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| Mirrors > Home > MPE Home > Th. List > metxmet | Structured version Visualization version GIF version | ||
| Description: A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| Ref | Expression |
|---|---|
| metxmet | ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ismet2 24343 | . 2 ⊢ (𝐷 ∈ (Met‘𝑋) ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐷:(𝑋 × 𝑋)⟶ℝ)) | |
| 2 | 1 | simplbi 497 | 1 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 × cxp 5683 ⟶wf 6557 ‘cfv 6561 ℝcr 11154 ∞Metcxmet 21349 Metcmet 21350 |
| 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-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-i2m1 11223 |
| 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-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 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-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-xadd 13155 df-xmet 21357 df-met 21358 |
| This theorem is referenced by: metdmdm 24346 meteq0 24349 mettri2 24351 met0 24353 metge0 24355 metsym 24360 metrtri 24367 metgt0 24369 metres2 24373 prdsmet 24380 imasf1omet 24386 blpnf 24407 bl2in 24410 isms2 24460 setsms 24492 tmsms 24500 metss2lem 24524 metss2 24525 methaus 24533 dscopn 24586 ngpocelbl 24725 cnxmet 24793 rexmet 24812 metdcn2 24861 metdsre 24875 metdscn2 24879 lebnumlem1 24993 lebnumlem2 24994 lebnumlem3 24995 lebnum 24996 xlebnum 24997 cmetcaulem 25322 cmetcau 25323 iscmet3lem1 25325 iscmet3lem2 25326 iscmet3 25327 equivcfil 25333 equivcau 25334 metsscmetcld 25349 cmetss 25350 relcmpcmet 25352 cmpcmet 25353 cncmet 25356 bcthlem2 25359 bcthlem3 25360 bcthlem4 25361 bcthlem5 25362 bcth2 25364 bcth3 25365 cmetcusp1 25387 cmetcusp 25388 minveclem3 25463 imsxmet 30711 blocni 30824 ubthlem1 30889 ubthlem2 30890 minvecolem4a 30896 hhxmet 31194 hilxmet 31214 fmcncfil 33930 blssp 37763 lmclim2 37765 geomcau 37766 caures 37767 caushft 37768 sstotbnd2 37781 equivtotbnd 37785 isbndx 37789 isbnd3 37791 ssbnd 37795 totbndbnd 37796 prdstotbnd 37801 prdsbnd2 37802 heibor1lem 37816 heibor1 37817 heiborlem3 37820 heiborlem6 37823 heiborlem8 37825 heiborlem9 37826 heiborlem10 37827 heibor 37828 bfplem1 37829 bfplem2 37830 rrncmslem 37839 ismrer1 37845 reheibor 37846 metpsmet 45096 qndenserrnbllem 46309 qndenserrnbl 46310 qndenserrnopnlem 46312 rrndsxmet 46318 hoiqssbllem2 46638 hoiqssbl 46640 opnvonmbllem2 46648 |
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