Theorem lineext 28036

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 28019	. . . 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 1003	. . 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 699	. 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 983	. . . . . . . . 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 1012	. . . . . . . . . 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 1011	. . . . . . . . . 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 529	. . . . . . . . 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 1016	. . . . . . . . . 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 1018	. . . . . . . . . 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 529	. . . . . . . . 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 1163	. . . . . . . 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 462	. . . . . . 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 23108	. . . . . . 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 757	. . . . . . . . . . 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 759	. . . . . . . . . . 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 815	. . . . . . . . . . . . 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 986	. . . . . . . . . . . . . . . . 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 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 ) ) -> A e. ( EE ` N ) )
20		simpl22 1062	. . . . . . . . . . . . . . . . 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 1038	. . . . . . . . . . . . . . . . 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 1039	. . . . . . . . . . . . . . . . 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 27958	. . . . . . . . . . . . . . . . 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 1222	. . . . . . . . . . . . . . . 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 699	. . . . . . . . . . . . . . 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 698	. . . . . . . . . . . . . 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 1063	. . . . . . . . . . . . . . 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 458	. . . . . . . . . . . . . . 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 27968	. . . . . . . . . . . . . . 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 1236	. . . . . . . . . . . . . 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 429	. . . . . . . . . . 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 1163	. . . . . . . . . 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 612	. . . . . . . . 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 27950	. . . . . . . . . . . 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 1222	. . . . . . . . . . 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 698	. . . . . . . . . 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 462	. . . . . . . . 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 987	. . . . . . . . . . 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 28006	. . . . . . . . . . 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 1225	. . . . . . . . . 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 462	. . . . . . . . 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 797	. . . . . . 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 2826	. . . . . 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 602	. . . 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 967	. . . . . . 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 27974	. . . . . . 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 472	. . . . . 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 1003	. . . . 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 985	. . . . . . . . 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 1017	. . . . . . . . 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 23108	. . . . . . . . 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 1217	. . . . . . . 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 462	. . . . . . 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 27968	. . . . . . . . . . . . 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 1225	. . . . . . . . . . . 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 748	. . . . . . . . . . . . . . 15 |- ( ( ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) /\ <. A , C >.Cgr <. D , f >. ) -> <. A , B >.Cgr <. D , E >. )
61		simpr 458	. . . . . . . . . . . . . . 15 |- ( ( ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , f >. ) /\ <. A , C >.Cgr <. D , f >. ) -> <. A , C >.Cgr <. D , f >. )
62		simplr 749	. . . . . . . . . . . . . . 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 1163	. . . . . . . . . . . . . 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 434	. . . . . . . . . . . . 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 463	. . . . . . . . . . . 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 815	. . . . . . . . . . . 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 27950	. . . . . . . . . . . . . . 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 1222	. . . . . . . . . . . . . 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 698	. . . . . . . . . . . . 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 698	. . . . . . . . . . . 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 437	. . . . . . . . 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 797	. . . . . . . 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 2826	. . . . . . 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 602	. . . . 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 28015	. . . . . . 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 1226	. . . . . 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 28008	. . . . . . . . . 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 1225	. . . . . . . . 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 464	. . . . . . 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 2826	. . . . . 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	exp3a 436	. . . 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 1277	. . 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	imp3a 431	. 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 369   \/ w3o 959   /\ w3a 960   e. wcel 1761   E.wrex 2714   <.cop 3880   class class class wbr 4289   ` cfv 5415   NNcn 10318   EEcee 23069   Btwn cbtwn 23070  Cgrccgr 23071  Cgr3ccgr3 27996   Colinear ccolin 27997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160  df-ee 23072  df-btwn 23073  df-cgr 23074  df-ofs 27943  df-colinear 27999  df-cgr3 28001

This theorem is referenced by:  brsegle2  28069