Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cuspusp Structured version   Visualization version   GIF version

Theorem cuspusp 21914
 Description: A complete uniform space is an uniform space. (Contributed by Thierry Arnoux, 3-Dec-2017.)
Assertion
Ref Expression
cuspusp (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)

Proof of Theorem cuspusp
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 iscusp 21913 . 2 (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
21simplbi 475 1 (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∅c0 3874  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  TopOpenctopn 15905  Filcfil 21459   fLim cflim 21548  UnifStcuss 21867  UnifSpcusp 21868  CauFiluccfilu 21900  CUnifSpccusp 21911 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-cusp 21912 This theorem is referenced by:  cnextucn  21917  ucnextcn  21918  rrhcn  29369  rrhre  29393
 Copyright terms: Public domain W3C validator