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Theorems · Theorem · group theory

Submonoid.list_prod_mem

∀ {M : Type u_1} [inst : Monoid M] (s : Submonoid M) {l : List M}, (∀ x ∈ l, x ∈ s) → l.prod ∈ s

Product of a list of elements in a submonoid is in the submonoid.

Defined in
Mathlib.Algebra.Group.Submonoid.BigOperators
Cited by
3 results in Mathlib
Foundations
Depth 60 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
Monoid
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Equiv.Perm.mclosure_isSwap · cited by 2Perm.mclosure_isSwapSubmonoid.multiset_noncommProd_mem · cited by 2Submonoid.multiset_noncom…jacobiSym.trichotomy · cited by 1jacobiSym.trichotomyMonoid · cited by 3887MonoidSubmonoid · cited by 3086Submonoidlist_prod_mem · cited by 9list_prod_memSubmonoid.list_prod_memCITED BYCITES

Cites3

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Cited by3

Results whose statement or proof uses this declaration.