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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 24288 | . 2 ⊢ (𝐷 ∈ (Met‘𝑋) ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐷:(𝑋 × 𝑋)⟶ℝ)) | |
2 | 1 | simplbi 496 | 1 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 × cxp 5676 ⟶wf 6545 ‘cfv 6549 ℝcr 11144 ∞Metcxmet 21286 Metcmet 21287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-mulcl 11207 ax-i2m1 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11287 df-mnf 11288 df-xr 11289 df-xadd 13133 df-xmet 21294 df-met 21295 |
This theorem is referenced by: metdmdm 24291 meteq0 24294 mettri2 24296 met0 24298 metge0 24300 metsym 24305 metrtri 24312 metgt0 24314 metres2 24318 prdsmet 24325 imasf1omet 24331 blpnf 24352 bl2in 24355 isms2 24405 setsms 24437 tmsms 24445 metss2lem 24469 metss2 24470 methaus 24478 dscopn 24531 ngpocelbl 24670 cnxmet 24738 rexmet 24756 metdcn2 24804 metdsre 24818 metdscn2 24822 lebnumlem1 24936 lebnumlem2 24937 lebnumlem3 24938 lebnum 24939 xlebnum 24940 cmetcaulem 25265 cmetcau 25266 iscmet3lem1 25268 iscmet3lem2 25269 iscmet3 25270 equivcfil 25276 equivcau 25277 metsscmetcld 25292 cmetss 25293 relcmpcmet 25295 cmpcmet 25296 cncmet 25299 bcthlem2 25302 bcthlem3 25303 bcthlem4 25304 bcthlem5 25305 bcth2 25307 bcth3 25308 cmetcusp1 25330 cmetcusp 25331 minveclem3 25406 imsxmet 30579 blocni 30692 ubthlem1 30757 ubthlem2 30758 minvecolem4a 30764 hhxmet 31062 hilxmet 31082 fmcncfil 33665 blssp 37362 lmclim2 37364 geomcau 37365 caures 37366 caushft 37367 sstotbnd2 37380 equivtotbnd 37384 isbndx 37388 isbnd3 37390 ssbnd 37394 totbndbnd 37395 prdstotbnd 37400 prdsbnd2 37401 heibor1lem 37415 heibor1 37416 heiborlem3 37419 heiborlem6 37422 heiborlem8 37424 heiborlem9 37425 heiborlem10 37426 heibor 37427 bfplem1 37428 bfplem2 37429 rrncmslem 37438 ismrer1 37444 reheibor 37445 metpsmet 44599 qndenserrnbllem 45822 qndenserrnbl 45823 qndenserrnopnlem 45825 rrndsxmet 45831 hoiqssbllem2 46151 hoiqssbl 46153 opnvonmbllem2 46161 |
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