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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 22138 | . 2 ⊢ (𝐷 ∈ (Met‘𝑋) ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐷:(𝑋 × 𝑋)⟶ℝ)) | |
2 | 1 | simplbi 476 | 1 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 × cxp 5112 ⟶wf 5884 ‘cfv 5888 ℝcr 9935 ∞Metcxmt 19731 Metcme 19732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-mulcl 9998 ax-i2m1 10004 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-xadd 11947 df-xmet 19739 df-met 19740 |
This theorem is referenced by: metdmdm 22141 meteq0 22144 mettri2 22146 met0 22148 metge0 22150 metsym 22155 metrtri 22162 metgt0 22164 metres2 22168 prdsmet 22175 imasf1omet 22181 blpnf 22202 bl2in 22205 isms2 22255 setsms 22285 tmsms 22292 metss2lem 22316 metss2 22317 methaus 22325 dscopn 22378 ngpocelbl 22508 cnxmet 22576 rexmet 22594 metdcn2 22642 metdsre 22656 metdscn2 22660 lebnumlem1 22760 lebnumlem2 22761 lebnumlem3 22762 lebnum 22763 xlebnum 22764 cmetcaulem 23086 cmetcau 23087 iscmet3lem1 23089 iscmet3lem2 23090 iscmet3 23091 equivcfil 23097 equivcau 23098 cmetss 23113 relcmpcmet 23115 cmpcmet 23116 cncmet 23119 bcthlem2 23122 bcthlem3 23123 bcthlem4 23124 bcthlem5 23125 bcth2 23127 bcth3 23128 cmetcusp1 23149 cmetcusp 23150 minveclem3 23200 imsxmet 27547 blocni 27660 ubthlem1 27726 ubthlem2 27727 minvecolem4a 27733 hhxmet 28032 hilxmet 28052 fmcncfil 29977 blssp 33552 lmclim2 33554 geomcau 33555 caures 33556 caushft 33557 sstotbnd2 33573 equivtotbnd 33577 isbndx 33581 isbnd3 33583 ssbnd 33587 totbndbnd 33588 prdstotbnd 33593 prdsbnd2 33594 heibor1lem 33608 heibor1 33609 heiborlem3 33612 heiborlem6 33615 heiborlem8 33617 heiborlem9 33618 heiborlem10 33619 heibor 33620 bfplem1 33621 bfplem2 33622 rrncmslem 33631 ismrer1 33637 reheibor 33638 metpsmet 39268 qndenserrnbllem 40514 qndenserrnbl 40515 qndenserrnopnlem 40517 rrndsxmet 40523 hoiqssbllem2 40837 hoiqssbl 40839 opnvonmbllem2 40847 |
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