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Mirrors > Home > ILE Home > Th. List > xblss2ps | Unicode version |
Description: One ball is contained in another if the center-to-center distance is less than the difference of the radii. In this version of blss2 12954 for extended metrics, we have to assume the balls are a finite distance apart, or else will not even be in the infinity ball around . (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
Ref | Expression |
---|---|
xblss2ps.1 | PsMet |
xblss2ps.2 | |
xblss2ps.3 | |
xblss2ps.4 | |
xblss2ps.5 | |
xblss2ps.6 | |
xblss2ps.7 |
Ref | Expression |
---|---|
xblss2ps |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xblss2ps.1 | . . . . . 6 PsMet | |
2 | xblss2ps.2 | . . . . . 6 | |
3 | xblss2ps.4 | . . . . . 6 | |
4 | elblps 12937 | . . . . . 6 PsMet | |
5 | 1, 2, 3, 4 | syl3anc 1227 | . . . . 5 |
6 | 5 | simprbda 381 | . . . 4 |
7 | 1 | adantr 274 | . . . . . . . 8 PsMet |
8 | xblss2ps.3 | . . . . . . . . 9 | |
9 | 8 | adantr 274 | . . . . . . . 8 |
10 | psmetcl 12873 | . . . . . . . 8 PsMet | |
11 | 7, 9, 6, 10 | syl3anc 1227 | . . . . . . 7 |
12 | 11 | adantr 274 | . . . . . 6 |
13 | xblss2ps.6 | . . . . . . . . . 10 | |
14 | 13 | adantr 274 | . . . . . . . . 9 |
15 | 14 | rexrd 7939 | . . . . . . . 8 |
16 | 3 | adantr 274 | . . . . . . . 8 |
17 | 15, 16 | xaddcld 9811 | . . . . . . 7 |
18 | 17 | adantr 274 | . . . . . 6 |
19 | xblss2ps.5 | . . . . . . 7 | |
20 | 19 | ad2antrr 480 | . . . . . 6 |
21 | 2 | adantr 274 | . . . . . . . . . 10 |
22 | psmetcl 12873 | . . . . . . . . . 10 PsMet | |
23 | 7, 21, 6, 22 | syl3anc 1227 | . . . . . . . . 9 |
24 | 15, 23 | xaddcld 9811 | . . . . . . . 8 |
25 | psmettri2 12875 | . . . . . . . . 9 PsMet | |
26 | 7, 21, 9, 6, 25 | syl13anc 1229 | . . . . . . . 8 |
27 | 5 | simplbda 382 | . . . . . . . . 9 |
28 | xltadd2 9804 | . . . . . . . . . 10 | |
29 | 23, 16, 14, 28 | syl3anc 1227 | . . . . . . . . 9 |
30 | 27, 29 | mpbid 146 | . . . . . . . 8 |
31 | 11, 24, 17, 26, 30 | xrlelttrd 9737 | . . . . . . 7 |
32 | 31 | adantr 274 | . . . . . 6 |
33 | 19 | adantr 274 | . . . . . . . . . 10 |
34 | 16 | xnegcld 9782 | . . . . . . . . . 10 |
35 | 33, 34 | xaddcld 9811 | . . . . . . . . 9 |
36 | xblss2ps.7 | . . . . . . . . . 10 | |
37 | 36 | adantr 274 | . . . . . . . . 9 |
38 | xleadd1a 9800 | . . . . . . . . 9 | |
39 | 15, 35, 16, 37, 38 | syl31anc 1230 | . . . . . . . 8 |
40 | 39 | adantr 274 | . . . . . . 7 |
41 | xnpcan 9799 | . . . . . . . 8 | |
42 | 33, 41 | sylan 281 | . . . . . . 7 |
43 | 40, 42 | breqtrd 4002 | . . . . . 6 |
44 | 12, 18, 20, 32, 43 | xrltletrd 9738 | . . . . 5 |
45 | 11 | adantr 274 | . . . . . . 7 |
46 | 13 | ad2antrr 480 | . . . . . . . . 9 |
47 | simpll 519 | . . . . . . . . . 10 | |
48 | simplr 520 | . . . . . . . . . . 11 | |
49 | simpr 109 | . . . . . . . . . . . 12 | |
50 | 49 | oveq2d 5852 | . . . . . . . . . . 11 |
51 | 48, 50 | eleqtrd 2243 | . . . . . . . . . 10 |
52 | xblpnfps 12945 | . . . . . . . . . . . 12 PsMet | |
53 | 1, 2, 52 | syl2anc 409 | . . . . . . . . . . 11 |
54 | 53 | simplbda 382 | . . . . . . . . . 10 |
