Theorem lineext 32183

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 32166	. . . 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 1081	. . 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 741	. 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 1061	. . . . . . . . 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 1090	. . . . . . . . . 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 1089	. . . . . . . . . 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 554	. . . . . . . . 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 1094	. . . . . . . . . 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 1096	. . . . . . . . . 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 554	. . . . . . . . 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 1242	. . . . . . . 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 481	. . . . . . 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 25807	. . . . . . 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 803	. . . . . . . . . . 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 805	. . . . . . . . . . 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 865	. . . . . . . . . . . . 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 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 ) ) -> N e. NN )
19		simpl21 1139	. . . . . . . . . . . . . . . . 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 1140	. . . . . . . . . . . . . . . . 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 1116	. . . . . . . . . . . . . . . . 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 1117	. . . . . . . . . . . . . . . . 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 32105	. . . . . . . . . . . . . . . . 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 1335	. . . . . . . . . . . . . . . 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 741	. . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . 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 1141	. . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . 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 32115	. . . . . . . . . . . . . . 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 1349	. . . . . . . . . . . . . 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 230	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 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 445	. . . . . . . . . . 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 1242	. . . . . . . . . 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 643	. . . . . . . . 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 32097	. . . . . . . . . . . 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 1335	. . . . . . . . . . 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 740	. . . . . . . . . 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 481	. . . . . . . . 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 1065	. . . . . . . . . . 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 32153	. . . . . . . . . . 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 1338	. . . . . . . . . 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 481	. . . . . . . . 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 283	. . . . . . . 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 846	. . . . . . 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 3017	. . . . . 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 631	. . . 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 1045	. . . . . . 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 32121	. . . . . . 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 492	. . . . . 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 1081	. . . . 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 1063	. . . . . . . . 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 1095	. . . . . . . . 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 25807	. . . . . . . . 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 1330	. . . . . . . 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 481	. . . . . . 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 32115	. . . . . . . . . . . . 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 1338	. . . . . . . . . . . 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 790	. . . . . . . . . . . . . . 15 |- ( ( ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) /\ <. A , C >.Cgr <. D , f >. ) -> <. A , B >.Cgr <. D , E >. )
61		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) /\ <. A , C >.Cgr <. D , f >. ) -> <. A , C >.Cgr <. D , f >. )
62		simplr 792	. . . . . . . . . . . . . . 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 1242	. . . . . . . . . . . . . 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 450	. . . . . . . . . . . . 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 482	. . . . . . . . . . . 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 865	. . . . . . . . . . . 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 32097	. . . . . . . . . . . . . . 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 1335	. . . . . . . . . . . . . 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 740	. . . . . . . . . . . . 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 740	. . . . . . . . . . . 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 272	. . . . . . . . . . 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 283	. . . . . . . . . 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 453	. . . . . . . . 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 846	. . . . . . . 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 3017	. . . . . . 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 631	. . . . 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 250	. . . 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 32162	. . . . . . 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 1339	. . . . . 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 32155	. . . . . . . . . 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 1338	. . . . . . . . 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 238	. . . . . . . 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 483	. . . . . . 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 3017	. . . . . 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 47	. . . . 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 452	. . . 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 1392	. . 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 447	. 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 230	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 196   /\ wa 384   \/ w3o 1036   /\ w3a 1037   e. wcel 1990   E.wrex 2913   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770  Cgr3ccgr3 32143   Colinear ccolin 32144

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-cgr3 32148

This theorem is referenced by:  brsegle2  32216