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Theorems · Theorem · group theory

AddSubmonoid.isAddCommutative_closure

∀ (M : Type u_1) [inst : AddMonoid M] {s : Set M},
  (∀ a ∈ s, ∀ b ∈ s, a + b = b + a) → IsAddCommutative ↥(AddSubmonoid.closure s)

If all the elements of a set s commute, then closure s forms an additive commutative monoid.

Defined in
Mathlib.GroupTheory.Submonoid.Centralizer
Cited by
0 results in Mathlib
Foundations
Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
AddMonoid
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Set · cited by 53352SetSetLike.coe · cited by 8199SetLike.coeAddMonoid · cited by 2864AddMonoidAddSubmonoid · cited by 1178AddSubmonoidAddSubmonoid.closure · cited by 224AddSubmonoid.closureIsAddCommutative · cited by 40IsAddCommutativeAddSubmonoid.centralizer · cited by 11AddSubmonoid.centralizerIsAddCommutative.of_setLike_add_comm · cited by 7IsAddCommutative.of_setLi…Set.addCentralizer_addCentralizer_comm_of_comm · cited by 3Set.addCentralizer_addCen…AddSubmonoid.closure_le_centralizer_centralizer · cited by 1AddSubmonoid.closure_le_c…AddSubmonoid.isAddCommutative…CITED BYCITES

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