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Mirrors > Home > ILE Home > Th. List > xmetres2 | Unicode version |
Description: Restriction of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmetres2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmetrel 14328 |
. . . . 5
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2 | relelfvdm 5569 |
. . . . 5
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3 | 1, 2 | mpan 424 |
. . . 4
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4 | 3 | adantr 276 |
. . 3
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5 | simpr 110 |
. . 3
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6 | 4, 5 | ssexd 4161 |
. 2
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7 | xmetf 14335 |
. . . 4
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8 | 7 | adantr 276 |
. . 3
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9 | xpss12 4754 |
. . . 4
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10 | 5, 9 | sylancom 420 |
. . 3
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11 | 8, 10 | fssresd 5414 |
. 2
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12 | ovres 6040 |
. . . . 5
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13 | 12 | adantl 277 |
. . . 4
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14 | 13 | eqeq1d 2198 |
. . 3
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15 | simpll 527 |
. . . 4
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16 | simplr 528 |
. . . . 5
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17 | simprl 529 |
. . . . 5
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18 | 16, 17 | sseldd 3171 |
. . . 4
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19 | simprr 531 |
. . . . 5
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20 | 16, 19 | sseldd 3171 |
. . . 4
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21 | xmeteq0 14344 |
. . . 4
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22 | 15, 18, 20, 21 | syl3anc 1249 |
. . 3
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23 | 14, 22 | bitrd 188 |
. 2
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24 | simpll 527 |
. . . 4
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25 | simplr 528 |
. . . . 5
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26 | simpr3 1007 |
. . . . 5
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27 | 25, 26 | sseldd 3171 |
. . . 4
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28 | 18 | 3adantr3 1160 |
. . . 4
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29 | 20 | 3adantr3 1160 |
. . . 4
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30 | xmettri2 14346 |
. . . 4
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31 | 24, 27, 28, 29, 30 | syl13anc 1251 |
. . 3
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32 | 13 | 3adantr3 1160 |
. . 3
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33 | simpr1 1005 |
. . . . 5
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34 | 26, 33 | ovresd 6041 |
. . . 4
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35 | simpr2 1006 |
. . . . 5
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36 | 26, 35 | ovresd 6041 |
. . . 4
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37 | 34, 36 | oveq12d 5918 |
. . 3
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38 | 31, 32, 37 | 3brtr4d 4053 |
. 2
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39 | 6, 11, 23, 38 | isxmetd 14332 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-cnex 7937 ax-resscn 7938 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-iun 3906 df-br 4022 df-opab 4083 df-mpt 4084 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-fv 5246 df-ov 5903 df-oprab 5904 df-mpo 5905 df-1st 6169 df-2nd 6170 df-map 6680 df-pnf 8029 df-mnf 8030 df-xr 8031 df-xmet 13882 |
This theorem is referenced by: metres2 14366 xmetres 14367 xmetresbl 14425 metrest 14491 divcnap 14540 cncfmet 14564 limcimolemlt 14618 cnplimcim 14621 cnplimclemr 14623 limccnpcntop 14629 limccnp2cntop 14631 |
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