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Theorem cuspusp 23452
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 23451 . 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 2943  wral 3064  c0 4256  cfv 6433  (class class class)co 7275  Basecbs 16912  TopOpenctopn 17132  Filcfil 22996   fLim cflim 23085  UnifStcuss 23405  UnifSpcusp 23406  CauFiluccfilu 23438  CUnifSpccusp 23449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-cusp 23450
This theorem is referenced by:  cnextucn  23455  ucnextcn  23456  rrhcn  31947  rrhre  31971
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