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Mirrors > Home > ILE Home > Th. List > isxmet2d | Unicode version |
Description: It is safe to only require the triangle inequality when the values are real (so that we can use the standard addition over the reals), but in this case the nonnegativity constraint cannot be deduced and must be provided separately. (Counterexample: satisfies all hypotheses except nonnegativity.) (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
isxmetd.0 | |
isxmetd.1 | |
isxmet2d.2 | |
isxmet2d.3 | |
isxmet2d.4 |
Ref | Expression |
---|---|
isxmet2d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isxmetd.0 | . 2 | |
2 | isxmetd.1 | . 2 | |
3 | 2 | fovrnda 5882 | . . . 4 |
4 | 0xr 7780 | . . . 4 | |
5 | xrletri3 9543 | . . . 4 | |
6 | 3, 4, 5 | sylancl 409 | . . 3 |
7 | isxmet2d.2 | . . . 4 | |
8 | 7 | biantrud 302 | . . 3 |
9 | isxmet2d.3 | . . 3 | |
10 | 6, 8, 9 | 3bitr2d 215 | . 2 |
11 | isxmet2d.4 | . . . . . . 7 | |
12 | 11 | 3expa 1166 | . . . . . 6 |
13 | rexadd 9590 | . . . . . . 7 | |
14 | 13 | adantl 275 | . . . . . 6 |
15 | 12, 14 | breqtrrd 3926 | . . . . 5 |
16 | 15 | anassrs 397 | . . . 4 |
17 | 3 | 3adantr3 1127 | . . . . . . 7 |
18 | pnfge 9530 | . . . . . . 7 | |
19 | 17, 18 | syl 14 | . . . . . 6 |
20 | 19 | ad2antrr 479 | . . . . 5 |
21 | oveq2 5750 | . . . . . 6 | |
22 | 2 | ffnd 5243 | . . . . . . . . . . 11 |
23 | elxrge0 9716 | . . . . . . . . . . . . 13 | |
24 | 3, 7, 23 | sylanbrc 413 | . . . . . . . . . . . 12 |
25 | 24 | ralrimivva 2491 | . . . . . . . . . . 11 |
26 | ffnov 5843 | . . . . . . . . . . 11 | |
27 | 22, 25, 26 | sylanbrc 413 | . . . . . . . . . 10 |
28 | 27 | adantr 274 | . . . . . . . . 9 |
29 | simpr3 974 | . . . . . . . . 9 | |
30 | simpr1 972 | . . . . . . . . 9 | |
31 | 28, 29, 30 | fovrnd 5883 | . . . . . . . 8 |
32 | elxrge0 9716 | . . . . . . . . 9 | |
33 | 32 | simplbi 272 | . . . . . . . 8 |
34 | 31, 33 | syl 14 | . . . . . . 7 |
35 | renemnf 7782 | . . . . . . 7 | |
36 | xaddpnf1 9584 | . . . . . . 7 | |
37 | 34, 35, 36 | syl2an 287 | . . . . . 6 |
38 | 21, 37 | sylan9eqr 2172 | . . . . 5 |
39 | 20, 38 | breqtrrd 3926 | . . . 4 |
40 | simpr2 973 | . . . . . . . . . . 11 | |
41 | 28, 29, 40 | fovrnd 5883 | . . . . . . . . . 10 |
42 | elxrge0 9716 | . . . . . . . . . . 11 | |
43 | 42 | simplbi 272 | . . . . . . . . . 10 |
44 | 41, 43 | syl 14 | . . . . . . . . 9 |
45 | 42 | simprbi 273 | . . . . . . . . . 10 |
46 | 41, 45 | syl 14 | . . . . . . . . 9 |
47 | ge0nemnf 9562 | . . . . . . . . 9 | |
48 | 44, 46, 47 | syl2anc 408 | . . . . . . . 8 |
49 | 48 | neneqd 2306 | . . . . . . 7 |
50 | 49 | pm2.21d 593 | . . . . . 6 |
51 | 50 | adantr 274 | . . . . 5 |
52 | 51 | imp 123 | . . . 4 |
53 | 44 | adantr 274 | . . . . 5 |
54 | elxr 9518 | . . . . 5 | |
55 | 53, 54 | sylib 121 | . . . 4 |
56 | 16, 39, 52, 55 | mpjao3dan 1270 | . . 3 |
57 | 19 | adantr 274 | . . . 4 |
58 | oveq1 5749 | . . . . 5 | |
59 | xaddpnf2 9585 | . . . . . 6 | |
60 | 44, 48, 59 | syl2anc 408 | . . . . 5 |
61 | 58, 60 | sylan9eqr 2172 | . . . 4 |
62 | 57, 61 | breqtrrd 3926 | . . 3 |
63 | 32 | simprbi 273 | . . . . . . . 8 |
64 | 31, 63 | syl 14 | . . . . . . 7 |
65 | ge0nemnf 9562 | . . . . . . 7 | |
66 | 34, 64, 65 | syl2anc 408 | . . . . . 6 |
67 | 66 | neneqd 2306 | . . . . 5 |
68 | 67 | pm2.21d 593 | . . . 4 |
69 | 68 | imp 123 | . . 3 |
70 | elxr 9518 | . . . 4 | |
71 | 34, 70 | sylib 121 | . . 3 |
72 | 56, 62, 69, 71 | mpjao3dan 1270 | . 2 |
73 | 1, 2, 10, 72 | isxmetd 12427 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3o 946 w3a 947 wceq 1316 wcel 1465 wne 2285 wral 2393 cvv 2660 class class class wbr 3899 cxp 4507 wfn 5088 wf 5089 cfv 5093 (class class class)co 5742 cr 7587 cc0 7588 caddc 7591 cpnf 7765 cmnf 7766 cxr 7767 cle 7769 cxad 9512 cicc 9629 cxmet 12060 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1re 7682 ax-addrcl 7685 ax-rnegex 7697 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-map 6512 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-xadd 9515 df-icc 9633 df-xmet 12068 |
This theorem is referenced by: (None) |
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