Theorem lineext 29954

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 29937	. . . 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 1014	. . 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 702	. 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 994	. . . . . . . . 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 1023	. . . . . . . . . 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 1022	. . . . . . . . . 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 530	. . . . . . . . 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 1027	. . . . . . . . . 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 1029	. . . . . . . . . 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 530	. . . . . . . . 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 1174	. . . . . . . 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 463	. . . . . . 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 24432	. . . . . . 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 16	. . . . . 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 762	. . . . . . . . . . 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 764	. . . . . . . . . . 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 822	. . . . . . . . . . . . 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 997	. . . . . . . . . . . . . . . . 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 1072	. . . . . . . . . . . . . . . . 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 1073	. . . . . . . . . . . . . . . . 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 1049	. . . . . . . . . . . . . . . . 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 1050	. . . . . . . . . . . . . . . . 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 29876	. . . . . . . . . . . . . . . . 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 1235	. . . . . . . . . . . . . . . 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 702	. . . . . . . . . . . . . . 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 701	. . . . . . . . . . . . . 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 1074	. . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . 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 29886	. . . . . . . . . . . . . . 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 1249	. . . . . . . . . . . . . 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 215	. . . . . . . . . . . . 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 217	. . . . . . . . . . . 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 427	. . . . . . . . . . 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 1174	. . . . . . . . . 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 613	. . . . . . . . 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 29868	. . . . . . . . . . . 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 1235	. . . . . . . . . . 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 701	. . . . . . . . . 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 463	. . . . . . . . 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 998	. . . . . . . . . . 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 29924	. . . . . . . . . . 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 1238	. . . . . . . . . 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 463	. . . . . . . . 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 268	. . . . . . . 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 802	. . . . . . 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 2929	. . . . . 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 603	. . . 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 978	. . . . . . 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 29892	. . . . . . 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 473	. . . . . 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 1014	. . . . 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 996	. . . . . . . . 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 1028	. . . . . . . . 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 24432	. . . . . . . . 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 1230	. . . . . . . 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 463	. . . . . . 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 29886	. . . . . . . . . . . . 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 1238	. . . . . . . . . . . 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 751	. . . . . . . . . . . . . . 15 |- ( ( ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) /\ <. A , C >.Cgr <. D , f >. ) -> <. A , B >.Cgr <. D , E >. )
61		simpr 459	. . . . . . . . . . . . . . 15 |- ( ( ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) /\ <. A , C >.Cgr <. D , f >. ) -> <. A , C >.Cgr <. D , f >. )
62		simplr 753	. . . . . . . . . . . . . . 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 1174	. . . . . . . . . . . . . 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 432	. . . . . . . . . . . . 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 464	. . . . . . . . . . . 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 29	. . . . . . . . . . 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 822	. . . . . . . . . . . 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 29868	. . . . . . . . . . . . . . 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 1235	. . . . . . . . . . . . . 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 701	. . . . . . . . . . . . 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 701	. . . . . . . . . . . 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 257	. . . . . . . . . . 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 268	. . . . . . . . . 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 435	. . . . . . . . 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 802	. . . . . . . 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 2929	. . . . . . 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 603	. . . . 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 235	. . . 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 29933	. . . . . . 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 1239	. . . . . 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 29926	. . . . . . . . . 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 1238	. . . . . . . . 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 223	. . . . . . . 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 465	. . . . . . 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 2929	. . . . . 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 44	. . . . 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 434	. . . 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 1290	. . 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 429	. 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 215	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 184   /\ wa 367   \/ w3o 970   /\ w3a 971   e. wcel 1823   E.wrex 2805   <.cop 4022   class class class wbr 4439   ` cfv 5570   NNcn 10531   EEcee 24393   Btwn cbtwn 24394  Cgrccgr 24395  Cgr3ccgr3 29914   Colinear ccolin 29915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-sum 13591  df-ee 24396  df-btwn 24397  df-cgr 24398  df-ofs 29861  df-colinear 29917  df-cgr3 29919

This theorem is referenced by:  brsegle2  29987