Theorem colinbtwnle 32225

Description: Given three colinear points A, B, and C, B falls in the middle iff the two segments to B are no longer than A C. Theorem 5.12 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
colinbtwnle	|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Colinear <. B , C >. -> ( B Btwn <. A , C >. <-> ( <. A , B >. Seg<_ <. A , C >. /\ <. B , C >. Seg<_ <. A , C >. ) ) ) )

Proof of Theorem colinbtwnle

Step	Hyp	Ref	Expression
1		btwnsegle 32224	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( B Btwn <. A , C >. -> <. A , B >. Seg<_ <. A , C >. ) )
2		3anrev 1049	. . . . . . 7 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) <-> ( C e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A e. ( EE ` N ) ) )
3		btwnsegle 32224	. . . . . . 7 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A e. ( EE ` N ) ) ) -> ( B Btwn <. C , A >. -> <. C , B >. Seg<_ <. C , A >. ) )
4	2, 3	sylan2b 492	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( B Btwn <. C , A >. -> <. C , B >. Seg<_ <. C , A >. ) )
5		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 ) ) )
6		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 >. ) )
7	5, 6	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 >. ) )
8		simpl 473	. . . . . . . 8 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> N e. NN )
9		simpr2 1068	. . . . . . . 8 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
10		simpr3 1069	. . . . . . . 8 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
11	8, 9, 10	cgrrflx2d 32091	. . . . . . 7 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> <. B , C >.Cgr <. C , B >. )
12		simpr1 1067	. . . . . . . 8 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
13	8, 12, 10	cgrrflx2d 32091	. . . . . . 7 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> <. A , C >.Cgr <. C , A >. )
14		seglecgr12 32218	. . . . . . . 8 |- ( ( ( N e. NN /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) /\ A e. ( EE ` N ) ) ) -> ( ( <. B , C >.Cgr <. C , B >. /\ <. A , C >.Cgr <. C , A >. ) -> ( <. B , C >. Seg<_ <. A , C >. <-> <. C , B >. Seg<_ <. C , A >. ) ) )
15	8, 9, 10, 12, 10, 10, 9, 10, 12, 14	syl333anc 1358	. . . . . . 7 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( <. B , C >.Cgr <. C , B >. /\ <. A , C >.Cgr <. C , A >. ) -> ( <. B , C >. Seg<_ <. A , C >. <-> <. C , B >. Seg<_ <. C , A >. ) ) )
16	11, 13, 15	mp2and 715	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( <. B , C >. Seg<_ <. A , C >. <-> <. C , B >. Seg<_ <. C , A >. ) )
17	4, 7, 16	3imtr4d 283	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( B Btwn <. A , C >. -> <. B , C >. Seg<_ <. A , C >. ) )
18	1, 17	jcad 555	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( B Btwn <. A , C >. -> ( <. A , B >. Seg<_ <. A , C >. /\ <. B , C >. Seg<_ <. A , C >. ) ) )
19	18	adantr 481	. . 3 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ A Colinear <. B , C >. ) -> ( B Btwn <. A , C >. -> ( <. A , B >. Seg<_ <. A , C >. /\ <. B , C >. Seg<_ <. A , C >. ) ) )
20		brcolinear 32166	. . . . 5 |- ( ( 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 >. ) ) )
21		simprl 794	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( A Btwn <. B , C >. /\ <. B , C >. Seg<_ <. A , C >. ) ) -> A Btwn <. B , C >. )
22	8, 12, 9, 10, 21	btwncomand 32122	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( A Btwn <. B , C >. /\ <. B , C >. Seg<_ <. A , C >. ) ) -> A Btwn <. C , B >. )
23	16	biimpa 501	. . . . . . . . . . . . 13 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ <. B , C >. Seg<_ <. A , C >. ) -> <. C , B >. Seg<_ <. C , A >. )
24	23	adantrl 752	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( A Btwn <. B , C >. /\ <. B , C >. Seg<_ <. A , C >. ) ) -> <. C , B >. Seg<_ <. C , A >. )
25		btwncom 32121	. . . . . . . . . . . . . . 15 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Btwn <. B , C >. <-> A Btwn <. C , B >. ) )
26		3anrot 1043	. . . . . . . . . . . . . . . 16 |- ( ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) <-> ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) )
