Theorems · Theorem · order theory
abs_eq_self
∀ {G : Type u_1} [inst : AddCommGroup G] [inst_1 : LinearOrder G] [IsOrderedAddMonoid G] {a : G}, |a| = a ↔ 0 ≤ a- Defined in
- Mathlib.Algebra.Order.Group.Abs
- Cited by
- 59 results in Mathlib
- Foundations
- Depth 19 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_left_iffproof · cited by 4
- neg_le_self_iffproof · cited by 4
- abs_eq_max_negproof · cited by 2
Cited by59
Results whose statement or proof uses this declaration.
- Real.abs_rpow_of_nonnegproof · cited by 11
- HasSum.nat_add_negproof · cited by 10
- ArchimedeanClass.lt_of_mk_lt_mk_of_nonnegproof · cited by 5
- Even.pow_absproof · cited by 5
- abs_casesproof · cited by 4
- integral_rpow_mul_exp_neg_rpowproof · cited by 4
- AddCircle.volume_closedBallproof · cited by 4
- round_leproof · cited by 3
- PeriodPair.summable_weierstrassPExceptSummandproof · cited by 3
- IsUnifLocDoublingMeasure.tendsto_closedBall_filterAtproof · cited by 3
- HahnSeries.orderTop_absproof · cited by 3
- ZLattice.exists_forall_abs_repr_le_normproof · cited by 2