Theorems · Theorem · order theory
le_of_forall_lt_imp_le_of_dense
∀ {α : Type u_2} [inst : LinearOrder α] [DenselyOrdered α] {a₁ a₂ : α}, (∀ a < a₂, a ≤ a₁) → a₂ ≤ a₁- Defined in
- Mathlib.Order.Basic
- Cited by
- 14 results in Mathlib
- Foundations
- Depth 12 from the axioms · uses propext, Quot.sound
- 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
- le_of_not_gtproof · cited by 430
- lt_irreflproof · cited by 190
- lt_of_lt_of_le'proof · cited by 15
- exists_between'proof · cited by 8
Cited by14
Results whose statement or proof uses this declaration.
- ENNReal.le_of_forall_nnreal_ltproof · cited by 16
- EReal.mul_le_of_forall_lt_of_nonnegproof · cited by 4
- Real.exists_isLUBproof · cited by 3
- EReal.add_le_of_forall_ltproof · cited by 3
- ENNReal.mul_le_of_forall_ltproof · cited by 3
- mul_le_of_forall_lt_of_nonnegproof · cited by 3
- add_le_of_forall_ltproof · cited by 2
- bsupr_limsup_dimHproof · cited by 1
- isLUB_Iioproof · cited by 1
- forall_lt_imp_le_iff_le_of_denseproof · cited by 1
- NNReal.le_of_forall_lt_one_mul_leproof · cited by 0
- eq_of_le_of_forall_gt_imp_ge_of_denseproof · cited by 0