Theorem cuspusp 24651

Description: A complete uniform space is a 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 24650	. 2 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
2	1	simplbi 502	1 ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∅c0 4281  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  TopOpenctopn 17624  Filcfil 24197   fLim cflim 24286  UnifStcuss 24605  UnifSpcusp 24606  CauFil_uccfilu 24637  CUnifSpccusp 24648

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6515  df-fv 6569  df-ov 7449  df-cusp 24649

This theorem is used by:  cnextucn  24654  ucnextcn  24655  rrhcn  34712  rrhre  34736