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Mirrors > Home > ILE Home > Th. List > xmettri2 | Unicode version |
Description: Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmettri2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmetrel 14295 |
. . . . . . . 8
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2 | relelfvdm 5566 |
. . . . . . . 8
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3 | 1, 2 | mpan 424 |
. . . . . . 7
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4 | isxmet 14297 |
. . . . . . 7
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5 | 3, 4 | syl 14 |
. . . . . 6
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6 | 5 | ibi 176 |
. . . . 5
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7 | simpr 110 |
. . . . . 6
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8 | 7 | 2ralimi 2554 |
. . . . 5
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9 | 6, 8 | simpl2im 386 |
. . . 4
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10 | oveq1 5902 |
. . . . . 6
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11 | oveq2 5903 |
. . . . . . 7
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12 | 11 | oveq1d 5910 |
. . . . . 6
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13 | 10, 12 | breq12d 4031 |
. . . . 5
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14 | oveq2 5903 |
. . . . . 6
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15 | oveq2 5903 |
. . . . . . 7
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16 | 15 | oveq2d 5911 |
. . . . . 6
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17 | 14, 16 | breq12d 4031 |
. . . . 5
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18 | oveq1 5902 |
. . . . . . 7
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19 | oveq1 5902 |
. . . . . . 7
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20 | 18, 19 | oveq12d 5913 |
. . . . . 6
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21 | 20 | breq2d 4030 |
. . . . 5
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22 | 13, 17, 21 | rspc3v 2872 |
. . . 4
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23 | 9, 22 | syl5 32 |
. . 3
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24 | 23 | 3comr 1213 |
. 2
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25 | 24 | impcom 125 |
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 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 |
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 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-map 6675 df-pnf 8023 df-mnf 8024 df-xr 8025 df-xmet 13854 |
This theorem is referenced by: mettri2 14314 xmetge0 14317 xmetsym 14320 xmetpsmet 14321 xmettri 14324 xmetres2 14331 xblss2 14357 xmstri2 14422 comet 14451 xmetxp 14459 |
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