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Mirrors > Home > ILE Home > Th. List > xmetsym | Unicode version |
Description: The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmetsym |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 981 | . . . 4 | |
2 | simp3 983 | . . . 4 | |
3 | simp2 982 | . . . 4 | |
4 | xmettri2 12519 | . . . 4 | |
5 | 1, 2, 3, 2, 4 | syl13anc 1218 | . . 3 |
6 | xmet0 12521 | . . . . . 6 | |
7 | 6 | 3adant2 1000 | . . . . 5 |
8 | 7 | oveq2d 5783 | . . . 4 |
9 | xmetcl 12510 | . . . . . 6 | |
10 | xaddid1 9638 | . . . . . 6 | |
11 | 9, 10 | syl 14 | . . . . 5 |
12 | 11 | 3com23 1187 | . . . 4 |
13 | 8, 12 | eqtrd 2170 | . . 3 |
14 | 5, 13 | breqtrd 3949 | . 2 |
15 | xmettri2 12519 | . . . 4 | |
16 | 1, 3, 2, 3, 15 | syl13anc 1218 | . . 3 |
17 | xmet0 12521 | . . . . . 6 | |
18 | 17 | 3adant3 1001 | . . . . 5 |
19 | 18 | oveq2d 5783 | . . . 4 |
20 | xmetcl 12510 | . . . . 5 | |
21 | xaddid1 9638 | . . . . 5 | |
22 | 20, 21 | syl 14 | . . . 4 |
23 | 19, 22 | eqtrd 2170 | . . 3 |
24 | 16, 23 | breqtrd 3949 | . 2 |
25 | 9 | 3com23 1187 | . . 3 |
26 | xrletri3 9581 | . . 3 | |
27 | 20, 25, 26 | syl2anc 408 | . 2 |
28 | 14, 24, 27 | mpbir2and 928 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 class class class wbr 3924 cfv 5118 (class class class)co 5767 cc0 7613 cxr 7792 cle 7794 cxad 9550 cxmet 12138 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1re 7707 ax-addrcl 7710 ax-0id 7721 ax-rnegex 7722 ax-pre-ltirr 7725 ax-pre-apti 7728 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-map 6537 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-xadd 9553 df-xmet 12146 |
This theorem is referenced by: xmettpos 12528 metsym 12529 xmettri 12530 xmettri3 12532 elbl3 12553 blss 12586 xmeter 12594 xmssym 12627 metcnp2 12671 |
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