Theorems · Theorem · order theory
ClosureOperator.isClosed_iff_closure_le
∀ {α : Type u_1} [inst : PartialOrder α] {c : ClosureOperator α} {x : α}, c.IsClosed x ↔ c x ≤ x- Defined in
- Mathlib.Order.Closure
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 15 from the axioms · uses propext, Quot.sound
- Assumes
- PartialOrder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- PartialOrderstatement and proof · cited by 6,410
- Eq.leproof · cited by 605
- LE.le.antisymmproof · cited by 507
- ClosureOperatorstatement and proof · cited by 371
- ClosureOperator.IsClosedstatement and proof · cited by 38
- ClosureOperator.le_closureproof · cited by 19
- ClosureOperator.isClosed_iffproof · cited by 14
- ClosureOperator.IsClosed.closure_eqproof · cited by 3
Cited by3
Results whose statement or proof uses this declaration.
- isClosed_restrictScalars_separableClosureproof · cited by 1
- LowerAdjoint.mem_closed_iff_closure_leproof · cited by 0
- ClosureOperator.sInf_isClosedproof · cited by 0