Theorem cuspusp 24325

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 24324	. 2 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
2	1	simplbi 497	1 ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)

Syntax hints:   → wi 4   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∅c0 4339  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  TopOpenctopn 17468  Filcfil 23869   fLim cflim 23958  UnifStcuss 24278  UnifSpcusp 24279  CauFil_uccfilu 24311  CUnifSpccusp 24322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-cusp 24323

This theorem is referenced by:  cnextucn  24328  ucnextcn  24329  rrhcn  33960  rrhre  33984