Theorems · Theorem · order theory
abs_eq_neg_self
∀ {G : Type u_1} [inst : AddCommGroup G] [inst_1 : LinearOrder G] [IsOrderedAddMonoid G] {a : G}, |a| = -a ↔ a ≤ 0- Defined in
- Mathlib.Algebra.Order.Group.Abs
- Cited by
- 14 results in Mathlib
- Foundations
- Depth 20 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- AddCommGroupstatement and proof · cited by 12,871
- LinearOrderstatement and proof · cited by 8,572
- absstatement · cited by 1,814
- IsOrderedAddMonoidstatement and proof · cited by 1,659
- max_eq_right_iffproof · cited by 5
- abs_eq_max_negproof · cited by 2
- le_neg_self_iffproof · cited by 1
Cited by14
Results whose statement or proof uses this declaration.
- ArchimedeanClass.lt_of_mk_lt_mk_of_nonposproof · cited by 6
- abs_casesproof · cited by 4
- round_leproof · cited by 3
- HahnSeries.orderTop_absproof · cited by 3
- integral_comp_absproof · cited by 2
- max_zero_add_max_neg_zero_eq_abs_selfproof · cited by 1
- AddCircle.coe_real_preimage_closedBall_inter_eqproof · cited by 1
- Nat.exists_mem_span_nat_finset_of_geproof · cited by 1
- MeasureTheory.integral_norm_eq_pos_sub_negproof · cited by 1
- HahnSeries.support_absproof · cited by 0
- Rat.surjective_padicValuationproof · cited by 0
- AddCircle.norm_coe_eq_abs_iffproof · cited by 0