Theorem lineext 30836

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 30819	. . . 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 1025	. . 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 709	. 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 1005	. . . . . . . . 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 1034	. . . . . . . . . 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 1033	. . . . . . . . . 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 534	. . . . . . . . 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 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 ) ) ) -> A e. ( EE ` N ) )
9		simp23 1040	. . . . . . . . . 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 534	. . . . . . . . 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 1185	. . . . . . . 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 466	. . . . . . 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 24944	. . . . . . 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 771	. . . . . . . . . . 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 773	. . . . . . . . . . 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 831	. . . . . . . . . . . . 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 1008	. . . . . . . . . . . . . . . . 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 1083	. . . . . . . . . . . . . . . . 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 1084	. . . . . . . . . . . . . . . . 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 1060	. . . . . . . . . . . . . . . . 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 1061	. . . . . . . . . . . . . . . . 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 30758	. . . . . . . . . . . . . . . . 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 1273	. . . . . . . . . . . . . . . 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 709	. . . . . . . . . . . . . . 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 708	. . . . . . . . . . . . . 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 1085	. . . . . . . . . . . . . . 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 462	. . . . . . . . . . . . . . 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 30768	. . . . . . . . . . . . . . 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 1287	. . . . . . . . . . . . . 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 218	. . . . . . . . . . . . 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 220	. . . . . . . . . . . 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 430	. . . . . . . . . . 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 1185	. . . . . . . . . 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 618	. . . . . . . . 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 30750	. . . . . . . . . . . 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 1273	. . . . . . . . . . 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 708	. . . . . . . . . 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 466	. . . . . . . . 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 1009	. . . . . . . . . . 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 30806	. . . . . . . . . . 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 1276	. . . . . . . . . 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 466	. . . . . . . . 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 271	. . . . . . . 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 811	. . . . . . 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 2900	. . . . . 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 608	. . . 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 989	. . . . . . 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 30774	. . . . . . 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 477	. . . . . 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 1025	. . . . 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 1007	. . . . . . . . 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 1039	. . . . . . . . 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 24944	. . . . . . . . 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 1268	. . . . . . . 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 466	. . . . . . 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 30768	. . . . . . . . . . . . 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 1276	. . . . . . . . . . . 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 758	. . . . . . . . . . . . . . 15 |- ( ( ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) /\ <. A , C >.Cgr <. D , f >. ) -> <. A , B >.Cgr <. D , E >. )
61		simpr 462	. . . . . . . . . . . . . . 15 |- ( ( ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) /\ <. A , C >.Cgr <. D , f >. ) -> <. A , C >.Cgr <. D , f >. )
62		simplr 760	. . . . . . . . . . . . . . 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 1185	. . . . . . . . . . . . . 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 435	. . . . . . . . . . . . 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 467	. . . . . . . . . . . 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 831	. . . . . . . . . . . 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 30750	. . . . . . . . . . . . . . 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 1273	. . . . . . . . . . . . . 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 708	. . . . . . . . . . . . 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 708	. . . . . . . . . . . 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 260	. . . . . . . . . . 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 271	. . . . . . . . . 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 438	. . . . . . . . 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 811	. . . . . . . 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 2900	. . . . . . 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 608	. . . . 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 238	. . . 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 30815	. . . . . . 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 1277	. . . . . 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 30808	. . . . . . . . . 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 1276	. . . . . . . . 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 226	. . . . . . . 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 468	. . . . . . 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 2900	. . . . . 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 437	. . . 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 1328	. . 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 432	. 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 218	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 187   /\ wa 370   \/ w3o 981   /\ w3a 982   e. wcel 1868   E.wrex 2776   <.cop 4002   class class class wbr 4420   ` cfv 5598   NNcn 10610   EEcee 24905   Btwn cbtwn 24906  Cgrccgr 24907  Cgr3ccgr3 30796   Colinear ccolin 30797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-1st 6804  df-2nd 6805  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 7959  df-oi 8028  df-card 8375  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-n0 10871  df-z 10939  df-uz 11161  df-rp 11304  df-ico 11642  df-icc 11643  df-fz 11786  df-fzo 11917  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-sum 13741  df-ee 24908  df-btwn 24909  df-cgr 24910  df-ofs 30743  df-colinear 30799  df-cgr3 30801

This theorem is referenced by:  brsegle2  30869