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Mirrors > Home > MPE Home > Th. List > cmetmet | Structured version Visualization version GIF version |
Description: A complete metric space is a metric space. (Contributed by NM, 18-Dec-2006.) (Revised by Mario Carneiro, 29-Jan-2014.) |
Ref | Expression |
---|---|
cmetmet | ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
2 | 1 | iscmet 23814 | . 2 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅)) |
3 | 2 | simplbi 498 | 1 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∅c0 4288 ‘cfv 6348 (class class class)co 7145 Metcmet 20459 MetOpencmopn 20463 fLim cflim 22470 CauFilccfil 23782 CMetccmet 23784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-cmet 23787 |
This theorem is referenced by: cmetmeti 23817 cmetcaulem 23818 cmetcau 23819 iscmet2 23824 metsscmetcld 23845 cmetss 23846 bcthlem2 23855 bcthlem3 23856 bcthlem4 23857 bcthlem5 23858 bcth2 23860 bcth3 23861 cmetcusp1 23883 cmetcusp 23884 minveclem3 23959 ubthlem1 28574 ubthlem2 28575 hlmet 28599 fmcncfil 31073 heiborlem3 34972 heiborlem6 34975 heiborlem8 34977 heiborlem9 34978 heiborlem10 34979 heibor 34980 bfplem1 34981 bfplem2 34982 bfp 34983 |
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