Theorem lineext 30855

Description: Extend a line with a missing point. Theorem 4.14 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 6-Oct-2013.)

Assertion

Ref	Expression
lineext	|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( ( A Colinear <. B , C >. /\ <. A , B >.Cgr <. D , E >. ) -> E. f e. ( EE ` N ) <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) )

Distinct variable groups:   f, N   A, f   B, f   C, f   D, f   f, E

Proof of Theorem lineext

Step	Hyp	Ref	Expression
1		brcolinear 30838	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Colinear <. B , C >. <-> ( A Btwn <. B , C >. \/ B Btwn <. C , A >. \/ C Btwn <. A , B >. ) ) )
2	1	3adant3 1029	. . 3 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( A Colinear <. B , C >. <-> ( A Btwn <. B , C >. \/ B Btwn <. C , A >. \/ C Btwn <. A , B >. ) ) )
3	2	anbi1d 712	. 2 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( ( A Colinear <. B , C >. /\ <. A , B >.Cgr <. D , E >. ) <-> ( ( A Btwn <. B , C >. \/ B Btwn <. C , A >. \/ C Btwn <. A , B >. ) /\ <. A , B >.Cgr <. D , E >. ) ) )
4		simp1 1009	. . . . . . . . 9 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> N e. NN )
5		simp3r 1038	. . . . . . . . . 10 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> E e. ( EE ` N ) )
6		simp3l 1037	. . . . . . . . . 10 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> D e. ( EE ` N ) )
7	5, 6	jca 535	. . . . . . . . 9 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( E e. ( EE ` N ) /\ D e. ( EE ` N ) ) )
8		simp21 1042	. . . . . . . . . 10 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
9		simp23 1044	. . . . . . . . . 10 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
10	8, 9	jca 535	. . . . . . . . 9 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) )
11	4, 7, 10	3jca 1189	. . . . . . . 8 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( N e. NN /\ ( E e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) )
12	11	adantr 467	. . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ ( A Btwn <. B , C >. /\ <. A , B >.Cgr <. D , E >. ) ) -> ( N e. NN /\ ( E e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) )
13		axsegcon 24969	. . . . . . 7 |- ( ( N e. NN /\ ( E e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> E. f e. ( EE ` N ) ( D Btwn <. E , f >. /\ <. D , f >.Cgr <. A , C >. ) )
14	12, 13	syl 17	. . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ ( A Btwn <. B , C >. /\ <. A , B >.Cgr <. D , E >. ) ) -> E. f e. ( EE ` N ) ( D Btwn <. E , f >. /\ <. D , f >.Cgr <. A , C >. ) )
15		simprlr 774	. . . . . . . . . . 11 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) /\ ( ( A Btwn <. B , C >. /\ <. A , B >.Cgr <. D , E >. ) /\ ( D Btwn <. E , f >. /\ <. A , C >.Cgr <. D , f >. ) ) ) -> <. A , B >.Cgr <. D , E >. )
16		simprrr 776	. . . . . . . . . . 11 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) /\ ( ( A Btwn <. B , C >. /\ <. A , B >.Cgr <. D , E >. ) /\ ( D Btwn <. E , f >. /\ <. A , C >.Cgr <. D , f >. ) ) ) -> <. A , C >.Cgr <. D , f >. )
17		an4 834	. . . . . . . . . . . . 13 |- ( ( ( A Btwn <. B , C >. /\ <. A , B >.Cgr <. D , E >. ) /\ ( D Btwn <. E , f >. /\ <. A , C >.Cgr <. D , f >. ) ) <-> ( ( A Btwn <. B , C >. /\ D Btwn <. E , f >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. A , C >.Cgr <. D , f >. ) ) )
18		simpl1 1012	. . . . . . . . . . . . . . . . 17 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> N e. NN )
19		simpl21 1087	. . . . . . . . . . . . . . . . 17 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> A e. ( EE ` N ) )
20		simpl22 1088	. . . . . . . . . . . . . . . . 17 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> B e. ( EE ` N ) )
21		simpl3l 1064	. . . . . . . . . . . . . . . . 17 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> D e. ( EE ` N ) )
22		simpl3r 1065	. . . . . . . . . . . . . . . . 17 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> E e. ( EE ` N ) )
