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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 2738 | . . 3 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
2 | 1 | iscmet 24448 | . 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 2106 ≠ wne 2943 ∀wral 3064 ∅c0 4256 ‘cfv 6433 (class class class)co 7275 Metcmet 20583 MetOpencmopn 20587 fLim cflim 23085 CauFilccfil 24416 CMetccmet 24418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-ov 7278 df-cmet 24421 |
This theorem is referenced by: cmetmeti 24451 cmetcaulem 24452 cmetcau 24453 iscmet2 24458 metsscmetcld 24479 cmetss 24480 bcthlem2 24489 bcthlem3 24490 bcthlem4 24491 bcthlem5 24492 bcth2 24494 bcth3 24495 cmetcusp1 24517 cmetcusp 24518 minveclem3 24593 ubthlem1 29232 ubthlem2 29233 hlmet 29257 fmcncfil 31881 heiborlem3 35971 heiborlem6 35974 heiborlem8 35976 heiborlem9 35977 heiborlem10 35978 heibor 35979 bfplem1 35980 bfplem2 35981 bfp 35982 |
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