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Mirrors > Home > MPE Home > Th. List > cuspusp | Structured version Visualization version GIF version |
Description: A complete uniform space is an uniform space. (Contributed by Thierry Arnoux, 3-Dec-2017.) |
Ref | Expression |
---|---|
cuspusp | ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscusp 23064 | . 2 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))) | |
2 | 1 | simplbi 501 | 1 ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ≠ wne 2935 ∀wral 3054 ∅c0 4221 ‘cfv 6350 (class class class)co 7183 Basecbs 16599 TopOpenctopn 16811 Filcfil 22609 fLim cflim 22698 UnifStcuss 23018 UnifSpcusp 23019 CauFiluccfilu 23051 CUnifSpccusp 23062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-ne 2936 df-ral 3059 df-rab 3063 df-v 3402 df-un 3858 df-in 3860 df-ss 3870 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-iota 6308 df-fv 6358 df-ov 7186 df-cusp 23063 |
This theorem is referenced by: cnextucn 23068 ucnextcn 23069 rrhcn 31530 rrhre 31554 |
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