23		cgrcomlr 30777	. . . . . . . . . . . . . . . . 17 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( <. A , B >.Cgr <. D , E >. <-> <. B , A >.Cgr <. E , D >. ) )
24	18, 19, 20, 21, 22, 23	syl122anc 1278	. . . . . . . . . . . . . . . 16 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( <. A , B >.Cgr <. D , E >. <-> <. B , A >.Cgr <. E , D >. ) )
25	24	anbi1d 712	. . . . . . . . . . . . . . 15 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( ( <. A , B >.Cgr <. D , E >. /\ <. A , C >.Cgr <. D , f >. ) <-> ( <. B , A >.Cgr <. E , D >. /\ <. A , C >.Cgr <. D , f >. ) ) )
26	25	anbi2d 711	. . . . . . . . . . . . . 14 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( ( ( A Btwn <. B , C >. /\ D Btwn <. E , f >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. A , C >.Cgr <. D , f >. ) ) <-> ( ( A Btwn <. B , C >. /\ D Btwn <. E , f >. ) /\ ( <. B , A >.Cgr <. E , D >. /\ <. A , C >.Cgr <. D , f >. ) ) ) )
27		simpl23 1089	. . . . . . . . . . . . . . 15 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> C e. ( EE ` N ) )
28		simpr 463	. . . . . . . . . . . . . . 15 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> f e. ( EE ` N ) )
29		cgrextend 30787	. . . . . . . . . . . . . . 15 |- ( ( N e. NN /\ ( B e. ( EE ` N ) /\ A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ D e. ( EE ` N ) /\ f e. ( EE ` N ) ) ) -> ( ( ( A Btwn <. B , C >. /\ D Btwn <. E , f >. ) /\ ( <. B , A >.Cgr <. E , D >. /\ <. A , C >.Cgr <. D , f >. ) ) -> <. B , C >.Cgr <. E , f >. ) )
30	18, 20, 19, 27, 22, 21, 28, 29	syl133anc 1292	. . . . . . . . . . . . . 14 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( ( ( A Btwn <. B , C >. /\ D Btwn <. E , f >. ) /\ ( <. B , A >.Cgr <. E , D >. /\ <. A , C >.Cgr <. D , f >. ) ) -> <. B , C >.Cgr <. E , f >. ) )
31	26, 30	sylbid 219	. . . . . . . . . . . . 13 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( ( ( A Btwn <. B , C >. /\ D Btwn <. E , f >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. A , C >.Cgr <. D , f >. ) ) -> <. B , C >.Cgr <. E , f >. ) )
32	17, 31	syl5bi 221	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( ( ( A Btwn <. B , C >. /\ <. A , B >.Cgr <. D , E >. ) /\ ( D Btwn <. E , f >. /\ <. A , C >.Cgr <. D , f >. ) ) -> <. B , C >.Cgr <. E , f >. ) )
33	32	imp 431	. . . . . . . . . . 11 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) /\ ( ( A Btwn <. B , C >. /\ <. A , B >.Cgr <. D , E >. ) /\ ( D Btwn <. E , f >. /\ <. A , C >.Cgr <. D , f >. ) ) ) -> <. B , C >.Cgr <. E , f >. )
34	15, 16, 33	3jca 1189	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) /\ ( ( A Btwn <. B , C >. /\ <. A , B >.Cgr <. D , E >. ) /\ ( D Btwn <. E , f >. /\ <. A , C >.Cgr <. D , f >. ) ) ) -> ( <. A , B >.Cgr <. D , E >. /\ <. A , C >.Cgr <. D , f >. /\ <. B , C >.Cgr <. E , f >. ) )
35	34	expr 620	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) /\ ( A Btwn <. B , C >. /\ <. A , B >.Cgr <. D , E >. ) ) -> ( ( D Btwn <. E , f >. /\ <. A , C >.Cgr <. D , f >. ) -> ( <. A , B >.Cgr <. D , E >. /\ <. A , C >.Cgr <. D , f >. /\ <. B , C >.Cgr <. E , f >. ) ) )
36		cgrcom 30769	. . . . . . . . . . . 12 |- ( ( N e. NN /\ ( D e. ( EE ` N ) /\ f e. ( EE ` N ) ) /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( <. D , f >.Cgr <. A , C >. <-> <. A , C >.Cgr <. D , f >. ) )
37	18, 21, 28, 19, 27, 36	syl122anc 1278	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( <. D , f >.Cgr <. A , C >. <-> <. A , C >.Cgr <. D , f >. ) )
38	37	anbi2d 711	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( ( D Btwn <. E , f >. /\ <. D , f >.Cgr <. A , C >. ) <-> ( D Btwn <. E , f >. /\ <. A , C >.Cgr <. D , f >. ) ) )
39	38	adantr 467	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) /\ ( A Btwn <. B , C >. /\ <. A , B >.Cgr <. D , E >. ) ) -> ( ( D Btwn <. E , f >. /\ <. D , f >.Cgr <. A , C >. ) <-> ( D Btwn <. E , f >. /\ <. A , C >.Cgr <. D , f >. ) ) )
