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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 | PsMet |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 986 | . . . 4 PsMet PsMet | |
2 | simp3 988 | . . . 4 PsMet | |
3 | simp2 987 | . . . 4 PsMet | |
4 | psmettri2 12869 | . . . 4 PsMet | |
5 | 1, 2, 3, 2, 4 | syl13anc 1229 | . . 3 PsMet |
6 | psmet0 12868 | . . . . . 6 PsMet | |
7 | 6 | 3adant2 1005 | . . . . 5 PsMet |
8 | 7 | oveq2d 5852 | . . . 4 PsMet |
9 | psmetcl 12867 | . . . . . 6 PsMet | |
10 | xaddid1 9789 | . . . . . 6 | |
11 | 9, 10 | syl 14 | . . . . 5 PsMet |
12 | 11 | 3com23 1198 | . . . 4 PsMet |
13 | 8, 12 | eqtrd 2197 | . . 3 PsMet |
14 | 5, 13 | breqtrd 4002 | . 2 PsMet |
15 | psmettri2 12869 | . . . 4 PsMet | |
16 | 1, 3, 2, 3, 15 | syl13anc 1229 | . . 3 PsMet |
17 | psmet0 12868 | . . . . . 6 PsMet | |
18 | 17 | 3adant3 1006 | . . . . 5 PsMet |
19 | 18 | oveq2d 5852 | . . . 4 PsMet |
20 | psmetcl 12867 | . . . . 5 PsMet | |
21 | xaddid1 9789 | . . . . 5 | |
22 | 20, 21 | syl 14 | . . . 4 PsMet |
23 | 19, 22 | eqtrd 2197 | . . 3 PsMet |
24 | 16, 23 | breqtrd 4002 | . 2 PsMet |
25 | 9 | 3com23 1198 | . . 3 PsMet |
26 | xrletri3 9731 | . . 3 | |
27 | 20, 25, 26 | syl2anc 409 | . 2 PsMet |
28 | 14, 24, 27 | mpbir2and 933 | 1 PsMet |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 class class class wbr 3976 cfv 5182 (class class class)co 5836 cc0 7744 cxr 7923 cle 7925 cxad 9697 PsMetcpsmet 12520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1re 7838 ax-addrcl 7841 ax-0id 7852 ax-rnegex 7853 ax-pre-ltirr 7856 ax-pre-apti 7859 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-map 6607 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-xadd 9700 df-psmet 12528 |
This theorem is referenced by: psmettri 12871 distspace 12876 elbl3ps 12935 blssps 12968 |
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