27		btwnsegle 32224	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( A Btwn <. C , B >. -> <. C , A >. Seg<_ <. C , B >. ) )
28	26, 27	sylan2br 493	. . . . . . . . . . . . . . 15 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Btwn <. C , B >. -> <. C , A >. Seg<_ <. C , B >. ) )
29	25, 28	sylbid 230	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Btwn <. B , C >. -> <. C , A >. Seg<_ <. C , B >. ) )
30	29	imp 445	. . . . . . . . . . . . 13 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ A Btwn <. B , C >. ) -> <. C , A >. Seg<_ <. C , B >. )
31	30	adantrr 753	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( A Btwn <. B , C >. /\ <. B , C >. Seg<_ <. A , C >. ) ) -> <. C , A >. Seg<_ <. C , B >. )
32		segleantisym 32222	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ A e. ( EE ` N ) ) ) -> ( ( <. C , B >. Seg<_ <. C , A >. /\ <. C , A >. Seg<_ <. C , B >. ) -> <. C , B >.Cgr <. C , A >. ) )
33	8, 10, 9, 10, 12, 32	syl122anc 1335	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( <. C , B >. Seg<_ <. C , A >. /\ <. C , A >. Seg<_ <. C , B >. ) -> <. C , B >.Cgr <. C , A >. ) )
34	33	adantr 481	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( A Btwn <. B , C >. /\ <. B , C >. Seg<_ <. A , C >. ) ) -> ( ( <. C , B >. Seg<_ <. C , A >. /\ <. C , A >. Seg<_ <. C , B >. ) -> <. C , B >.Cgr <. C , A >. ) )
35	24, 31, 34	mp2and 715	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( A Btwn <. B , C >. /\ <. B , C >. Seg<_ <. A , C >. ) ) -> <. C , B >.Cgr <. C , A >. )
36	8, 10, 9, 12, 22, 35	endofsegidand 32193	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( A Btwn <. B , C >. /\ <. B , C >. Seg<_ <. A , C >. ) ) -> B = A )
37		btwntriv1 32123	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ A e. ( EE ` N ) /\ C e. ( EE ` N ) ) -> A Btwn <. A , C >. )
38	37	3adant3r2 1275	. . . . . . . . . . . 12 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> A Btwn <. A , C >. )
39		breq1 4656	. . . . . . . . . . . 12 |- ( B = A -> ( B Btwn <. A , C >. <-> A Btwn <. A , C >. ) )
40	38, 39	syl5ibrcom 237	. . . . . . . . . . 11 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( B = A -> B Btwn <. A , C >. ) )
41	40	adantr 481	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( A Btwn <. B , C >. /\ <. B , C >. Seg<_ <. A , C >. ) ) -> ( B = A -> B Btwn <. A , C >. ) )
42	36, 41	mpd 15	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( A Btwn <. B , C >. /\ <. B , C >. Seg<_ <. A , C >. ) ) -> B Btwn <. A , C >. )
43	42	expr 643	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ A Btwn <. B , C >. ) -> ( <. B , C >. Seg<_ <. A , C >. -> B Btwn <. A , C >. ) )
44	43	adantld 483	. . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ A Btwn <. B , C >. ) -> ( ( <. A , B >. Seg<_ <. A , C >. /\ <. B , C >. Seg<_ <. A , C >. ) -> B Btwn <. A , C >. ) )
45	44	ex 450	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Btwn <. B , C >. -> ( ( <. A , B >. Seg<_ <. A , C >. /\ <. B , C >. Seg<_ <. A , C >. ) -> B Btwn <. A , C >. ) ) )
46	7	biimprd 238	. . . . . . 7 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( B Btwn <. C , A >. -> B Btwn <. A , C >. ) )
47	46	a1dd 50	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( B Btwn <. C , A >. -> ( ( <. A , B >. Seg<_ <. A , C >. /\ <. B , C >. Seg<_ <. A , C >. ) -> B Btwn <. A , C >. ) ) )
48		simprl 794	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( C Btwn <. A , B >. /\ <. A , B >. Seg<_ <. A , C >. ) ) -> C Btwn <. A , B >. )
49		simprr 796	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( C Btwn <. A , B >. /\ <. A , B >. Seg<_ <. A , C >. ) ) -> <. A , B >. Seg<_ <. A , C >. )
50		3ancomb 1047	. . . . . . . . . . . . . . 15 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) <-> ( A e. ( EE ` N ) /\ C e. ( EE ` N ) /\ B e. ( EE ` N ) ) )
51		btwnsegle 32224	. . . . . . . . . . . . . . 15 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( C Btwn <. A , B >. -> <. A , C >. Seg<_ <. A , B >. ) )