40		simpl2 1013	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) )
41		brcgr3 30825	. . . . . . . . . . 11 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ f e. ( EE ` N ) ) ) -> ( <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. <-> ( <. A , B >.Cgr <. D , E >. /\ <. A , C >.Cgr <. D , f >. /\ <. B , C >.Cgr <. E , f >. ) ) )
42	18, 40, 21, 22, 28, 41	syl113anc 1281	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. <-> ( <. A , B >.Cgr <. D , E >. /\ <. A , C >.Cgr <. D , f >. /\ <. B , C >.Cgr <. E , f >. ) ) )
43	42	adantr 467	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) /\ ( A Btwn <. B , C >. /\ <. A , B >.Cgr <. D , E >. ) ) -> ( <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. <-> ( <. A , B >.Cgr <. D , E >. /\ <. A , C >.Cgr <. D , f >. /\ <. B , C >.Cgr <. E , f >. ) ) )
44	35, 39, 43	3imtr4d 272	. . . . . . . 8 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) /\ ( A Btwn <. B , C >. /\ <. A , B >.Cgr <. D , E >. ) ) -> ( ( D Btwn <. E , f >. /\ <. D , f >.Cgr <. A , C >. ) -> <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) )
45	44	an32s 814	. . . . . . 7 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ ( A Btwn <. B , C >. /\ <. A , B >.Cgr <. D , E >. ) ) /\ f e. ( EE ` N ) ) -> ( ( D Btwn <. E , f >. /\ <. D , f >.Cgr <. A , C >. ) -> <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) )
46	45	reximdva 2864	. . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ ( A Btwn <. B , C >. /\ <. A , B >.Cgr <. D , E >. ) ) -> ( E. f e. ( EE ` N ) ( D Btwn <. E , f >. /\ <. D , f >.Cgr <. A , C >. ) -> E. f e. ( EE ` N ) <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) )
47	14, 46	mpd 15	. . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ ( A Btwn <. B , C >. /\ <. A , B >.Cgr <. D , E >. ) ) -> E. f e. ( EE ` N ) <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. )
48	47	exp32 610	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( A Btwn <. B , C >. -> ( <. A , B >.Cgr <. D , E >. -> E. f e. ( EE ` N ) <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) ) )
49		3ancoma 993	. . . . . . 7 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) <-> ( B e. ( EE ` N ) /\ A e. ( EE ` N ) /\ C e. ( EE ` N ) ) )
50		btwncom 30793	. . . . . . 7 |- ( ( N e. NN /\ ( B e. ( EE ` N ) /\ A e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( B Btwn <. A , C >. <-> B Btwn <. C , A >. ) )
51	49, 50	sylan2b 478	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( B Btwn <. A , C >. <-> B Btwn <. C , A >. ) )
52	51	3adant3 1029	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( B Btwn <. A , C >. <-> B Btwn <. C , A >. ) )
53		simp3 1011	. . . . . . . . 9 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) )
54		simp22 1043	. . . . . . . . 9 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
55		axsegcon 24969	. . . . . . . . 9 |- ( ( N e. NN /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> E. f e. ( EE ` N ) ( E Btwn <. D , f >. /\ <. E , f >.Cgr <. B , C >. ) )
56	4, 53, 54, 9, 55	syl112anc 1273	. . . . . . . 8 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> E. f e. ( EE ` N ) ( E Btwn <. D , f >. /\ <. E , f >.Cgr <. B , C >. ) )
57	56	adantr 467	. . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ <. A , B >.Cgr <. D , E >. ) ) -> E. f e. ( EE ` N ) ( E Btwn <. D , f >. /\ <. E , f >.Cgr <. B , C >. ) )
58		cgrextend 30787	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ f e. ( EE ` N ) ) ) -> ( ( ( B Btwn <. A , C >. /\ E Btwn <. D , f >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) ) -> <. A , C >.Cgr <. D , f >. ) )
59	18, 40, 21, 22, 28, 58	syl113anc 1281	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( ( ( B Btwn <. A , C >. /\ E Btwn <. D , f >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) ) -> <. A , C >.Cgr <. D , f >. ) )
