Theorems · Theorem · order theory
ClosureOperator.le_closure_iff
∀ {α : Type u_1} [inst : Preorder α] {c : ClosureOperator α} {x y : α}, x ≤ c y ↔ c x ≤ c y- Defined in
- Mathlib.Order.Closure
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 15 from the axioms · uses propext, Quot.sound
- Assumes
- Preorder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- LE.le.transproof · cited by 3,151
- ClosureOperatorstatement and proof · cited by 371
- ClosureOperator.le_closureproof · cited by 19
- ClosureOperator.monotoneproof · cited by 18
- ClosureOperator.idempotentproof · cited by 12
Cited by5
Results whose statement or proof uses this declaration.
- ClosureOperator.IsClosed.closure_le_iffproof · cited by 6
- ClosureOperator.closure_sup_closure_leftproof · cited by 5
- ClosureOperator.closure_iSup_closureproof · cited by 2
- ClosureOperator.closure_iSup₂_closureproof · cited by 1
- LowerAdjoint.le_closure_iffproof · cited by 0