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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 949 |
. . . 4
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2 | simp3 951 |
. . . 4
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3 | simp2 950 |
. . . 4
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4 | psmettri2 12256 |
. . . 4
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5 | 1, 2, 3, 2, 4 | syl13anc 1186 |
. . 3
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6 | psmet0 12255 |
. . . . . 6
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7 | 6 | 3adant2 968 |
. . . . 5
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8 | 7 | oveq2d 5722 |
. . . 4
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9 | psmetcl 12254 |
. . . . . 6
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10 | xaddid1 9486 |
. . . . . 6
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11 | 9, 10 | syl 14 |
. . . . 5
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12 | 11 | 3com23 1155 |
. . . 4
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13 | 8, 12 | eqtrd 2132 |
. . 3
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14 | 5, 13 | breqtrd 3899 |
. 2
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15 | psmettri2 12256 |
. . . 4
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16 | 1, 3, 2, 3, 15 | syl13anc 1186 |
. . 3
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17 | psmet0 12255 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | 17 | 3adant3 969 |
. . . . 5
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19 | 18 | oveq2d 5722 |
. . . 4
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20 | psmetcl 12254 |
. . . . 5
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21 | xaddid1 9486 |
. . . . 5
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22 | 20, 21 | syl 14 |
. . . 4
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23 | 19, 22 | eqtrd 2132 |
. . 3
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24 | 16, 23 | breqtrd 3899 |
. 2
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25 | 9 | 3com23 1155 |
. . 3
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26 | xrletri3 9429 |
. . 3
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27 | 20, 25, 26 | syl2anc 406 |
. 2
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28 | 14, 24, 27 | mpbir2and 896 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1re 7589 ax-addrcl 7592 ax-0id 7603 ax-rnegex 7604 ax-pre-ltirr 7607 ax-pre-apti 7610 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-if 3422 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-map 6474 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-xadd 9401 df-psmet 11938 |
This theorem is referenced by: psmettri 12258 distspace 12263 elbl3ps 12322 blssps 12355 |
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