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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 2737 | . . 3 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
| 2 | 1 | iscmet 25318 | . 2 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅)) |
| 3 | 2 | simplbi 497 | 1 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 Metcmet 21350 MetOpencmopn 21354 fLim cflim 23942 CauFilccfil 25286 CMetccmet 25288 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-cmet 25291 |
| This theorem is referenced by: cmetmeti 25321 cmetcaulem 25322 cmetcau 25323 iscmet2 25328 metsscmetcld 25349 cmetss 25350 bcthlem2 25359 bcthlem3 25360 bcthlem4 25361 bcthlem5 25362 bcth2 25364 bcth3 25365 cmetcusp1 25387 cmetcusp 25388 minveclem3 25463 ubthlem1 30889 ubthlem2 30890 hlmet 30914 fmcncfil 33930 heiborlem3 37820 heiborlem6 37823 heiborlem8 37825 heiborlem9 37826 heiborlem10 37827 heibor 37828 bfplem1 37829 bfplem2 37830 bfp 37831 |
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