Theorems · Theorem · general topology
ContractingWith.efixedPoint_isFixedPt
∀ {α : Type u_1} [inst : EMetricSpace α] {K : NNReal} {f : α → α} [inst_1 : CompleteSpace α] (hf : ContractingWith K f)
{x : α} (hx : edist x (f x) ≠ ⊤), Function.IsFixedPt f (ContractingWith.efixedPoint f hf x hx)- Defined in
- Mathlib.Topology.MetricSpace.Contracting
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 161 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- EMetricSpaceCompleteSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- ENNRealstatement · cited by 9,879
- Top.topstatement and proof · cited by 9,680
- NNRealstatement and proof · cited by 4,310
- CompleteSpacestatement and proof · cited by 2,532
- EDist.ediststatement and proof · cited by 735
- EMetricSpacestatement and proof · cited by 242
- Function.IsFixedPtstatement · cited by 84
- ContractingWithstatement and proof · cited by 39
- ContractingWith.efixedPointstatement · cited by 7
- ContractingWith.exists_fixedPointproof · cited by 4
Cited by3
Results whose statement or proof uses this declaration.
- ContractingWith.fixedPoint_isFixedPtproof · cited by 4
- ContractingWith.tendsto_iterate_fixedPointproof · cited by 1
- ContractingWith.efixedPoint_eq_of_edist_lt_topproof · cited by 0