Theorems · Theorem · order theory
abs_cases
∀ {G : Type u_1} [inst : AddCommGroup G] [inst_1 : LinearOrder G] [IsOrderedAddMonoid G] (a : G),
|a| = a ∧ 0 ≤ a ∨ |a| = -a ∧ a < 0For an element a of an additive linear ordered group,
either |a| = a and 0 ≤ a, or |a| = -a and a < 0.
Use cases on this lemma to automate linarith in inequalities
- Defined in
- Mathlib.Algebra.Order.Group.Abs
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 21 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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
- le_of_ltproof · cited by 1,175
- le_or_gtproof · cited by 269
- abs_eq_selfproof · cited by 59
- abs_eq_neg_selfproof · cited by 14
Cited by4
Results whose statement or proof uses this declaration.
- ModularGroup.exists_smul_mem_fdproof · cited by 2
- ModularGroup.three_le_four_mul_im_sq_of_mem_fdproof · cited by 2
- zmultiples_absproof · cited by 1
- ModularGroup.three_lt_four_mul_im_sq_of_mem_fdoproof · cited by 0