Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 5907 | . . . 4 |
4 | 0xr 7805 | . . . 4 | |
5 | xrletri3 9581 | . . . 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 1181 | . . . . . 6 |
13 | rexadd 9628 | . . . . . . 7 | |
14 | 13 | adantl 275 | . . . . . 6 |
15 | 12, 14 | breqtrrd 3951 | . . . . 5 |
16 | 15 | anassrs 397 | . . . 4 |
17 | 3 | 3adantr3 1142 | . . . . . . 7 |
18 | pnfge 9568 | . . . . . . 7 | |
19 | 17, 18 | syl 14 | . . . . . 6 |
20 | 19 | ad2antrr 479 | . . . . 5 |
21 | oveq2 5775 | . . . . . 6 | |
22 | 2 | ffnd 5268 | . . . . . . . . . . 11 |
23 | elxrge0 9754 | . . . . . . . . . . . . 13 | |
24 | 3, 7, 23 | sylanbrc 413 | . . . . . . . . . . . 12 |
25 | 24 | ralrimivva 2512 | . . . . . . . . . . 11 |
26 | ffnov 5868 | . . . . . . . . . . 11 | |
27 | 22, 25, 26 | sylanbrc 413 | . . . . . . . . . 10 |
28 | 27 | adantr 274 | . . . . . . . . 9 |
29 | simpr3 989 | . . . . . . . . 9 | |
30 | simpr1 987 | . . . . . . . . 9 | |
31 | 28, 29, 30 | fovrnd 5908 | . . . . . . . 8 |
32 | elxrge0 9754 | . . . . . . . . 9 | |
33 | 32 | simplbi 272 | . . . . . . . 8 |
34 | 31, 33 | syl 14 | . . . . . . 7 |
35 | renemnf 7807 | . . . . . . 7 | |
36 | xaddpnf1 9622 | . . . . . . 7 | |
37 | 34, 35, 36 | syl2an 287 | . . . . . 6 |
38 | 21, 37 | sylan9eqr 2192 | . . . . 5 |
39 | 20, 38 | breqtrrd 3951 | . . . 4 |
40 | simpr2 988 | . . . . . . . . . . 11 | |
41 | 28, 29, 40 | fovrnd 5908 | . . . . . . . . . 10 |
42 | elxrge0 9754 | . . . . . . . . . . 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 9600 | . . . . . . . . 9 | |
48 | 44, 46, 47 | syl2anc 408 | . . . . . . . 8 |
49 | 48 | neneqd 2327 | . . . . . . 7 |
50 | 49 | pm2.21d 608 | . . . . . 6 |
51 | 50 | adantr 274 | . . . . 5 |
52 | 51 | imp 123 | . . . 4 |
53 | 44 | adantr 274 | . . . . 5 |
54 | elxr 9556 | . . . . 5 | |
55 | 53, 54 | sylib 121 | . . . 4 |
56 | 16, 39, 52, 55 | mpjao3dan 1285 | . . 3 |
57 | 19 | adantr 274 | . . . 4 |
58 | oveq1 5774 | . . . . 5 | |
59 | xaddpnf2 9623 | . . . . . 6 | |
60 | 44, 48, 59 | syl2anc 408 | . . . . 5 |
61 | 58, 60 | sylan9eqr 2192 | . . . 4 |
62 | 57, 61 | breqtrrd 3951 | . . 3 |
63 | 32 | simprbi 273 | . . . . . . . 8 |
64 | 31, 63 | syl 14 | . . . . . . 7 |
65 | ge0nemnf 9600 | . . . . . . 7 | |
66 | 34, 64, 65 | syl2anc 408 | . . . . . 6 |
67 | 66 | neneqd 2327 | . . . . 5 |
68 | 67 | pm2.21d 608 | . . . 4 |
69 | 68 | imp 123 | . . 3 |
70 | elxr 9556 | . . . 4 | |
71 | 34, 70 | sylib 121 | . . 3 |
72 | 56, 62, 69, 71 | mpjao3dan 1285 | . 2 |
73 | 1, 2, 10, 72 | isxmetd 12505 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3o 961 w3a 962 wceq 1331 wcel 1480 wne 2306 wral 2414 cvv 2681 class class class wbr 3924 cxp 4532 wfn 5113 wf 5114 cfv 5118 (class class class)co 5767 cr 7612 cc0 7613 caddc 7616 cpnf 7790 cmnf 7791 cxr 7792 cle 7794 cxad 9550 cicc 9667 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-rnegex 7722 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 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-po 4213 df-iso 4214 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-icc 9671 df-xmet 12146 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |