Theorem cuspusp 24255

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 24254	. 2 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
2	1	simplbi 496	1 ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)

Syntax hints:   → wi 4   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∅c0 4287  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  TopOpenctopn 17353  Filcfil 23801   fLim cflim 23890  UnifStcuss 24209  UnifSpcusp 24210  CauFil_uccfilu 24241  CUnifSpccusp 24252

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-cusp 24253

This theorem is referenced by:  cnextucn  24258  ucnextcn  24259  rrhcn  34174  rrhre  34198