Theorems · Theorem · order theory
le_isGLB_iff
∀ {α : Type u_1} [inst : Preorder α] {s : Set α} {a b : α}, IsGLB s a → (b ≤ a ↔ b ∈ lowerBounds s)- Defined in
- Mathlib.Order.Bounds.Basic
- Cited by
- 17 results in Mathlib
- Foundations
- Depth 8 from the axioms · uses propext, Quot.sound
- Assumes
- Preorder
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
- Preorderstatement and proof · cited by 7,952
- IsGLBstatement and proof · cited by 213
- lowerBoundsstatement · cited by 212
- IsGLB.lowerBounds_eqproof · cited by 11
Cited by17
Results whose statement or proof uses this declaration.
- le_iInf_iffproof · cited by 30
- le_sInf_iffproof · cited by 13
- isGLB_lt_iffproof · cited by 6
- le_ciInf_iffproof · cited by 5
- le_csInf_iffproof · cited by 3
- isGLB_iff_le_iffproof · cited by 3
- Finset.le_min'_iffproof · cited by 2
- Sion.minimaxproof · cited by 2
- le_of_isGLB_Ioiproof · cited by 2
- Directed.le_ciInf_iffproof · cited by 2
- lowerBounds_countableInfClosureproof · cited by 1
- LinearLocallyFiniteOrder.le_succFnproof · cited by 1