Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 24036 | . 2 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅)) |
3 | 2 | simplbi 501 | 1 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ≠ wne 2934 ∀wral 3053 ∅c0 4211 ‘cfv 6339 (class class class)co 7170 Metcmet 20203 MetOpencmopn 20207 fLim cflim 22685 CauFilccfil 24004 CMetccmet 24006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-iota 6297 df-fun 6341 df-fv 6347 df-ov 7173 df-cmet 24009 |
This theorem is referenced by: cmetmeti 24039 cmetcaulem 24040 cmetcau 24041 iscmet2 24046 metsscmetcld 24067 cmetss 24068 bcthlem2 24077 bcthlem3 24078 bcthlem4 24079 bcthlem5 24080 bcth2 24082 bcth3 24083 cmetcusp1 24105 cmetcusp 24106 minveclem3 24181 ubthlem1 28805 ubthlem2 28806 hlmet 28830 fmcncfil 31453 heiborlem3 35594 heiborlem6 35597 heiborlem8 35599 heiborlem9 35600 heiborlem10 35601 heibor 35602 bfplem1 35603 bfplem2 35604 bfp 35605 |
Copyright terms: Public domain | W3C validator |