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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 21913 | . 2 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))) | |
2 | 1 | simplbi 475 | 1 ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 TopOpenctopn 15905 Filcfil 21459 fLim cflim 21548 UnifStcuss 21867 UnifSpcusp 21868 CauFiluccfilu 21900 CUnifSpccusp 21911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-cusp 21912 |
This theorem is referenced by: cnextucn 21917 ucnextcn 21918 rrhcn 29369 rrhre 29393 |
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