Theorems · Theorem · order theory
ClosureOperator.IsClosed.closure_le_iff
∀ {α : Type u_1} [inst : Preorder α] {c : ClosureOperator α} {x y : α}, c.IsClosed y → (c x ≤ y ↔ x ≤ y)- Defined in
- Mathlib.Order.Closure
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 16 from the axioms · uses propext, Quot.sound
- Assumes
- Preorder
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.
- DFunLike.coestatement and proof · cited by 62,936
- Preorderstatement and proof · cited by 7,952
- ClosureOperatorstatement and proof · cited by 371
- ClosureOperator.IsClosedstatement and proof · cited by 38
- ClosureOperator.le_closure_iffproof · cited by 5
- ClosureOperator.IsClosed.closure_eqproof · cited by 3
Cited by6
Results whose statement or proof uses this declaration.
- ClosureOperator.closure_minproof · cited by 12
- Convexity.IsConvexSet.convexHull_subset_iffproof · cited by 2
- Convex.convexHull_subset_iffproof · cited by 1
- separableClosure_le_separableClosure_iffproof · cited by 1
- LowerAdjoint.closure_le_closed_iff_leproof · cited by 1
- AbsConvex.absConvexHull_subset_iffproof · cited by 0