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Theorem cuspusp 23736
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 23735 . 2 (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
21simplbi 498 1 (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wne 2940  wral 3061  c0 4319  cfv 6533  (class class class)co 7394  Basecbs 17128  TopOpenctopn 17351  Filcfil 23280   fLim cflim 23369  UnifStcuss 23689  UnifSpcusp 23690  CauFiluccfilu 23722  CUnifSpccusp 23733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-iota 6485  df-fv 6541  df-ov 7397  df-cusp 23734
This theorem is referenced by:  cnextucn  23739  ucnextcn  23740  rrhcn  32872  rrhre  32896
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