Theorem segcon2 32212

Description: Generalization of axsegcon 25807. This time, we generate an endpoint for a segment on the ray Q A congruent to B C and starting at Q, as opposed to axsegcon 25807, where the segment starts at A (Contributed by Scott Fenton, 14-Oct-2013.) (Removed unneeded inequality, 15-Oct-2013.)

Assertion

Ref	Expression
segcon2	|- ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> E. x e. ( EE ` N ) ( ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) /\ <. Q , x >.Cgr <. B , C >. ) )

Distinct variable groups:   x, Q   x, N   x, A   x, B   x, C

Proof of Theorem segcon2

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4656	. . . . 5 |- ( A = Q -> ( A Btwn <. Q , x >. <-> Q Btwn <. Q , x >. ) )
2	1	orbi1d 739	. . . 4 |- ( A = Q -> ( ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) <-> ( Q Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) ) )
3	2	anbi1d 741	. . 3 |- ( A = Q -> ( ( ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) /\ <. Q , x >.Cgr <. B , C >. ) <-> ( ( Q Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) /\ <. Q , x >.Cgr <. B , C >. ) ) )
4	3	rexbidv 3052	. 2 |- ( A = Q -> ( E. x e. ( EE ` N ) ( ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) /\ <. Q , x >.Cgr <. B , C >. ) <-> E. x e. ( EE ` N ) ( ( Q Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) /\ <. Q , x >.Cgr <. B , C >. ) ) )
5		simp1 1061	. . . . 5 |- ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> N e. NN )
6		simp2 1062	. . . . . 6 |- ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) )
7	6	ancomd 467	. . . . 5 |- ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A e. ( EE ` N ) /\ Q e. ( EE ` N ) ) )
8		axsegcon 25807	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ Q e. ( EE ` N ) ) /\ ( A e. ( EE ` N ) /\ Q e. ( EE ` N ) ) ) -> E. a e. ( EE ` N ) ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) )
9	5, 7, 7, 8	syl3anc 1326	. . . 4 |- ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> E. a e. ( EE ` N ) ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) )
10	9	adantr 481	. . 3 |- ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ A =/= Q ) -> E. a e. ( EE ` N ) ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) )
11		simpl1 1064	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ a e. ( EE ` N ) ) -> N e. NN )
12		simpr 477	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ a e. ( EE ` N ) ) -> a e. ( EE ` N ) )
13		simpl2l 1114	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ a e. ( EE ` N ) ) -> Q e. ( EE ` N ) )
14		simpl3 1066	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ a e. ( EE ` N ) ) -> ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) )
15		axsegcon 25807	. . . . . . . . 9 |- ( ( N e. NN /\ ( a e. ( EE ` N ) /\ Q e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> E. x e. ( EE ` N ) ( Q Btwn <. a , x >. /\ <. Q , x >.Cgr <. B , C >. ) )
16	11, 12, 13, 14, 15	syl121anc 1331	. . . . . . . 8 |- ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ a e. ( EE ` N ) ) -> E. x e. ( EE ` N ) ( Q Btwn <. a , x >. /\ <. Q , x >.Cgr <. B , C >. ) )
17	16	adantr 481	. . . . . . 7 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ a e. ( EE ` N ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) ) ) -> E. x e. ( EE ` N ) ( Q Btwn <. a , x >. /\ <. Q , x >.Cgr <. B , C >. ) )
18		anass 681	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ a e. ( EE ` N ) ) /\ x e. ( EE ` N ) ) <-> ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) )
19		df-3an 1039	. . . . . . . . . . . . 13 |- ( ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) /\ Q Btwn <. a , x >. ) <-> ( ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) ) /\ Q Btwn <. a , x >. ) )
20		simpr1 1067	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) /\ Q Btwn <. a , x >. ) ) -> A =/= Q )
21		simpr2r 1121	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) /\ Q Btwn <. a , x >. ) ) -> <. Q , a >.Cgr <. A , Q >. )
22		simpl1 1064	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) -> N e. NN )
23		simpl2l 1114	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) -> Q e. ( EE ` N ) )
24		simprl 794	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) -> a e. ( EE ` N ) )
25		simpl2r 1115	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
26		cgrdegen 32111	. . . . . . . . . . . . . . . . . . . 20 |- ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ a e. ( EE ` N ) ) /\ ( A e. ( EE ` N ) /\ Q e. ( EE ` N ) ) ) -> ( <. Q , a >.Cgr <. A , Q >. -> ( Q = a <-> A = Q ) ) )
27	22, 23, 24, 25, 23, 26	syl122anc 1335	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) -> ( <. Q , a >.Cgr <. A , Q >. -> ( Q = a <-> A = Q ) ) )
28	27	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) /\ Q Btwn <. a , x >. ) ) -> ( <. Q , a >.Cgr <. A , Q >. -> ( Q = a <-> A = Q ) ) )
29	21, 28	mpd 15	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) /\ Q Btwn <. a , x >. ) ) -> ( Q = a <-> A = Q ) )
30	29	necon3bid 2838	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) /\ Q Btwn <. a , x >. ) ) -> ( Q =/= a <-> A =/= Q ) )
31	20, 30	mpbird 247	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) /\ Q Btwn <. a , x >. ) ) -> Q =/= a )
32	31	necomd 2849	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) /\ Q Btwn <. a , x >. ) ) -> a =/= Q )
33		simpr2l 1120	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) /\ Q Btwn <. a , x >. ) ) -> Q Btwn <. A , a >. )
34	22, 23, 25, 24, 33	btwncomand 32122	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) /\ Q Btwn <. a , x >. ) ) -> Q Btwn <. a , A >. )
35		simpr3 1069	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) /\ Q Btwn <. a , x >. ) ) -> Q Btwn <. a , x >. )
36		simprr 796	. . . . . . . . . . . . . . . 16 |- ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) -> x e. ( EE ` N ) )
