Theorem cuspusp 23804

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.

Step	Hyp	Ref	Expression
1		iscusp 23803	. 2 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
2	1	simplbi 498	1 ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)

Syntax hints:   → wi 4   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∅c0 4322  ‘cfv 6543  (class class class)co 7408  Basecbs 17143  TopOpenctopn 17366  Filcfil 23348   fLim cflim 23437  UnifStcuss 23757  UnifSpcusp 23758  CauFil_uccfilu 23790  CUnifSpccusp 23801

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 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411  df-cusp 23802

This theorem is referenced by:  cnextucn  23807  ucnextcn  23808  rrhcn  32972  rrhre  32996