outsideoftr - Mathbox for Scott Fenton

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Theorem outsideoftr 32236

Description: Transitivity law for outsideness. Theorem 6.7 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Assertion**
Ref	Expression
outsideoftr	$/\$ $/\$ $/\$ $/\$ OutsideOf $/\$ OutsideOf OutsideOf

**Proof of Theorem outsideoftr**
Step	Hyp	Ref	Expression
1		simpll 790	. . . . 5 $/\$ $/\$ $/\$
2		simplr 792	. . . . 5 $/\$ $/\$ $/\$
3		simprr 796	. . . . 5 $/\$ $/\$ $/\$
4	1, 2, 3	3jca 1242	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
5		simplr1 1103	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
6		simplr3 1105	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
7		df-3an 1039	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simp1 1061	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		simp3r 1090	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
10		simp2l 1087	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
11		simp2r 1088	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12		simp3l 1089	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
13		simpr2 1068	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14		simpr3 1069	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15	8, 9, 10, 11, 12, 13, 14	btwnexchand 32133	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16	15	orcd 407	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
17	7, 16	sylan2br 493	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
18	17	expr 643	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
19		simprlr 803	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		simprr 796	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21		btwnconn3 32210	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
22	8, 9, 10, 12, 11, 21	syl122anc 1335	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
23	22	adantr 481	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
24	19, 20, 23	mp2and 715	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
25	24	expr 643	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
26	18, 25	jaod 395	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
27	26	expr 643	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
28		simpll2 1101	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
29	28	adantl 482	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	29	necomd 2849	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31		simprlr 803	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		simprr 796	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		btwnconn1 32208	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
34	8, 9, 11, 10, 12, 33	syl122anc 1335	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
35	34	adantr 481	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
36	30, 31, 32, 35	mp3and 1427	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
37	36	expr 643	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
38		df-3an 1039	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		simpr3 1069	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40		simpr2 1068	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	8, 9, 12, 11, 10, 39, 40	btwnexchand 32133	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	olcd 408	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
43	38, 42	sylan2br 493	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
44	43	expr 643	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
45	37, 44	jaod 395	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
46	45	expr 643	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
47	27, 46	jaod 395	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$ $\/$
48	47	imp32 449	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $\/$
49	5, 6, 48	3jca 1242	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$
50	49	exp31 630	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$
51	4, 50	syl5 34	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$
52	51	impd 447	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$
53		broutsideof2 32229	. . . . 5 $/\$ $/\$ $/\$ OutsideOf $/\$ $/\$ $\/$
54	8, 9, 10, 11, 53	syl13anc 1328	. . . 4 $/\$ $/\$ $/\$ $/\$ OutsideOf $/\$ $/\$ $\/$
55		broutsideof2 32229	. . . . 5 $/\$ $/\$ $/\$ OutsideOf $/\$ $/\$ $\/$
56	8, 9, 11, 12, 55	syl13anc 1328	. . . 4 $/\$ $/\$ $/\$ $/\$ OutsideOf $/\$ $/\$ $\/$
57	54, 56	anbi12d 747	. . 3 $/\$ $/\$ $/\$ $/\$ OutsideOf $/\$ OutsideOf $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $\/$
58		df-3an 1039	. . . . 5 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$
59		df-3an 1039	. . . . 5 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$
60	58, 59	anbi12i 733	. . . 4 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $\/$
61		an4 865	. . . 4 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $\/$
62	60, 61	bitr4i 267	. . 3 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
63	57, 62	syl6bb 276	. 2 $/\$ $/\$ $/\$ $/\$ OutsideOf $/\$ OutsideOf $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
64		broutsideof2 32229	. . 3 $/\$ $/\$ $/\$ OutsideOf $/\$ $/\$ $\/$
65	8, 9, 10, 12, 64	syl13anc 1328	. 2 $/\$ $/\$ $/\$ $/\$ OutsideOf $/\$ $/\$ $\/$
66	52, 63, 65	3imtr4d 283	1 $/\$ $/\$ $/\$ $/\$ OutsideOf $/\$ OutsideOf OutsideOf

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384 $/\$ w3a 1037

wcel 1990

wne 2794

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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