Theorems · Theorem · group theory
prod_mem
∀ {B : Type u_3} {S : B} {M : Type u_4} [inst : CommMonoid M] [inst_1 : SetLike B M] [SubmonoidClass B M] {ι : Type u_5}
{t : Finset ι} {f : ι → M}, (∀ c ∈ t, f c ∈ S) → ∏ c ∈ t, f c ∈ SProduct of elements of a submonoid of a CommMonoid indexed by a Finset is in the
submonoid.
- Cited by
- 16 results in Mathlib
- Foundations
- Depth 61 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Finsetstatement and proof · cited by 13,712
- Finset.prodstatement · cited by 2,356
- CommMonoidstatement and proof · cited by 2,264
- SetLikestatement and proof · cited by 1,084
- Multiset.mapproof · cited by 876
- Finset.valproof · cited by 438
- Multiset.mem_mapproof · cited by 72
- SubmonoidClassstatement and proof · cited by 60
- multiset_prod_memproof · cited by 11
Cited by16
Results whose statement or proof uses this declaration.
- Polynomial.Splits.prodproof · cited by 5
- Subalgebra.prod_memproof · cited by 4
- Subgroup.prod_memproof · cited by 3
- Subsemiring.prod_memproof · cited by 1
- MvPolynomial.eval₂_memproof · cited by 1
- Localization.exists_awayMap_bijective_of_localRingHom_bijectiveproof · cited by 1
- IsLocalization.iff_map_piEvalRingHomproof · cited by 1
- IntermediateField.prod_memproof · cited by 0
- SubmonoidClass.finsuppProd_memproof · cited by 0
- tprod_memproof · cited by 0
- Subfield.prod_memproof · cited by 0
- Subring.prod_memproof · cited by 0