37		btwnconn2 32209	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN /\ ( a e. ( EE ` N ) /\ Q e. ( EE ` N ) ) /\ ( A e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) -> ( ( a =/= Q /\ Q Btwn <. a , A >. /\ Q Btwn <. a , x >. ) -> ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) ) )
38	22, 24, 23, 25, 36, 37	syl122anc 1335	. . . . . . . . . . . . . . 15 |- ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) -> ( ( a =/= Q /\ Q Btwn <. a , A >. /\ Q Btwn <. a , x >. ) -> ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) ) )
39	38	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) /\ Q Btwn <. a , x >. ) ) -> ( ( a =/= Q /\ Q Btwn <. a , A >. /\ Q Btwn <. a , x >. ) -> ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) ) )
40	32, 34, 35, 39	mp3and 1427	. . . . . . . . . . . . 13 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) /\ Q Btwn <. a , x >. ) ) -> ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) )
41	19, 40	sylan2br 493	. . . . . . . . . . . 12 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) /\ ( ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) ) /\ Q Btwn <. a , x >. ) ) -> ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) )
42	41	expr 643	. . . . . . . . . . 11 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) ) ) -> ( Q Btwn <. a , x >. -> ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) ) )
43	42	anim1d 588	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( a e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) ) ) -> ( ( Q Btwn <. a , x >. /\ <. Q , x >.Cgr <. B , C >. ) -> ( ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) /\ <. Q , x >.Cgr <. B , C >. ) ) )
44	18, 43	sylanb 489	. . . . . . . . 9 |- ( ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ a e. ( EE ` N ) ) /\ x e. ( EE ` N ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) ) ) -> ( ( Q Btwn <. a , x >. /\ <. Q , x >.Cgr <. B , C >. ) -> ( ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) /\ <. Q , x >.Cgr <. B , C >. ) ) )
45	44	an32s 846	. . . . . . . 8 |- ( ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ a e. ( EE ` N ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) ) ) /\ x e. ( EE ` N ) ) -> ( ( Q Btwn <. a , x >. /\ <. Q , x >.Cgr <. B , C >. ) -> ( ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) /\ <. Q , x >.Cgr <. B , C >. ) ) )
46	45	reximdva 3017	. . . . . . 7 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ a e. ( EE ` N ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) ) ) -> ( E. x e. ( EE ` N ) ( Q Btwn <. a , x >. /\ <. Q , x >.Cgr <. B , C >. ) -> E. x e. ( EE ` N ) ( ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) /\ <. Q , x >.Cgr <. B , C >. ) ) )
47	17, 46	mpd 15	. . . . . 6 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ a e. ( EE ` N ) ) /\ ( A =/= Q /\ ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) ) ) -> E. x e. ( EE ` N ) ( ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) /\ <. Q , x >.Cgr <. B , C >. ) )
48	47	expr 643	. . . . 5 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ a e. ( EE ` N ) ) /\ A =/= Q ) -> ( ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) -> E. x e. ( EE ` N ) ( ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) /\ <. Q , x >.Cgr <. B , C >. ) ) )
49	48	an32s 846	. . . 4 |- ( ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ A =/= Q ) /\ a e. ( EE ` N ) ) -> ( ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) -> E. x e. ( EE ` N ) ( ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) /\ <. Q , x >.Cgr <. B , C >. ) ) )
50	49	rexlimdva 3031	. . 3 |- ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ A =/= Q ) -> ( E. a e. ( EE ` N ) ( Q Btwn <. A , a >. /\ <. Q , a >.Cgr <. A , Q >. ) -> E. x e. ( EE ` N ) ( ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) /\ <. Q , x >.Cgr <. B , C >. ) ) )
51	10, 50	mpd 15	. 2 |- ( ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ A =/= Q ) -> E. x e. ( EE ` N ) ( ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) /\ <. Q , x >.Cgr <. B , C >. ) )
52		simp2l 1087	. . . 4 |- ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> Q e. ( EE ` N ) )
53		simp3 1063	. . . 4 |- ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) )
54		axsegcon 25807	. . . 4 |- ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ Q e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> E. x e. ( EE ` N ) ( Q Btwn <. Q , x >. /\ <. Q , x >.Cgr <. B , C >. ) )
55	5, 52, 52, 53, 54	syl121anc 1331	. . 3 |- ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> E. x e. ( EE ` N ) ( Q Btwn <. Q , x >. /\ <. Q , x >.Cgr <. B , C >. ) )
56		orc 400	. . . . 5 |- ( Q Btwn <. Q , x >. -> ( Q Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) )
57	56	anim1i 592	. . . 4 |- ( ( Q Btwn <. Q , x >. /\ <. Q , x >.Cgr <. B , C >. ) -> ( ( Q Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) /\ <. Q , x >.Cgr <. B , C >. ) )
58	57	reximi 3011	. . 3 |- ( E. x e. ( EE ` N ) ( Q Btwn <. Q , x >. /\ <. Q , x >.Cgr <. B , C >. ) -> E. x e. ( EE ` N ) ( ( Q Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) /\ <. Q , x >.Cgr <. B , C >. ) )
59	55, 58	syl 17	. 2 |- ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> E. x e. ( EE ` N ) ( ( Q Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) /\ <. Q , x >.Cgr <. B , C >. ) )
60	4, 51, 59	pm2.61ne 2879	1 |- ( ( N e. NN /\ ( Q e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> E. x e. ( EE ` N ) ( ( A Btwn <. Q , x >. \/ x Btwn <. Q , A >. ) /\ <. Q , x >.Cgr <. B , C >. ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770

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

This theorem is referenced by:  seglelin  32223  outsideofeu  32238