Theorems · Theorem · ring theory
mul_neg_one
∀ {α : Type u} [inst : MulOneClass α] [inst_1 : HasDistribNeg α] (a : α), a * -1 = -aAn element of a ring multiplied by the additive inverse of one is the element's additive inverse.
- Defined in
- Mathlib.Algebra.Ring.Defs
- Cited by
- 30 results in Mathlib
- Foundations
- Depth 6 from the axioms · uses propext
- Assumes
- MulOneClassHasDistribNeg
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- mul_oneproof · cited by 3,885
- MulOneClassstatement and proof · cited by 1,018
- mul_negproof · cited by 590
- HasDistribNegstatement and proof · cited by 114
Cited by30
Results whose statement or proof uses this declaration.
- Complex.sin_sq_add_cos_sqproof · cited by 11
- Complex.cos_addproof · cited by 11
- Complex.sinh_mul_Iproof · cited by 10
- Complex.hasStrictDerivAt_sinproof · cited by 5
- Complex.hasStrictDerivAt_coshproof · cited by 4
- Complex.hasStrictDerivAt_sinhproof · cited by 4
- neg_pow'proof · cited by 3
- CliffordAlgebra.map_mul_map_of_isOrtho_of_mem_evenOddproof · cited by 3
- Int.normalize_of_nonposproof · cited by 2
- one_add_mul_self_lt_rpow_one_addproof · cited by 2
- NumberField.exists_ne_zero_mem_ideal_of_norm_le_mul_sqrt_discrproof · cited by 2