Theorem outsideofeq 29755

Description: Uniqueness law for OutsideOf. Analog of segconeq 29635. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
outsideofeq	|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( ( AOutsideOf <. X , R >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( AOutsideOf <. Y , R >. /\ <. A , Y >.Cgr <. B , C >. ) ) -> X = Y ) )

Proof of Theorem outsideofeq

Step	Hyp	Ref	Expression
1		simp1 997	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> N e. NN )
2		simp21 1030	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
3		simp32 1034	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> X e. ( EE ` N ) )
4		simp22 1031	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> R e. ( EE ` N ) )
5		broutsideof2 29747	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ X e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( AOutsideOf <. X , R >. <-> ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) ) )
6	1, 2, 3, 4, 5	syl13anc 1231	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( AOutsideOf <. X , R >. <-> ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) ) )
7	6	anbi1d 704	. . 3 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( AOutsideOf <. X , R >. /\ <. A , X >.Cgr <. B , C >. ) <-> ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) ) )
8		simp33 1035	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> Y e. ( EE ` N ) )
9		broutsideof2 29747	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ Y e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( AOutsideOf <. Y , R >. <-> ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) ) )
10	1, 2, 8, 4, 9	syl13anc 1231	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( AOutsideOf <. Y , R >. <-> ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) ) )
11	10	anbi1d 704	. . 3 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( AOutsideOf <. Y , R >. /\ <. A , Y >.Cgr <. B , C >. ) <-> ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) )
12	7, 11	anbi12d 710	. 2 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( ( AOutsideOf <. X , R >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( AOutsideOf <. Y , R >. /\ <. A , Y >.Cgr <. B , C >. ) ) <-> ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) )
13		simpll3 1038	. . . . . . 7 |- ( ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) -> ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) )
14		simprl3 1044	. . . . . . 7 |- ( ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) -> ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) )
15	13, 14	jca 532	. . . . . 6 |- ( ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) -> ( ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) )
16	15	adantl 466	. . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> ( ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) )
17		simpll2 1037	. . . . . 6 |- ( ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) -> R =/= A )
18	17	adantl 466	. . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> R =/= A )
19		simp23 1032	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
20		simp31 1033	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
21		simprlr 764	. . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> <. A , X >.Cgr <. B , C >. )
22		simprrr 766	. . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> <. A , Y >.Cgr <. B , C >. )
23	1, 2, 3, 2, 8, 19, 20, 21, 22	cgrtr3and 29620	. . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> <. A , X >.Cgr <. A , Y >. )
24	16, 18, 23	jca32 535	. . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> ( ( ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) )
25		simprll 763	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X Btwn <. A , R >. )
26		simprlr 764	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> Y Btwn <. A , R >. )
27		simprrr 766	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> <. A , X >.Cgr <. A , Y >. )
28		midofsegid 29729	. . . . . . . . . 10 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) )
29	1, 2, 4, 3, 8, 28	syl122anc 1238	. . . . . . . . 9 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) )
30	29	adantr 465	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) )
31	25, 26, 27, 30	mp3and 1328	. . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X = Y )
32	31	exp32 605	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. ) -> ( ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) ) )
33		simprlr 764	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> Y Btwn <. A , R >. )
34		simprll 763	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> R Btwn <. A , X >. )
35	1, 2, 8, 4, 3, 33, 34	btwnexchand 29651	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> Y Btwn <. A , X >. )
36		simprrr 766	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> <. A , X >.Cgr <. A , Y >. )
37	1, 2, 3, 8, 35, 36	endofsegidand 29711	. . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X = Y )
38	37	exp32 605	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( R Btwn <. A , X >. /\ Y Btwn <. A , R >. ) -> ( ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) ) )
