : Type u_7
V : Type u_8
inst✝² : Semiring R
inst✝¹ : AddCommMonoid V
inst✝ : Module R V
s : SummableFamily Γ R α
t : SummableFamily Γ' V β
g : Γ'
gh : Γ × Γ'
x✝ : gh ∈ VAddAntidiagonal ⋯ ⋯ g
hgh : gh ∉ (fun a => a) '' ↑(VAddAntidiagonal ⋯ ⋯ g)
⊢ s.hsum.coeff gh.1 • t.hsum.coeff gh.2 = 0

Proof state:
case coeff.h.refine_1.right
Γ : Type u_1
Γ' : Type u_2
α : Type u_5
β : Type u_6
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
R : Type u_7
V : Type u_8
inst✝² : Semiring R
inst✝¹ : AddCommMonoid V
inst✝ : Module R V
s : SummableFamily Γ R α
t : SummableFamily Γ' V β
g : Γ'
x✝ : Γ × Γ'
hgh : ¬s.hsum.coeff x✝.1 = 0 ∧ ¬t.hsum.coeff x✝.2 = 0 ∧ x✝.1 +ᵥ x✝.2 = g
⊢ ∃ i, ¬(t i).coeff x✝.2 = 0

Tactic:
by_contra hi

Result:
case coeff.h.refine_1.right
Γ : Type u_1
Γ' : Type u_2
α : Type u_5
β : Type u_6
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
R : Type u_7
V : Type u_8
inst✝² : Semiring R
inst✝¹ : AddCommMonoid V
inst✝ : Module R V
s : SummableFamily Γ R α
t : SummableFamily Γ' V β
g : Γ'
x✝ : Γ × Γ'
hgh : ¬s.hsum.coeff x✝.1 = 0 ∧ ¬t.hsum.coeff x✝.2 = 0 ∧ x✝.1 +ᵥ x✝.2 = g
hi : ¬∃ i, ¬(t i).coeff x✝.2 = 0
⊢ False