Theorems · Theorem · order theory
ArchimedeanClass.mk_le_mk_iff_denselyOrdered
∀ {R : Type u_1} {S : Type u_2} [inst : LinearOrder R] [inst_1 : LinearOrder S] [inst_2 : CommRing R]
[IsStrictOrderedRing R] [inst_4 : Ring S] [inst_5 : IsStrictOrderedRing S] [DenselyOrdered R] [Archimedean R]
{x y : S} (f : R →+* S),
StrictMono ⇑f → (ArchimedeanClass.mk x ≤ ArchimedeanClass.mk y ↔ ∃ q, 0 < f q ∧ f q * |y| ≤ |x|)- Defined in
- Mathlib.Algebra.Order.Ring.Archimedean
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 38 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites41
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- RingHomstatement and proof · cited by 10,189
- LinearOrderstatement and proof · cited by 8,572
- Ringstatement and proof · cited by 7,463
- one_mulproof · cited by 2,841
- Nat.cast_oneproof · cited by 2,501
- IsStrictOrderedRingstatement and proof · cited by 2,490
- mul_commproof · cited by 2,262
- LT.lt.leproof · cited by 2,189
- MulZeroClass.mul_zeroproof · cited by 2,091
- absstatement and proof · cited by 1,814
Cited by1
Results whose statement or proof uses this declaration.
- ArchimedeanClass.mk_le_mk_iff_ratCastproof · cited by 0