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Mirrors > Home > ILE Home > Th. List > psmetsym | Unicode version |
Description: The distance function of a pseudometric is symmetrical. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
Ref | Expression |
---|---|
psmetsym |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 999 |
. . . 4
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2 | simp3 1001 |
. . . 4
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3 | simp2 1000 |
. . . 4
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4 | psmettri2 14313 |
. . . 4
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5 | 1, 2, 3, 2, 4 | syl13anc 1251 |
. . 3
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6 | psmet0 14312 |
. . . . . 6
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7 | 6 | 3adant2 1018 |
. . . . 5
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8 | 7 | oveq2d 5916 |
. . . 4
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9 | psmetcl 14311 |
. . . . . 6
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10 | xaddid1 9898 |
. . . . . 6
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11 | 9, 10 | syl 14 |
. . . . 5
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12 | 11 | 3com23 1211 |
. . . 4
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13 | 8, 12 | eqtrd 2222 |
. . 3
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14 | 5, 13 | breqtrd 4047 |
. 2
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15 | psmettri2 14313 |
. . . 4
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16 | 1, 3, 2, 3, 15 | syl13anc 1251 |
. . 3
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17 | psmet0 14312 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | 17 | 3adant3 1019 |
. . . . 5
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19 | 18 | oveq2d 5916 |
. . . 4
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20 | psmetcl 14311 |
. . . . 5
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21 | xaddid1 9898 |
. . . . 5
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22 | 20, 21 | syl 14 |
. . . 4
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23 | 19, 22 | eqtrd 2222 |
. . 3
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24 | 16, 23 | breqtrd 4047 |
. 2
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25 | 9 | 3com23 1211 |
. . 3
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26 | xrletri3 9840 |
. . 3
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27 | 20, 25, 26 | syl2anc 411 |
. 2
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28 | 14, 24, 27 | mpbir2and 946 |
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-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-map 6680 df-pnf 8029 df-mnf 8030 df-xr 8031 df-ltxr 8032 df-le 8033 df-xadd 9809 df-psmet 13881 |
This theorem is referenced by: psmettri 14315 distspace 14320 elbl3ps 14379 blssps 14412 |
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