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Theorems · Theorem · group theory

AddCommute.eq

∀ {S : Type u_3} [inst : Add S] {a b : S}, AddCommute a b → a + b = b + a

Equality behind AddCommute a b; useful for rewriting.

Defined in
Mathlib.Algebra.Group.Commute.Defs
Cited by
18 results in Mathlib
Foundations
Depth 6 from the axioms · uses no axioms
Assumes
Add
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Around this declaration

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

AddCommute.left_comm · cited by 3AddCommute.left_commFinsupp.induction₂ · cited by 3Finsupp.induction₂AddCommute.add_neg_cancel · cited by 1AddCommute.add_neg_cancelIsAddCentral.right_comm · cited by 1IsAddCentral.right_commAddCommute.neg_add_cancel · cited by 1AddCommute.neg_add_cancelAddEquivClass.apply_mem_center · cited by 1AddEquivClass.apply_mem_c…Finsupp.induction_on_max₂ · cited by 1Finsupp.induction_on_max₂AddCommute.of_map · cited by 1AddCommute.of_mapAddMonoidAlgebra.single_commute_single · cited by 1AddMonoidAlgebra.single_c…AddCommute.right_comm · cited by 1AddCommute.right_commAddCommute.sub_add_sub_comm · cited by 1AddCommute.sub_add_sub_co…AddCommute.zsmul_add · cited by 1AddCommute.zsmul_addOrdinal.addCommute_iff_eq_mul_natCast · cited by 0Ordinal.addCommute_iff_eq…AddCommute.neg · cited by 0AddCommute.negAddCommute.addOrderOf_add_eq_left_of_forall_prime_mul_dvd · cited by 0AddCommute.addOrderOf_add…AddCommute · cited by 185AddCommuteAddCommute.eqCITED BYCITES

Cites1

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by18

Results whose statement or proof uses this declaration.