60		simpll 761	. . . . . . . . . . . . . . 15 |- ( ( ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) /\ <. A , C >.Cgr <. D , f >. ) -> <. A , B >.Cgr <. D , E >. )
61		simpr 463	. . . . . . . . . . . . . . 15 |- ( ( ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) /\ <. A , C >.Cgr <. D , f >. ) -> <. A , C >.Cgr <. D , f >. )
62		simplr 763	. . . . . . . . . . . . . . 15 |- ( ( ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) /\ <. A , C >.Cgr <. D , f >. ) -> <. B , C >.Cgr <. E , f >. )
63	60, 61, 62	3jca 1189	. . . . . . . . . . . . . 14 |- ( ( ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) /\ <. A , C >.Cgr <. D , f >. ) -> ( <. A , B >.Cgr <. D , E >. /\ <. A , C >.Cgr <. D , f >. /\ <. B , C >.Cgr <. E , f >. ) )
64	63	ex 436	. . . . . . . . . . . . 13 |- ( ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) -> ( <. A , C >.Cgr <. D , f >. -> ( <. A , B >.Cgr <. D , E >. /\ <. A , C >.Cgr <. D , f >. /\ <. B , C >.Cgr <. E , f >. ) ) )
65	64	adantl 468	. . . . . . . . . . . 12 |- ( ( ( B Btwn <. A , C >. /\ E Btwn <. D , f >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) ) -> ( <. A , C >.Cgr <. D , f >. -> ( <. A , B >.Cgr <. D , E >. /\ <. A , C >.Cgr <. D , f >. /\ <. B , C >.Cgr <. E , f >. ) ) )
66	59, 65	sylcom 30	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( ( ( B Btwn <. A , C >. /\ E Btwn <. D , f >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) ) -> ( <. A , B >.Cgr <. D , E >. /\ <. A , C >.Cgr <. D , f >. /\ <. B , C >.Cgr <. E , f >. ) ) )
67		an4 834	. . . . . . . . . . . 12 |- ( ( ( B Btwn <. A , C >. /\ <. A , B >.Cgr <. D , E >. ) /\ ( E Btwn <. D , f >. /\ <. E , f >.Cgr <. B , C >. ) ) <-> ( ( B Btwn <. A , C >. /\ E Btwn <. D , f >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. E , f >.Cgr <. B , C >. ) ) )
68		cgrcom 30769	. . . . . . . . . . . . . . 15 |- ( ( N e. NN /\ ( E e. ( EE ` N ) /\ f e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( <. E , f >.Cgr <. B , C >. <-> <. B , C >.Cgr <. E , f >. ) )
69	18, 22, 28, 20, 27, 68	syl122anc 1278	. . . . . . . . . . . . . 14 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( <. E , f >.Cgr <. B , C >. <-> <. B , C >.Cgr <. E , f >. ) )
70	69	anbi2d 711	. . . . . . . . . . . . 13 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( ( <. A , B >.Cgr <. D , E >. /\ <. E , f >.Cgr <. B , C >. ) <-> ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) ) )
71	70	anbi2d 711	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( ( ( B Btwn <. A , C >. /\ E Btwn <. D , f >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. E , f >.Cgr <. B , C >. ) ) <-> ( ( B Btwn <. A , C >. /\ E Btwn <. D , f >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) ) ) )
72	67, 71	syl5bb 261	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( ( ( B Btwn <. A , C >. /\ <. A , B >.Cgr <. D , E >. ) /\ ( E Btwn <. D , f >. /\ <. E , f >.Cgr <. B , C >. ) ) <-> ( ( B Btwn <. A , C >. /\ E Btwn <. D , f >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) ) ) )
73	66, 72, 42	3imtr4d 272	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( ( ( B Btwn <. A , C >. /\ <. A , B >.Cgr <. D , E >. ) /\ ( E Btwn <. D , f >. /\ <. E , f >.Cgr <. B , C >. ) ) -> <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) )
74	73	expdimp 439	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) /\ ( B Btwn <. A , C >. /\ <. A , B >.Cgr <. D , E >. ) ) -> ( ( E Btwn <. D , f >. /\ <. E , f >.Cgr <. B , C >. ) -> <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) )
75	74	an32s 814	. . . . . . . 8 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ <. A , B >.Cgr <. D , E >. ) ) /\ f e. ( EE ` N ) ) -> ( ( E Btwn <. D , f >. /\ <. E , f >.Cgr <. B , C >. ) -> <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) )
76	75	reximdva 2864	. . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ <. A , B >.Cgr <. D , E >. ) ) -> ( E. f e. ( EE ` N ) ( E Btwn <. D , f >. /\ <. E , f >.Cgr <. B , C >. ) -> E. f e. ( EE ` N ) <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) )
