Theorem outsideofeq 30897

Description: Uniqueness law for OutsideOf. Analogue of segconeq 30777. (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 1008	. . . . 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 1041	. . . . 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 1045	. . . . 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 1042	. . . . 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 30889	. . . . 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 1270	. . . 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 711	. . 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 1046	. . . . 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 30889	. . . . 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 1270	. . . 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 711	. . 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 717	. 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 1049	. . . . . . 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 1055	. . . . . . 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 535	. . . . . 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 468	. . . . 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 1048	. . . . . 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 468	. . . . 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 1043	. . . . . 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 1044	. . . . . 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 773	. . . . . 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 775	. . . . . 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 30762	. . . . 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 538	. . . 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 772	. . . . . . . 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 773	. . . . . . . 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 775	. . . . . . . 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 30871	. . . . . . . . . 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 1277	. . . . . . . . 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 467	. . . . . . . 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 1367	. . . . . . 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 610	. . . . . 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 773	. . . . . . . . 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 772	. . . . . . . . 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 30793	. . . . . . . 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 775	. . . . . . . 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 30853	. . . . . . 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 610	. . . . . 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 772	. . . . . . . . . 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 773	. . . . . . . . . 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 30793	. . . . . . . . 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 775	. . . . . . . . . 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 30758	. . . . . . . . 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 30853	. . . . . . . 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 2457	. . . . . . 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 610	. . . . . 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 766	. . . . . . . . . . 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 771	. . . . . . . . . . . . 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 468	. . . . . . . . . . . 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 30758	. . . . . . . . . . 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 30853	. . . . . . . . . 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 2457	. . . . . . . . 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 620	. . . . . . . 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 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 ) ) ) /\ ( ( ( 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 771	. . . . . . . . . . 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 468	. . . . . . . . . 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 30853	. . . . . . . . 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 620	. . . . . . . 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 774	. . . . . . . . . 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 2679	. . . . . . . . 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 772	. . . . . . . . 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 773	. . . . . . . . 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 30868	. . . . . . . . . . 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 1277	. . . . . . . . . 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 467	. . . . . . . . 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 1367	. . . . . . . 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 383	. . . . . . 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 610	. . . . . 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 958	. . . . 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 435	. . . 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 473	. . 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 436	. 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 219	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 188   \/ wo 370   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   <.cop 3974   class class class wbr 4402   ` cfv 5582   NNcn 10609   EEcee 24918   Btwn cbtwn 24919  Cgrccgr 24920  OutsideOfcoutsideof 30886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-ee 24921  df-btwn 24922  df-cgr 24923  df-ofs 30750  df-colinear 30806  df-ifs 30807  df-cgr3 30808  df-fs 30809  df-outsideof 30887

This theorem is referenced by:  outsideofeu  30898  outsidele  30899