39		simprll 763	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X Btwn <. A , R >. )
40		simprlr 764	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> R Btwn <. A , Y >. )
41	1, 2, 3, 4, 8, 39, 40	btwnexchand 29651	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X Btwn <. A , Y >. )
42		simprrr 766	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> <. A , X >.Cgr <. A , Y >. )
43	1, 2, 3, 2, 8, 42	cgrcomand 29616	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> <. A , Y >.Cgr <. A , X >. )
44	1, 2, 8, 3, 41, 43	endofsegidand 29711	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> Y = X )
45	44	eqcomd 2451	. . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X = Y )
46	45	exp32 605	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) -> ( ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) ) )
47		simprr 757	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ X Btwn <. A , Y >. ) ) -> X Btwn <. A , Y >. )
48		simplrr 762	. . . . . . . . . . . . 13 |- ( ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ X Btwn <. A , Y >. ) -> <. A , X >.Cgr <. A , Y >. )
49	48	adantl 466	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ X Btwn <. A , Y >. ) ) -> <. A , X >.Cgr <. A , Y >. )
50	1, 2, 3, 2, 8, 49	cgrcomand 29616	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ X Btwn <. A , Y >. ) ) -> <. A , Y >.Cgr <. A , X >. )
51	1, 2, 8, 3, 47, 50	endofsegidand 29711	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ X Btwn <. A , Y >. ) ) -> Y = X )
52	51	eqcomd 2451	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ X Btwn <. A , Y >. ) ) -> X = Y )
53	52	expr 615	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> ( X Btwn <. A , Y >. -> X = Y ) )
54		simprr 757	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ Y Btwn <. A , X >. ) ) -> Y Btwn <. A , X >. )
55		simplrr 762	. . . . . . . . . . 11 |- ( ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ Y Btwn <. A , X >. ) -> <. A , X >.Cgr <. A , Y >. )
56	55	adantl 466	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ Y Btwn <. A , X >. ) ) -> <. A , X >.Cgr <. A , Y >. )
57	1, 2, 3, 8, 54, 56	endofsegidand 29711	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ Y Btwn <. A , X >. ) ) -> X = Y )
58	57	expr 615	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> ( Y Btwn <. A , X >. -> X = Y ) )
59		simprrl 765	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> R =/= A )
60	59	necomd 2714	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> A =/= R )
61		simprll 763	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> R Btwn <. A , X >. )
62		simprlr 764	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> R Btwn <. A , Y >. )
63		btwnconn1 29726	. . . . . . . . . . 11 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( A =/= R /\ R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) -> ( X Btwn <. A , Y >. \/ Y Btwn <. A , X >. ) ) )
64	1, 2, 4, 3, 8, 63	syl122anc 1238	. . . . . . . . . 10 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( A =/= R /\ R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) -> ( X Btwn <. A , Y >. \/ Y Btwn <. A , X >. ) ) )
65	64	adantr 465	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> ( ( A =/= R /\ R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) -> ( X Btwn <. A , Y >. \/ Y Btwn <. A , X >. ) ) )
66	60, 61, 62, 65	mp3and 1328	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> ( X Btwn <. A , Y >. \/ Y Btwn <. A , X >. ) )
67	53, 58, 66	mpjaod 381	. . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X = Y )
68	67	exp32 605	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) -> ( ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) ) )
69	32, 38, 46, 68	ccased 947	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) -> ( ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) ) )
70	69	imp32 433	. . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X = Y )
71	24, 70	syldan 470	. . 3 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> X = Y )
72	71	ex 434	. 2 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) -> X = Y ) )
73	12, 72	sylbid 215	1 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( ( AOutsideOf <. X , R >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( AOutsideOf <. Y , R >. /\ <. A , Y >.Cgr <. B , C >. ) ) -> X = Y ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 974   = wceq 1383   e. wcel 1804   =/= wne 2638   <.cop 4020   class class class wbr 4437   ` cfv 5578   NNcn 10542   EEcee 24063   Btwn cbtwn 24064  Cgrccgr 24065  OutsideOfcoutsideof 29744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-n0 10802  df-z 10871  df-uz 11091  df-rp 11230  df-ico 11544  df-icc 11545  df-fz 11682  df-fzo 11804  df-seq 12087  df-exp 12146  df-hash 12385  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-clim 13290  df-sum 13488  df-ee 24066  df-btwn 24067  df-cgr 24068  df-ofs 29608  df-colinear 29664  df-ifs 29665  df-cgr3 29666  df-fs 29667  df-outsideof 29745

This theorem is referenced by:  outsideofeu  29756  outsidele  29757