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Mirrors > Home > ILE Home > Th. List > xmettri | Unicode version |
Description: Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmettri |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 109 |
. . 3
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2 | simpr3 1007 |
. . 3
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3 | simpr1 1005 |
. . 3
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4 | simpr2 1006 |
. . 3
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5 | xmettri2 14346 |
. . 3
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6 | 1, 2, 3, 4, 5 | syl13anc 1251 |
. 2
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7 | xmetsym 14353 |
. . . 4
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8 | 1, 2, 3, 7 | syl3anc 1249 |
. . 3
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9 | 8 | oveq1d 5915 |
. 2
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10 | 6, 9 | breqtrd 4047 |
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 ax-1re 7940 ax-addrcl 7943 ax-0id 7954 ax-rnegex 7955 ax-pre-ltirr 7958 ax-pre-apti 7961 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 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-nel 2456 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-if 3550 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-ltxr 8032 df-le 8033 df-xadd 9809 df-xmet 13882 |
This theorem is referenced by: xmettri3 14359 xmetrtri 14361 xmeter 14421 xmstri 14457 |
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