Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > comet | Unicode version |
Description: The composition of an extended metric with a monotonic subadditive function is an extended metric. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
comet.1 | |
comet.2 | |
comet.3 | |
comet.4 | |
comet.5 |
Ref | Expression |
---|---|
comet |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmetrel 12423 | . . . 4 | |
2 | comet.1 | . . . 4 | |
3 | relelfvdm 5421 | . . . 4 | |
4 | 1, 2, 3 | sylancr 410 | . . 3 |
5 | 4 | elexd 2673 | . 2 |
6 | comet.2 | . . 3 | |
7 | xmetf 12430 | . . . . . 6 | |
8 | 2, 7 | syl 14 | . . . . 5 |
9 | 8 | ffnd 5243 | . . . 4 |
10 | xmetcl 12432 | . . . . . . . 8 | |
11 | xmetge0 12445 | . . . . . . . 8 | |
12 | elxrge0 9716 | . . . . . . . 8 | |
13 | 10, 11, 12 | sylanbrc 413 | . . . . . . 7 |
14 | 13 | 3expb 1167 | . . . . . 6 |
15 | 2, 14 | sylan 281 | . . . . 5 |
16 | 15 | ralrimivva 2491 | . . . 4 |
17 | ffnov 5843 | . . . 4 | |
18 | 9, 16, 17 | sylanbrc 413 | . . 3 |
19 | fco 5258 | . . 3 | |
20 | 6, 18, 19 | syl2anc 408 | . 2 |
21 | opelxpi 4541 | . . . . . 6 | |
22 | fvco3 5460 | . . . . . 6 | |
23 | 8, 21, 22 | syl2an 287 | . . . . 5 |
24 | df-ov 5745 | . . . . 5 | |
25 | df-ov 5745 | . . . . . 6 | |
26 | 25 | fveq2i 5392 | . . . . 5 |
27 | 23, 24, 26 | 3eqtr4g 2175 | . . . 4 |
28 | 27 | eqeq1d 2126 | . . 3 |
29 | fveq2 5389 | . . . . . 6 | |
30 | 29 | eqeq1d 2126 | . . . . 5 |
31 | eqeq1 2124 | . . . . 5 | |
32 | 30, 31 | bibi12d 234 | . . . 4 |
33 | comet.3 | . . . . . 6 | |
34 | 33 | ralrimiva 2482 | . . . . 5 |
35 | 34 | adantr 274 | . . . 4 |
36 | 32, 35, 15 | rspcdva 2768 | . . 3 |
37 | xmeteq0 12439 | . . . . 5 | |
38 | 37 | 3expb 1167 | . . . 4 |
39 | 2, 38 | sylan 281 | . . 3 |
40 | 28, 36, 39 | 3bitrd 213 | . 2 |
41 | 6 | adantr 274 | . . . . 5 |
42 | 15 | 3adantr3 1127 | . . . . 5 |
43 | 41, 42 | ffvelrnd 5524 | . . . 4 |
44 | 18 | adantr 274 | . . . . . . 7 |
45 | simpr3 974 | . . . . . . 7 | |
46 | simpr1 972 | . . . . . . 7 | |
47 | 44, 45, 46 | fovrnd 5883 | . . . . . 6 |
48 | simpr2 973 | . . . . . . 7 | |
49 | 44, 45, 48 | fovrnd 5883 | . . . . . 6 |
50 | ge0xaddcl 9721 | . . . . . 6 | |
51 | 47, 49, 50 | syl2anc 408 | . . . . 5 |
52 | 41, 51 | ffvelrnd 5524 | . . . 4 |
53 | 41, 47 | ffvelrnd 5524 | . . . . 5 |
54 | 41, 49 | ffvelrnd 5524 | . . . . 5 |
55 | 53, 54 | xaddcld 9622 | . . . 4 |
56 | 3anrot 952 | . . . . . . 7 | |
57 | xmettri2 12441 | . . . . . . 7 | |
58 | 56, 57 | sylan2br 286 | . . . . . 6 |
59 | 2, 58 | sylan 281 | . . . . 5 |
