Theorems · Theorem · order theory
le_of_forall_gt_imp_ge_of_dense
∀ {α : Type u_2} [inst : LinearOrder α] [DenselyOrdered α] {a₁ a₂ : α}, (∀ (a : α), a₂ < a → a₁ ≤ a) → a₁ ≤ a₂- Defined in
- Mathlib.Order.Basic
- Cited by
- 21 results in Mathlib
- Foundations
- Depth 12 from the axioms · uses no axioms
- Assumes
- LinearOrderDenselyOrdered
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- LinearOrderstatement and proof · cited by 8,572
- DenselyOrderedstatement and proof · cited by 471
- lt_of_lt_of_leproof · cited by 438
- le_of_not_gtproof · cited by 430
- lt_irreflproof · cited by 190
- exists_betweenproof · cited by 102
Cited by21
Results whose statement or proof uses this declaration.
- ENNReal.inv_topproof · cited by 36
- le_of_forall_pos_le_addproof · cited by 15
- EReal.le_mul_of_forall_ltproof · cited by 4
- eq_of_le_of_forall_lt_imp_le_of_denseproof · cited by 4
- le_of_forall_one_lt_le_mulproof · cited by 3
- forall_gt_imp_ge_iff_le_of_denseproof · cited by 3
- PhragmenLindelof.horizontal_stripproof · cited by 3
- isGLB_Ioiproof · cited by 2
- le_mul_of_forall_lt₀proof · cited by 2
- le_add_of_forall_ltproof · cited by 2
- MeasureTheory.Measure.mkMetric_le_liminf_tsumproof · cited by 2
- LSeries.abscissaOfAbsConv_le_of_forall_lt_LSeriesSummableproof · cited by 2