Theorems · Theorem · order theory
le_bot_iff
∀ {α : Type u} [inst : PartialOrder α] [inst_1 : OrderBot α] {a : α}, a ≤ ⊥ ↔ a = ⊥- Defined in
- Mathlib.Order.BoundedOrder.Basic
- Cited by
- 116 results in Mathlib
- Foundations
- Depth 8 from the axioms, rests on 16 definitions · uses no axioms
- Assumes
- PartialOrderOrderBot
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- PartialOrderstatement and proof · cited by 6,410
- Bot.botstatement · cited by 4,720
- OrderBotstatement and proof · cited by 1,055
- bot_leproof · cited by 306
- LE.le.ge_iff_eq'proof · cited by 50
Cited by116
Results whose statement or proof uses this declaration.
- eq_bot_iffproof · cited by 159
- disjoint_iffproof · cited by 76
- sdiff_selfproof · cited by 38
- Submodule.bot_smulproof · cited by 11
- sdiff_eq_bot_iffproof · cited by 10
- Multiset.disjoint_leftproof · cited by 9
- LinearMap.range_eq_botproof · cited by 7
- sSup_eq_botproof · cited by 7
- GaloisConnection.l_eq_botproof · cited by 6
- Group.isSolvable_of_ker_le_rangeproof · cited by 5
- Ideal.comap_bot_of_injectiveproof · cited by 5
- Subgroup.lowerCentralSeries_length_eq_nilpotencyClassproof · cited by 5