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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 22908 | . 2 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))) | |
2 | 1 | simplbi 500 | 1 ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∅c0 4291 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 TopOpenctopn 16695 Filcfil 22453 fLim cflim 22542 UnifStcuss 22862 UnifSpcusp 22863 CauFiluccfilu 22895 CUnifSpccusp 22906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-cusp 22907 |
This theorem is referenced by: cnextucn 22912 ucnextcn 22913 rrhcn 31238 rrhre 31262 |
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