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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 12884 | . . . 4 | |
2 | comet.1 | . . . 4 | |
3 | relelfvdm 5512 | . . . 4 | |
4 | 1, 2, 3 | sylancr 411 | . . 3 |
5 | 4 | elexd 2734 | . 2 |
6 | comet.2 | . . 3 | |
7 | xmetf 12891 | . . . . . 6 | |
8 | 2, 7 | syl 14 | . . . . 5 |
9 | 8 | ffnd 5332 | . . . 4 |
10 | xmetcl 12893 | . . . . . . . 8 | |
11 | xmetge0 12906 | . . . . . . . 8 | |
12 | elxrge0 9905 | . . . . . . . 8 | |
13 | 10, 11, 12 | sylanbrc 414 | . . . . . . 7 |
14 | 13 | 3expb 1193 | . . . . . 6 |
15 | 2, 14 | sylan 281 | . . . . 5 |
16 | 15 | ralrimivva 2546 | . . . 4 |
17 | ffnov 5937 | . . . 4 | |
18 | 9, 16, 17 | sylanbrc 414 | . . 3 |
19 | fco 5347 | . . 3 | |
20 | 6, 18, 19 | syl2anc 409 | . 2 |
21 | opelxpi 4630 | . . . . . 6 | |
22 | fvco3 5551 | . . . . . 6 | |
23 | 8, 21, 22 | syl2an 287 | . . . . 5 |
24 | df-ov 5839 | . . . . 5 | |
25 | df-ov 5839 | . . . . . 6 | |
26 | 25 | fveq2i 5483 | . . . . 5 |
27 | 23, 24, 26 | 3eqtr4g 2222 | . . . 4 |
28 | 27 | eqeq1d 2173 | . . 3 |
29 | fveq2 5480 | . . . . . 6 | |
30 | 29 | eqeq1d 2173 | . . . . 5 |
31 | eqeq1 2171 | . . . . 5 | |
32 | 30, 31 | bibi12d 234 | . . . 4 |
33 | comet.3 | . . . . . 6 | |
34 | 33 | ralrimiva 2537 | . . . . 5 |
35 | 34 | adantr 274 | . . . 4 |
36 | 32, 35, 15 | rspcdva 2830 | . . 3 |
37 | xmeteq0 12900 | . . . . 5 | |
38 | 37 | 3expb 1193 | . . . 4 |
39 | 2, 38 | sylan 281 | . . 3 |
40 | 28, 36, 39 | 3bitrd 213 | . 2 |
41 | 6 | adantr 274 | . . . . 5 |
42 | 15 | 3adantr3 1147 | . . . . 5 |
43 | 41, 42 | ffvelrnd 5615 | . . . 4 |
44 | 18 | adantr 274 | . . . . . . 7 |
45 | simpr3 994 | . . . . . . 7 | |
46 | simpr1 992 | . . . . . . 7 | |
47 | 44, 45, 46 | fovrnd 5977 | . . . . . 6 |
48 | simpr2 993 | . . . . . . 7 | |
49 | 44, 45, 48 | fovrnd 5977 | . . . . . 6 |
50 | ge0xaddcl 9910 | . . . . . 6 | |
51 | 47, 49, 50 | syl2anc 409 | . . . . 5 |
52 | 41, 51 | ffvelrnd 5615 | . . . 4 |
53 | 41, 47 | ffvelrnd 5615 | . . . . 5 |
54 | 41, 49 | ffvelrnd 5615 | . . . . 5 |
55 | 53, 54 | xaddcld 9811 | . . . 4 |
56 | 3anrot 972 | . . . . . . 7 | |
57 | xmettri2 12902 | . . . . . . 7 | |
58 | 56, 57 | sylan2br 286 | . . . . . 6 |
59 | 2, 58 | sylan 281 | . . . . 5 |
60 | comet.4 | . . . . . . . 8 | |
61 | 60 | ralrimivva 2546 | . . . . . . 7 |
62 | 61 | adantr 274 | . . . . . 6 |
63 | breq1 3979 | . . . . . . . 8 | |
64 | 29 | breq1d 3986 | . . . . . . . 8 |
65 | 63, 64 | imbi12d 233 | . . . . . . 7 |
66 | breq2 3980 | . . . . . . . 8 | |
67 | fveq2 5480 | . . . . . . . . 9 | |
68 | 67 | breq2d 3988 | . . . . . . . 8 |
69 | 66, 68 | imbi12d 233 | . . . . . . 7 |
70 | 65, 69 | rspc2va 2839 | . . . . . 6 |
71 | 42, 51, 62, 70 | syl21anc 1226 | . . . . 5 |
72 | 59, 71 | mpd 13 | . . . 4 |
73 | comet.5 | . . . . . . 7 | |
74 | 73 | ralrimivva 2546 | . . . . . 6 |
75 | 74 | adantr 274 | . . . . 5 |
76 | fvoveq1 5859 | . . . . . . 7 | |
77 | fveq2 5480 | . . . . . . . 8 | |
78 | 77 | oveq1d 5851 | . . . . . . 7 |
79 | 76, 78 | breq12d 3989 | . . . . . 6 |
80 | oveq2 5844 | . . . . . . . 8 | |
81 | 80 | fveq2d 5484 | . . . . . . 7 |
82 | fveq2 5480 | . . . . . . . 8 | |
83 | 82 | oveq2d 5852 | . . . . . . 7 |
84 | 81, 83 | breq12d 3989 | . . . . . 6 |
85 | 79, 84 | rspc2va 2839 | . . . . 5 |
86 | 47, 49, 75, 85 | syl21anc 1226 | . . . 4 |
87 | 43, 52, 55, 72, 86 | xrletrd 9739 | . . 3 |
88 | 27 | 3adantr3 1147 | . . 3 |
89 | 8 | adantr 274 | . . . . . 6 |
90 | 45, 46 | opelxpd 4631 | . . . . . 6 |
91 | fvco3 5551 | . . . . . 6 | |
92 | 89, 90, 91 | syl2anc 409 | . . . . 5 |
93 | df-ov 5839 | . . . . 5 | |
94 | df-ov 5839 | . . . . . 6 | |
95 | 94 | fveq2i 5483 | . . . . 5 |
96 | 92, 93, 95 | 3eqtr4g 2222 | . . . 4 |
97 | 45, 48 | opelxpd 4631 | . . . . . 6 |
98 | fvco3 5551 | . . . . . 6 | |
99 | 89, 97, 98 | syl2anc 409 | . . . . 5 |
100 | df-ov 5839 | . . . . 5 | |
101 | df-ov 5839 | . . . . . 6 | |
102 | 101 | fveq2i 5483 | . . . . 5 |
103 | 99, 100, 102 | 3eqtr4g 2222 | . . . 4 |
104 | 96, 103 | oveq12d 5854 | . . 3 |
105 | 87, 88, 104 | 3brtr4d 4008 | . 2 |
106 | 5, 20, 40, 105 | isxmetd 12888 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 wral 2442 cop 3573 class class class wbr 3976 cxp 4596 cdm 4598 ccom 4602 wrel 4603 wfn 5177 wf 5178 cfv 5182 (class class class)co 5836 cc0 7744 cpnf 7921 cxr 7923 cle 7925 cxad 9697 cicc 9818 cxmet 12521 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-map 6607 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-2 8907 df-xadd 9700 df-icc 9822 df-xmet 12529 |
This theorem is referenced by: bdxmet 13042 |
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