60 | comet.4 | . . . . . . . 8 | |
61 | 60 | ralrimivva 2491 | . . . . . . 7 |
62 | 61 | adantr 274 | . . . . . 6 |
63 | breq1 3902 | . . . . . . . 8 | |
64 | 29 | breq1d 3909 | . . . . . . . 8 |
65 | 63, 64 | imbi12d 233 | . . . . . . 7 |
66 | breq2 3903 | . . . . . . . 8 | |
67 | fveq2 5389 | . . . . . . . . 9 | |
68 | 67 | breq2d 3911 | . . . . . . . 8 |
69 | 66, 68 | imbi12d 233 | . . . . . . 7 |
70 | 65, 69 | rspc2va 2777 | . . . . . 6 |
71 | 42, 51, 62, 70 | syl21anc 1200 | . . . . 5 |
72 | 59, 71 | mpd 13 | . . . 4 |
73 | comet.5 | . . . . . . 7 | |
74 | 73 | ralrimivva 2491 | . . . . . 6 |
75 | 74 | adantr 274 | . . . . 5 |
76 | fvoveq1 5765 | . . . . . . 7 | |
77 | fveq2 5389 | . . . . . . . 8 | |
78 | 77 | oveq1d 5757 | . . . . . . 7 |
79 | 76, 78 | breq12d 3912 | . . . . . 6 |
80 | oveq2 5750 | . . . . . . . 8 | |
81 | 80 | fveq2d 5393 | . . . . . . 7 |
82 | fveq2 5389 | . . . . . . . 8 | |
83 | 82 | oveq2d 5758 | . . . . . . 7 |
84 | 81, 83 | breq12d 3912 | . . . . . 6 |
85 | 79, 84 | rspc2va 2777 | . . . . 5 |
86 | 47, 49, 75, 85 | syl21anc 1200 | . . . 4 |
87 | 43, 52, 55, 72, 86 | xrletrd 9550 | . . 3 |
88 | 27 | 3adantr3 1127 | . . 3 |
89 | 8 | adantr 274 | . . . . . 6 |
90 | 45, 46 | opelxpd 4542 | . . . . . 6 |
91 | fvco3 5460 | . . . . . 6 | |
92 | 89, 90, 91 | syl2anc 408 | . . . . 5 |
93 | df-ov 5745 | . . . . 5 | |
94 | df-ov 5745 | . . . . . 6 | |
95 | 94 | fveq2i 5392 | . . . . 5 |
96 | 92, 93, 95 | 3eqtr4g 2175 | . . . 4 |
97 | 45, 48 | opelxpd 4542 | . . . . . 6 |
98 | fvco3 5460 | . . . . . 6 | |
99 | 89, 97, 98 | syl2anc 408 | . . . . 5 |
100 | df-ov 5745 | . . . . 5 | |
101 | df-ov 5745 | . . . . . 6 | |
102 | 101 | fveq2i 5392 | . . . . 5 |
103 | 99, 100, 102 | 3eqtr4g 2175 | . . . 4 |
104 | 96, 103 | oveq12d 5760 | . . 3 |
105 | 87, 88, 104 | 3brtr4d 3930 | . 2 |
106 | 5, 20, 40, 105 | isxmetd 12427 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 947 wceq 1316 wcel 1465 wral 2393 cop 3500 class class class wbr 3899 cxp 4507 cdm 4509 ccom 4513 wrel 4514 wfn 5088 wf 5089 cfv 5093 (class class class)co 5742 cc0 7588 cpnf 7765 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-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 |
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-reu 2400 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-riota 5698 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-sub 7903 df-neg 7904 df-2 8743 df-xadd 9515 df-icc 9633 df-xmet 12068 |
This theorem is referenced by: bdxmet 12581 |
Copyright terms: Public domain | W3C validator |