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Mirrors > Home > MPE Home > Th. List > Mathboxes > broutsideof3 | Structured version Visualization version Unicode version |
Description: Characterization of outsideness in terms of relationship to a fourth point. Theorem 6.3 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
broutsideof3 | OutsideOf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | broutsideof2 32229 | . 2 OutsideOf | |
2 | simpl 473 | . . . . . . . . . 10 | |
3 | simpr3 1069 | . . . . . . . . . 10 | |
4 | simpr1 1067 | . . . . . . . . . 10 | |
5 | btwndiff 32134 | . . . . . . . . . 10 | |
6 | 2, 3, 4, 5 | syl3anc 1326 | . . . . . . . . 9 |
7 | 6 | adantr 481 | . . . . . . . 8 |
8 | df-3an 1039 | . . . . . . . . . . . 12 | |
9 | 3anass 1042 | . . . . . . . . . . . 12 | |
10 | simpr3 1069 | . . . . . . . . . . . . . 14 | |
11 | 10 | necomd 2849 | . . . . . . . . . . . . 13 |
12 | simp1 1061 | . . . . . . . . . . . . . 14 | |
13 | simp23 1096 | . . . . . . . . . . . . . 14 | |
14 | simp22 1095 | . . . . . . . . . . . . . 14 | |
15 | simp21 1094 | . . . . . . . . . . . . . 14 | |
16 | simp3 1063 | . . . . . . . . . . . . . 14 | |
17 | simpr1r 1119 | . . . . . . . . . . . . . . 15 | |
18 | 12, 14, 15, 13, 17 | btwncomand 32122 | . . . . . . . . . . . . . 14 |
19 | simpr2 1068 | . . . . . . . . . . . . . 14 | |
20 | 12, 13, 14, 15, 16, 18, 19 | btwnexch3and 32128 | . . . . . . . . . . . . 13 |
21 | 11, 20, 19 | 3jca 1242 | . . . . . . . . . . . 12 |
22 | 8, 9, 21 | syl2anbr 497 | . . . . . . . . . . 11 |
23 | 22 | expr 643 | . . . . . . . . . 10 |
24 | 23 | an32s 846 | . . . . . . . . 9 |
25 | 24 | reximdva 3017 | . . . . . . . 8 |
26 | 7, 25 | mpd 15 | . . . . . . 7 |
27 | 26 | expr 643 | . . . . . 6 |
28 | simpr2 1068 | . . . . . . . . . 10 | |
29 | btwndiff 32134 | . . . . . . . . . 10 | |
30 | 2, 28, 4, 29 | syl3anc 1326 | . . . . . . . . 9 |
31 | 30 | adantr 481 | . . . . . . . 8 |
32 | 3anass 1042 | . . . . . . . . . . . 12 | |
33 | simpr3 1069 | . . . . . . . . . . . . . 14 | |
34 | 33 | necomd 2849 | . . . . . . . . . . . . 13 |
35 | simpr2 1068 | . . . . . . . . . . . . 13 | |
36 | simpr1r 1119 | . . . . . . . . . . . . . . 15 | |
37 | 12, 13, 15, 14, 36 | btwncomand 32122 | . . . . . . . . . . . . . 14 |
38 | 12, 14, 13, 15, 16, 37, 35 | btwnexch3and 32128 | . . . . . . . . . . . . 13 |
39 | 34, 35, 38 | 3jca 1242 | . . . . . . . . . . . 12 |
40 | 8, 32, 39 | syl2anbr 497 | . . . . . . . . . . 11 |
41 | 40 | expr 643 | . . . . . . . . . 10 |
42 | 41 | an32s 846 | . . . . . . . . 9 |
43 | 42 | reximdva 3017 | . . . . . . . 8 |
44 | 31, 43 | mpd 15 | . . . . . . 7 |
45 | 44 | expr 643 | . . . . . 6 |
46 | 27, 45 | jaod 395 | . . . . 5 |
47 | simprr1 1109 | . . . . . . . . 9 | |
48 | simpll 790 | . . . . . . . . . 10 | |
49 | simplr1 1103 | . . . . . . . . . 10 | |
50 | simplr2 1104 | . . . . . . . . . 10 | |
51 | simpr 477 | . . . . . . . . . 10 | |
52 | simprr2 1110 | . . . . . . . . . 10 | |
53 | 48, 49, 50, 51, 52 | btwncomand 32122 | . . . . . . . . 9 |
54 | simplr3 1105 | . . . . . . . . . 10 | |
55 | simprr3 1111 | . . . . . . . . . 10 | |
56 | 48, 49, 54, 51, 55 | btwncomand 32122 | . . . . . . . . 9 |
57 | btwnconn2 32209 | . . . . . . . . . . 11 | |
58 | 48, 51, 49, 50, 54, 57 | syl122anc 1335 | . . . . . . . . . 10 |
59 | 58 | adantr 481 | . . . . . . . . 9 |
60 | 47, 53, 56, 59 | mp3and 1427 | . . . . . . . 8 |
61 | 60 | expr 643 | . . . . . . 7 |
62 | 61 | an32s 846 | . . . . . 6 |
63 | 62 | rexlimdva 3031 | . . . . 5 |
64 | 46, 63 | impbid 202 | . . . 4 |
65 | 64 | pm5.32da 673 | . . 3 |
66 | df-3an 1039 | . . 3 | |
67 | df-3an 1039 | . . 3 | |
68 | 65, 66, 67 | 3bitr4g 303 | . 2 |
69 | 1, 68 | bitrd 268 | 1 OutsideOf |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wo 383 wa 384 w3a 1037 wcel 1990 wne 2794 wrex 2913 cop 4183 class class class wbr 4653 cfv 5888 cn 11020 cee 25768 cbtwn 25769 OutsideOfcoutsideof 32226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-ee 25771 df-btwn 25772 df-cgr 25773 df-ofs 32090 df-colinear 32146 df-ifs 32147 df-cgr3 32148 df-fs 32149 df-outsideof 32227 |
This theorem is referenced by: (None) |
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