52	50, 51	sylan2b 492	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( C Btwn <. A , B >. -> <. A , C >. Seg<_ <. A , B >. ) )
53	52	imp 445	. . . . . . . . . . . . 13 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ C Btwn <. A , B >. ) -> <. A , C >. Seg<_ <. A , B >. )
54	53	adantrr 753	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( C Btwn <. A , B >. /\ <. A , B >. Seg<_ <. A , C >. ) ) -> <. A , C >. Seg<_ <. A , B >. )
55		segleantisym 32222	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( <. A , B >. Seg<_ <. A , C >. /\ <. A , C >. Seg<_ <. A , B >. ) -> <. A , B >.Cgr <. A , C >. ) )
56	8, 12, 9, 12, 10, 55	syl122anc 1335	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( <. A , B >. Seg<_ <. A , C >. /\ <. A , C >. Seg<_ <. A , B >. ) -> <. A , B >.Cgr <. A , C >. ) )
57	56	adantr 481	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( C Btwn <. A , B >. /\ <. A , B >. Seg<_ <. A , C >. ) ) -> ( ( <. A , B >. Seg<_ <. A , C >. /\ <. A , C >. Seg<_ <. A , B >. ) -> <. A , B >.Cgr <. A , C >. ) )
58	49, 54, 57	mp2and 715	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( C Btwn <. A , B >. /\ <. A , B >. Seg<_ <. A , C >. ) ) -> <. A , B >.Cgr <. A , C >. )
59	8, 12, 9, 10, 48, 58	endofsegidand 32193	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( C Btwn <. A , B >. /\ <. A , B >. Seg<_ <. A , C >. ) ) -> B = C )
60		btwntriv2 32119	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ A e. ( EE ` N ) /\ C e. ( EE ` N ) ) -> C Btwn <. A , C >. )
61	60	3adant3r2 1275	. . . . . . . . . . . 12 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> C Btwn <. A , C >. )
62		breq1 4656	. . . . . . . . . . . 12 |- ( B = C -> ( B Btwn <. A , C >. <-> C Btwn <. A , C >. ) )
63	61, 62	syl5ibrcom 237	. . . . . . . . . . 11 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( B = C -> B Btwn <. A , C >. ) )
64	63	adantr 481	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( C Btwn <. A , B >. /\ <. A , B >. Seg<_ <. A , C >. ) ) -> ( B = C -> B Btwn <. A , C >. ) )
65	59, 64	mpd 15	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( C Btwn <. A , B >. /\ <. A , B >. Seg<_ <. A , C >. ) ) -> B Btwn <. A , C >. )
66	65	expr 643	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ C Btwn <. A , B >. ) -> ( <. A , B >. Seg<_ <. A , C >. -> B Btwn <. A , C >. ) )
67	66	adantrd 484	. . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ C Btwn <. A , B >. ) -> ( ( <. A , B >. Seg<_ <. A , C >. /\ <. B , C >. Seg<_ <. A , C >. ) -> B Btwn <. A , C >. ) )
68	67	ex 450	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( C Btwn <. A , B >. -> ( ( <. A , B >. Seg<_ <. A , C >. /\ <. B , C >. Seg<_ <. A , C >. ) -> B Btwn <. A , C >. ) ) )
69	45, 47, 68	3jaod 1392	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( A Btwn <. B , C >. \/ B Btwn <. C , A >. \/ C Btwn <. A , B >. ) -> ( ( <. A , B >. Seg<_ <. A , C >. /\ <. B , C >. Seg<_ <. A , C >. ) -> B Btwn <. A , C >. ) ) )
70	20, 69	sylbid 230	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Colinear <. B , C >. -> ( ( <. A , B >. Seg<_ <. A , C >. /\ <. B , C >. Seg<_ <. A , C >. ) -> B Btwn <. A , C >. ) ) )
71	70	imp 445	. . 3 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ A Colinear <. B , C >. ) -> ( ( <. A , B >. Seg<_ <. A , C >. /\ <. B , C >. Seg<_ <. A , C >. ) -> B Btwn <. A , C >. ) )
72	19, 71	impbid 202	. 2 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ A Colinear <. B , C >. ) -> ( B Btwn <. A , C >. <-> ( <. A , B >. Seg<_ <. A , C >. /\ <. B , C >. Seg<_ <. A , C >. ) ) )
73	72	ex 450	1 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Colinear <. B , C >. -> ( B Btwn <. A , C >. <-> ( <. A , B >. Seg<_ <. A , C >. /\ <. B , C >. Seg<_ <. A , C >. ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770   Colinear ccolin 32144   Seg<_ csegle 32213

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-ifs 32147  df-cgr3 32148  df-segle 32214

This theorem is referenced by: (None)