77	57, 76	mpd 15	. . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ ( B Btwn <. A , C >. /\ <. A , B >.Cgr <. D , E >. ) ) -> E. f e. ( EE ` N ) <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. )
78	77	exp32 610	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( B Btwn <. A , C >. -> ( <. A , B >.Cgr <. D , E >. -> E. f e. ( EE ` N ) <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) ) )
79	52, 78	sylbird 239	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( B Btwn <. C , A >. -> ( <. A , B >.Cgr <. D , E >. -> E. f e. ( EE ` N ) <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) ) )
80		cgrxfr 30834	. . . . . . 7 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( ( C Btwn <. A , B >. /\ <. A , B >.Cgr <. D , E >. ) -> E. f e. ( EE ` N ) ( f Btwn <. D , E >. /\ <. A , <. C , B >. >.Cgr3 <. D , <. f , E >. >. ) ) )
81	4, 8, 9, 54, 53, 80	syl131anc 1282	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( ( C Btwn <. A , B >. /\ <. A , B >.Cgr <. D , E >. ) -> E. f e. ( EE ` N ) ( f Btwn <. D , E >. /\ <. A , <. C , B >. >.Cgr3 <. D , <. f , E >. >. ) ) )
82		cgr3permute1 30827	. . . . . . . . . 10 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ f e. ( EE ` N ) ) ) -> ( <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. <-> <. A , <. C , B >. >.Cgr3 <. D , <. f , E >. >. ) )
83	18, 40, 21, 22, 28, 82	syl113anc 1281	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. <-> <. A , <. C , B >. >.Cgr3 <. D , <. f , E >. >. ) )
84	83	biimprd 227	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( <. A , <. C , B >. >.Cgr3 <. D , <. f , E >. >. -> <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) )
85	84	adantld 469	. . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ f e. ( EE ` N ) ) -> ( ( f Btwn <. D , E >. /\ <. A , <. C , B >. >.Cgr3 <. D , <. f , E >. >. ) -> <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) )
86	85	reximdva 2864	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( E. f e. ( EE ` N ) ( f Btwn <. D , E >. /\ <. A , <. C , B >. >.Cgr3 <. D , <. f , E >. >. ) -> E. f e. ( EE ` N ) <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) )
87	81, 86	syld 45	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( ( C Btwn <. A , B >. /\ <. A , B >.Cgr <. D , E >. ) -> E. f e. ( EE ` N ) <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) )
88	87	expd 438	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( C Btwn <. A , B >. -> ( <. A , B >.Cgr <. D , E >. -> E. f e. ( EE ` N ) <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) ) )
89	48, 79, 88	3jaod 1334	. . 3 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( ( A Btwn <. B , C >. \/ B Btwn <. C , A >. \/ C Btwn <. A , B >. ) -> ( <. A , B >.Cgr <. D , E >. -> E. f e. ( EE ` N ) <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) ) )
90	89	impd 433	. 2 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( ( ( A Btwn <. B , C >. \/ B Btwn <. C , A >. \/ C Btwn <. A , B >. ) /\ <. A , B >.Cgr <. D , E >. ) -> E. f e. ( EE ` N ) <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) )
91	3, 90	sylbid 219	1 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( ( A Colinear <. B , C >. /\ <. A , B >.Cgr <. D , E >. ) -> E. f e. ( EE ` N ) <. A , <. B , C >. >.Cgr3 <. D , <. E , f >. >. ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   \/ w3o 985   /\ w3a 986   e. wcel 1889   E.wrex 2740   <.cop 3976   class class class wbr 4405   ` cfv 5585   NNcn 10616   EEcee 24930   Btwn cbtwn 24931  Cgrccgr 24932  Cgr3ccgr3 30815   Colinear ccolin 30816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7961  df-oi 8030  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-sum 13765  df-ee 24933  df-btwn 24934  df-cgr 24935  df-ofs 30762  df-colinear 30818  df-cgr3 30820

This theorem is referenced by:  brsegle2  30888