Theorems · Theorem · order theory
csSup_le
∀ {α : Type u_1} [inst : ConditionallyCompleteLattice α] {s : Set α} {a : α}, s.Nonempty → (∀ b ∈ s, b ≤ a) → sSup s ≤ a- Cited by
- 35 results in Mathlib
- Foundations
- Depth 10 from the axioms · uses no axioms
- Assumes
- ConditionallyCompleteLattice
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.
- Setstatement and proof · cited by 53,352
- Set.Nonemptystatement and proof · cited by 2,627
- SupSet.sSupstatement · cited by 954
- ConditionallyCompleteLatticestatement and proof · cited by 364
- isLUB_csSupproof · cited by 34
Cited by35
Results whose statement or proof uses this declaration.
- ciSup_leproof · cited by 56
- exists_lt_of_lt_csSupproof · cited by 13
- Monotone.leftLim_leproof · cited by 13
- csSup_eq_of_forall_le_of_forall_lt_exists_gtproof · cited by 8
- Filter.liminf_le_of_leproof · cited by 7
- csSup_le_csSupproof · cited by 6
- cauchySeq_iff_le_tendsto_0proof · cited by 5
- IsCompact.sSup_lt_iff_of_continuousproof · cited by 3
- MeasureTheory.upcrossingsBefore_leproof · cited by 3
- Monotone.csSup_image_leproof · cited by 2
- le_csSup_iffproof · cited by 2
- ConvexOn.leftDeriv_le_rightDeriv_of_mem_interiorproof · cited by 2