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Theorems · Theorem · commutative algebra

IsIdempotentElem.add

∀ {R : Type u_1} [inst : NonUnitalNonAssocSemiring R] {a b : R},
  IsIdempotentElem a → IsIdempotentElem b → a * b + b * a = 0 → IsIdempotentElem (a + b)

a + b is idempotent when a and b anti-commute.

Defined in
Mathlib.Algebra.Ring.Idempotent
Cited by
2 results in Mathlib
Foundations
Depth 8 from the axioms · uses propext
Assumes
NonUnitalNonAssocSemiring
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Around this declaration

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

IsIdempotentElem.add_iff · cited by 1IsIdempotentElem.add_iffIsStarProjection.add · cited by 0IsStarProjection.addzero_add · cited by 2366zero_addNonUnitalNonAssocSemiring · cited by 1081NonUnitalNonAssocSemiringadd_assoc · cited by 746add_assocmul_add · cited by 413mul_addadd_mul · cited by 363add_mulIsIdempotentElem · cited by 217IsIdempotentElemadd_add_add_comm · cited by 56add_add_add_commIsIdempotentElem.eq · cited by 42IsIdempotentElem.eqIsIdempotentElem.addCITED BYCITES

Cites8

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

Cited by2

Results whose statement or proof uses this declaration.