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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 22103 | . 2 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))) | |
2 | 1 | simplbi 476 | 1 ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∅c0 3915 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 TopOpenctopn 16082 Filcfil 21649 fLim cflim 21738 UnifStcuss 22057 UnifSpcusp 22058 CauFiluccfilu 22090 CUnifSpccusp 22101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-cusp 22102 |
This theorem is referenced by: cnextucn 22107 ucnextcn 22108 rrhcn 30041 rrhre 30065 |
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