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[Merged by Bors] - fix: make an ideal lemma more constructive #6080

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[Merged by Bors] - fix: make an ideal lemma more constructive #6080

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6 changes: 4 additions & 2 deletions Mathlib/RingTheory/Ideal/Basic.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -36,7 +36,7 @@ variable {α : Type u} {β : Type v}

open Set Function

open Classical BigOperators Pointwise
open BigOperators Pointwise

/-- A (left) ideal in a semiring `R` is an additive submonoid `s` such that
`a * b ∈ s` whenever `b ∈ s`. If `R` is a ring, then `s` is an additive subgroup. -/
Expand Down Expand Up @@ -582,7 +582,8 @@ theorem IsPrime.pow_mem_iff_mem {I : Ideal α} (hI : I.IsPrime) {r : α} (n :
⟨hI.mem_of_pow_mem n, fun hr => I.pow_mem_of_mem hr n hn⟩
#align ideal.is_prime.pow_mem_iff_mem Ideal.IsPrime.pow_mem_iff_mem

theorem pow_multiset_sum_mem_span_pow (s : Multiset α) (n : ℕ) : s.sum ^ (Multiset.card s * n + 1) ∈
theorem pow_multiset_sum_mem_span_pow [DecidableEq α] (s : Multiset α) (n : ℕ) :
s.sum ^ (Multiset.card s * n + 1) ∈
span ((s.map fun (x:α) ↦ x ^ (n + 1)).toFinset : Set α) := by
induction' s using Multiset.induction_on with a s hs
· simp
Expand Down Expand Up @@ -610,6 +611,7 @@ theorem pow_multiset_sum_mem_span_pow (s : Multiset α) (n : ℕ) : s.sum ^ (Mul

theorem sum_pow_mem_span_pow {ι} (s : Finset ι) (f : ι → α) (n : ℕ) :
(∑ i in s, f i) ^ (s.card * n + 1) ∈ span ((fun i => f i ^ (n + 1)) '' s) := by
classical
simpa only [Multiset.card_map, Multiset.map_map, comp_apply, Multiset.toFinset_map,
Finset.coe_image, Finset.val_toFinset] using pow_multiset_sum_mem_span_pow (s.1.map f) n
#align ideal.sum_pow_mem_span_pow Ideal.sum_pow_mem_span_pow
Expand Down
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