Theorem: IsNilpotent.finsum (DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean)

State before:
α : Type u_1
inst✝ : CommSemiring α
β : Type u_2
f : β → α
hf : ∀ (b : β), IsNilpotent (f b)
⊢ IsNilpotent (_root_.finsum f)
Tactic:
by_cases h : Set.Finite f.support
State after:
case pos
α : Type u_1
inst✝ : CommSemiring α
β : Type u_2
f : β → α
hf : ∀ (b : β), IsNilpotent (f b)
h : (Function.support f).Finite
⊢ IsNilpotent (_root_.finsum f)

case neg
α : Type u_1
inst✝ : CommSemiring α
β : Type u_2
f : β → α
hf : ∀ (b : β), IsNilpotent (f b)
h : ¬(Function.support f).Finite
⊢ IsNilpotent (_root_.finsum f)

State before:
case pos
α : Type u_1
inst✝ : CommSemiring α
β : Type u_2
f : β → α
hf : ∀ (b : β), IsNilpotent (f b)
h : (Function.support f).Finite
⊢ IsNilpotent (_root_.finsum f)
Tactic:
rw [finsum_def, dif_pos h]
State after:
case pos
α : Type u_1
inst✝ : CommSemiring α
β : Type u_2
f : β → α
hf : ∀ (b : β), IsNilpotent (f b)
h : (Function.support f).Finite
⊢ IsNilpotent (∑ i ∈ h.toFinset, f i)

State before:
case pos
α : Type u_1
inst✝ : CommSemiring α
β : Type u_2
f : β → α
hf : ∀ (b : β), IsNilpotent (f b)
h : (Function.support f).Finite
⊢ IsNilpotent (∑ i ∈ h.toFinset, f i)
Tactic:
exact IsNilpotent.finset_sum _ (fun b _ ↦ hf b)
State after:
no goals

State before:
case neg
α : Type u_1
inst✝ : CommSemiring α
β : Type u_2
f : β → α
hf : ∀ (b : β), IsNilpotent (f b)
h : ¬(Function.support f).Finite
⊢ IsNilpotent (_root_.finsum f)
Tactic:
simp only [finsum_def, dif_neg h, IsNilpotent.zero]
State after:
no goals