Theorem cuspusp 24218

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 24217	. 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 2099   ≠ wne 2937  ∀wral 3058  ∅c0 4323  ‘cfv 6548  (class class class)co 7420  Basecbs 17180  TopOpenctopn 17403  Filcfil 23762   fLim cflim 23851  UnifStcuss 24171  UnifSpcusp 24172  CauFil_uccfilu 24204  CUnifSpccusp 24215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-cusp 24216

This theorem is referenced by:  cnextucn  24221  ucnextcn  24222  rrhcn  33598  rrhre  33622

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