55 | 47, 51, 54 | syl2anc 409 | . . . . . . . . 9 |
56 | 46, 55 | readdcld 7919 | . . . . . . . 8 |
57 | 56 | rexrd 7939 | . . . . . . 7 |
58 | pnfxr 7942 | . . . . . . . 8 | |
59 | 58 | a1i 9 | . . . . . . 7 |
60 | 1 | ad2antrr 480 | . . . . . . . . 9 PsMet |
61 | 2 | ad2antrr 480 | . . . . . . . . 9 |
62 | 8 | ad2antrr 480 | . . . . . . . . 9 |
63 | 6 | adantr 274 | . . . . . . . . 9 |
64 | 60, 61, 62, 63, 25 | syl13anc 1229 | . . . . . . . 8 |
65 | 46, 55 | rexaddd 9781 | . . . . . . . 8 |
66 | 64, 65 | breqtrd 4002 | . . . . . . 7 |
67 | ltpnf 9707 | . . . . . . . 8 | |
68 | 56, 67 | syl 14 | . . . . . . 7 |
69 | 45, 57, 59, 66, 68 | xrlelttrd 9737 | . . . . . 6 |
70 | 19 | ad2antrr 480 | . . . . . . . 8 |
71 | xrpnfdc 9769 | . . . . . . . 8 DECID | |
72 | 70, 71 | syl 14 | . . . . . . 7 DECID |
73 | 0xr 7936 | . . . . . . . . . . 11 | |
74 | 73 | a1i 9 | . . . . . . . . . 10 |
75 | psmetge0 12878 | . . . . . . . . . . 11 PsMet | |
76 | 7, 21, 9, 75 | syl3anc 1227 | . . . . . . . . . 10 |
77 | 74, 15, 35, 76, 37 | xrletrd 9739 | . . . . . . . . 9 |
78 | ge0nemnf 9751 | . . . . . . . . 9 | |
79 | 35, 77, 78 | syl2anc 409 | . . . . . . . 8 |
80 | 79 | adantr 274 | . . . . . . 7 |
81 | xaddmnf1 9775 | . . . . . . . . . . . 12 | |
82 | 81 | ex 114 | . . . . . . . . . . 11 |
83 | 70, 82 | syl 14 | . . . . . . . . . 10 |
84 | xnegeq 9754 | . . . . . . . . . . . . . 14 | |
85 | 49, 84 | syl 14 | . . . . . . . . . . . . 13 |
86 | xnegpnf 9755 | . . . . . . . . . . . . 13 | |
87 | 85, 86 | eqtrdi 2213 | . . . . . . . . . . . 12 |
88 | 87 | oveq2d 5852 | . . . . . . . . . . 11 |
89 | 88 | eqeq1d 2173 | . . . . . . . . . 10 |
90 | 83, 89 | sylibrd 168 | . . . . . . . . 9 |
91 | 90 | a1d 22 | . . . . . . . 8 DECID |
92 | 91 | necon1ddc 2412 | . . . . . . 7 DECID |
93 | 72, 80, 92 | mp2d 47 | . . . . . 6 |
94 | 69, 93 | breqtrrd 4004 | . . . . 5 |
95 | psmetge0 12878 | . . . . . . . . . . 11 PsMet | |
96 | 7, 21, 6, 95 | syl3anc 1227 | . . . . . . . . . 10 |
97 | 74, 23, 16, 96, 27 | xrlelttrd 9737 | . . . . . . . . 9 |
98 | 74, 16, 97 | xrltled 9726 | . . . . . . . 8 |
99 | ge0nemnf 9751 | . . . . . . . 8 | |
100 | 16, 98, 99 | syl2anc 409 | . . . . . . 7 |
101 | 16, 100 | jca 304 | . . . . . 6 |
102 | xrnemnf 9704 | . . . . . 6 | |
103 | 101, 102 | sylib 121 | . . . . 5 |
104 | 44, 94, 103 | mpjaodan 788 | . . . 4 |
105 | elblps 12937 | . . . . 5 PsMet | |
106 | 7, 9, 33, 105 | syl3anc 1227 | . . . 4 |
107 | 6, 104, 106 | mpbir2and 933 | . . 3 |
108 | 107 | ex 114 | . 2 |
109 | 108 | ssrdv 3143 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 DECID wdc 824 wceq 1342 wcel 2135 wne 2334 wss 3111 class class class wbr 3976 cfv 5182 (class class class)co 5836 cr 7743 cc0 7744 caddc 7747 cpnf 7921 cmnf 7922 cxr 7923 clt 7924 cle 7925 cxne 9696 cxad 9697 PsMetcpsmet 12526 cbl 12529 |
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-stab 821 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-xneg 9699 df-xadd 9700 df-psmet 12534 df-bl 12537 |
This theorem is referenced by: blss2ps 12953 ssblps 12972 |
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