Theorems · Theorem · order theory
eq_bot_iff
∀ {α : Type u} [inst : PartialOrder α] [inst_1 : OrderBot α] {a : α}, a = ⊥ ↔ a ≤ ⊥- Defined in
- Mathlib.Order.BoundedOrder.Basic
- Cited by
- 159 results in Mathlib
- Foundations
- Depth 9 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.
Cites4
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
- le_bot_iffproof · cited by 116
Cited by159
Results whose statement or proof uses this declaration.
- Ideal.dvd_iff_leproof · cited by 33
- Submodule.eq_bot_iffproof · cited by 31
- LinearMap.ker_eq_bot_of_injectiveproof · cited by 20
- rank_subsingleton'proof · cited by 13
- Ideal.map_quotient_selfproof · cited by 12
- Ideal.eq_bot_of_comap_eq_botproof · cited by 9
- Submodule.mkQ_map_selfproof · cited by 8
- Algebra.IsCentral.center_eq_botproof · cited by 8
- AddSubgroup.addCommutator_eq_bot_iff_le_centralizerproof · cited by 7
- Disjoint.eq_bot_of_leproof · cited by 7
- Submodule.inf_orthogonal_eq_botproof · cited by 6
- LieSubmodule.eq_bot_iffproof · cited by 6