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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 13423 | . . . 4 | |
2 | comet.1 | . . . 4 | |
3 | relelfvdm 5539 | . . . 4 | |
4 | 1, 2, 3 | sylancr 414 | . . 3 |
5 | 4 | elexd 2748 | . 2 |
6 | comet.2 | . . 3 | |
7 | xmetf 13430 | . . . . . 6 | |
8 | 2, 7 | syl 14 | . . . . 5 |
9 | 8 | ffnd 5358 | . . . 4 |
10 | xmetcl 13432 | . . . . . . . 8 | |
11 | xmetge0 13445 | . . . . . . . 8 | |
12 | elxrge0 9949 | . . . . . . . 8 | |
13 | 10, 11, 12 | sylanbrc 417 | . . . . . . 7 |
14 | 13 | 3expb 1204 | . . . . . 6 |
15 | 2, 14 | sylan 283 | . . . . 5 |
16 | 15 | ralrimivva 2557 | . . . 4 |
17 | ffnov 5969 | . . . 4 | |
18 | 9, 16, 17 | sylanbrc 417 | . . 3 |
19 | fco 5373 | . . 3 | |
20 | 6, 18, 19 | syl2anc 411 | . 2 |
21 | opelxpi 4652 | . . . . . 6 | |
22 | fvco3 5579 | . . . . . 6 | |
23 | 8, 21, 22 | syl2an 289 | . . . . 5 |
24 | df-ov 5868 | . . . . 5 | |
25 | df-ov 5868 | . . . . . 6 | |
26 | 25 | fveq2i 5510 | . . . . 5 |
27 | 23, 24, 26 | 3eqtr4g 2233 | . . . 4 |
28 | 27 | eqeq1d 2184 | . . 3 |
29 | fveq2 5507 | . . . . . 6 | |
30 | 29 | eqeq1d 2184 | . . . . 5 |
31 | eqeq1 2182 | . . . . 5 | |
32 | 30, 31 | bibi12d 235 | . . . 4 |
33 | comet.3 | . . . . . 6 | |
34 | 33 | ralrimiva 2548 | . . . . 5 |
35 | 34 | adantr 276 | . . . 4 |
36 | 32, 35, 15 | rspcdva 2844 | . . 3 |
37 | xmeteq0 13439 | . . . . 5 | |
38 | 37 | 3expb 1204 | . . . 4 |
39 | 2, 38 | sylan 283 | . . 3 |
40 | 28, 36, 39 | 3bitrd 214 | . 2 |
41 | 6 | adantr 276 | . . . . 5 |
42 | 15 | 3adantr3 1158 | . . . . 5 |
43 | 41, 42 | ffvelcdmd 5644 | . . . 4 |
44 | 18 | adantr 276 | . . . . . . 7 |
45 | simpr3 1005 | . . . . . . 7 | |
46 | simpr1 1003 | . . . . . . 7 | |
47 | 44, 45, 46 | fovcdmd 6009 | . . . . . 6 |
48 | simpr2 1004 | . . . . . . 7 | |
49 | 44, 45, 48 | fovcdmd 6009 | . . . . . 6 |
50 | ge0xaddcl 9954 | . . . . . 6 | |
51 | 47, 49, 50 | syl2anc 411 | . . . . 5 |
52 | 41, 51 | ffvelcdmd 5644 | . . . 4 |
53 | 41, 47 | ffvelcdmd 5644 | . . . . 5 |
54 | 41, 49 | ffvelcdmd 5644 | . . . . 5 |
55 | 53, 54 | xaddcld 9855 | . . . 4 |
56 | 3anrot 983 | . . . . . . 7 | |
57 | xmettri2 13441 | . . . . . . 7 | |
58 | 56, 57 | sylan2br 288 | . . . . . 6 |
59 | 2, 58 | sylan 283 | . . . . 5 |
60 | comet.4 | . . . . . . . 8 | |
61 | 60 | ralrimivva 2557 | . . . . . . 7 |
62 | 61 | adantr 276 | . . . . . 6 |
63 | breq1 4001 | . . . . . . . 8 | |
64 | 29 | breq1d 4008 | . . . . . . . 8 |
65 | 63, 64 | imbi12d 234 | . . . . . . 7 |
66 | breq2 4002 | . . . . . . . 8 | |
67 | fveq2 5507 | . . . . . . . . 9 | |
68 | 67 | breq2d 4010 | . . . . . . . 8 |
69 | 66, 68 | imbi12d 234 | . . . . . . 7 |
70 | 65, 69 | rspc2va 2853 | . . . . . 6 |
71 | 42, 51, 62, 70 | syl21anc 1237 | . . . . 5 |
72 | 59, 71 | mpd 13 | . . . 4 |
73 | comet.5 | . . . . . . 7 | |
74 | 73 | ralrimivva 2557 | . . . . . 6 |
75 | 74 | adantr 276 | . . . . 5 |
76 | fvoveq1 5888 | . . . . . . 7 | |
77 | fveq2 5507 | . . . . . . . 8 | |
78 | 77 | oveq1d 5880 | . . . . . . 7 |
79 | 76, 78 | breq12d 4011 | . . . . . 6 |
80 | oveq2 5873 | . . . . . . . 8 | |
81 | 80 | fveq2d 5511 | . . . . . . 7 |
82 | fveq2 5507 | . . . . . . . 8 | |
83 | 82 | oveq2d 5881 | . . . . . . 7 |
84 | 81, 83 | breq12d 4011 | . . . . . 6 |
85 | 79, 84 | rspc2va 2853 | . . . . 5 |
86 | 47, 49, 75, 85 | syl21anc 1237 | . . . 4 |
87 | 43, 52, 55, 72, 86 | xrletrd 9783 | . . 3 |
88 | 27 | 3adantr3 1158 | . . 3 |
89 | 8 | adantr 276 | . . . . . 6 |
90 | 45, 46 | opelxpd 4653 | . . . . . 6 |
91 | fvco3 5579 | . . . . . 6 | |
92 | 89, 90, 91 | syl2anc 411 | . . . . 5 |
93 | df-ov 5868 | . . . . 5 | |
94 | df-ov 5868 | . . . . . 6 | |
95 | 94 | fveq2i 5510 | . . . . 5 |
96 | 92, 93, 95 | 3eqtr4g 2233 | . . . 4 |
97 | 45, 48 | opelxpd 4653 | . . . . . 6 |
98 | fvco3 5579 | . . . . . 6 | |
99 | 89, 97, 98 | syl2anc 411 | . . . . 5 |
100 | df-ov 5868 | . . . . 5 | |
101 | df-ov 5868 | . . . . . 6 | |
102 | 101 | fveq2i 5510 | . . . . 5 |
103 | 99, 100, 102 | 3eqtr4g 2233 | . . . 4 |
104 | 96, 103 | oveq12d 5883 | . . 3 |
105 | 87, 88, 104 | 3brtr4d 4030 | . 2 |
106 | 5, 20, 40, 105 | isxmetd 13427 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 w3a 978 wceq 1353 wcel 2146 wral 2453 cop 3592 class class class wbr 3998 cxp 4618 cdm 4620 ccom 4624 wrel 4625 wfn 5203 wf 5204 cfv 5208 (class class class)co 5865 cc0 7786 cpnf 7963 cxr 7965 cle 7967 cxad 9741 cicc 9862 cxmet 13060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-po 4290 df-iso 4291 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-map 6640 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-2 8951 df-xadd 9744 df-icc 9866 df-xmet 13068 |
This theorem is referenced by: bdxmet 13581 |
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