Theorem cgrextend 32115

Description: Link congruence over a pair of line segments. Theorem 2.11 of [Schwabhauser] p. 29. (Contributed by Scott Fenton, 12-Jun-2013.)

Assertion

Ref	Expression
cgrextend	|- ( ( 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 >. ) )

Proof of Theorem cgrextend

Step	Hyp	Ref	Expression
1		opeq1 4402	. . . . . . . . 9 |- ( A = B -> <. A , B >. = <. B , B >. )
2	1	breq1d 4663	. . . . . . . 8 |- ( A = B -> ( <. A , B >.Cgr <. D , E >. <-> <. B , B >.Cgr <. D , E >. ) )
3	2	adantr 481	. . . . . . 7 |- ( ( A = B /\ ( 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 , 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 ) /\ F e. ( EE ` N ) ) ) -> N e. NN )
5		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 ) /\ F e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
6		simp31 1097	. . . . . . . . 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 ) ) ) -> D e. ( EE ` N ) )
7		simp32 1098	. . . . . . . . 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 ) ) ) -> E e. ( EE ` N ) )
8		cgrid2 32110	. . . . . . . . 9 |- ( ( N e. NN /\ ( B e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( <. B , B >.Cgr <. D , E >. -> D = E ) )
9	4, 5, 6, 7, 8	syl13anc 1328	. . . . . . . 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 ) ) ) -> ( <. B , B >.Cgr <. D , E >. -> D = E ) )
10	9	adantl 482	. . . . . . 7 |- ( ( A = B /\ ( 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 , B >.Cgr <. D , E >. -> D = E ) )
11	3, 10	sylbid 230	. . . . . 6 |- ( ( A = B /\ ( 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 >. -> D = E ) )
12		opeq1 4402	. . . . . . . . 9 |- ( A = B -> <. A , C >. = <. B , C >. )
13		opeq1 4402	. . . . . . . . 9 |- ( D = E -> <. D , F >. = <. E , F >. )
14	12, 13	breqan12d 4669	. . . . . . . 8 |- ( ( A = B /\ D = E ) -> ( <. A , C >.Cgr <. D , F >. <-> <. B , C >.Cgr <. E , F >. ) )
15	14	exbiri 652	. . . . . . 7 |- ( A = B -> ( D = E -> ( <. B , C >.Cgr <. E , F >. -> <. A , C >.Cgr <. D , F >. ) ) )
16	15	adantr 481	. . . . . 6 |- ( ( A = B /\ ( 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 -> ( <. B , C >.Cgr <. E , F >. -> <. A , C >.Cgr <. D , F >. ) ) )
17	11, 16	syld 47	. . . . 5 |- ( ( A = B /\ ( 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 , C >.Cgr <. E , F >. -> <. A , C >.Cgr <. D , F >. ) ) )
18	17	impd 447	. . . 4 |- ( ( A = B /\ ( 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 , C >.Cgr <. E , F >. ) -> <. A , C >.Cgr <. D , F >. ) )
19	18	adantld 483	. . 3 |- ( ( A = B /\ ( 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 >. ) )
20	19	ex 450	. 2 |- ( A = B -> ( ( 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 >. ) ) )
21		simpl1 1064	. . . . . . . 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 =/= B /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> N e. NN )
22		simpl21 1139	. . . . . . . 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 =/= B /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> A e. ( EE ` N ) )
23		simpl22 1140	. . . . . . . 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 =/= B /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> B e. ( EE ` N ) )
24	21, 22, 23	3jca 1242	. . . . . . 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 ) ) ) /\ ( A =/= B /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) )
25		simpl23 1141	. . . . . . . 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 =/= B /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> C e. ( EE ` N ) )
26		simpl31 1142	. . . . . . . 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 =/= B /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> D e. ( EE ` N ) )
27	25, 22, 26	3jca 1242	. . . . . . 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 ) ) ) /\ ( A =/= B /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ D e. ( EE ` N ) ) )
28		simpl32 1143	. . . . . . . 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 =/= B /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> E e. ( EE ` N ) )
29		simpl33 1144	. . . . . . . 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 =/= B /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> F e. ( EE ` N ) )
30	28, 29, 26	3jca 1242	. . . . . . 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 ) ) ) /\ ( A =/= B /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ D e. ( EE ` N ) ) )
31	24, 27, 30	3jca 1242	. . . . . 6 |- ( ( ( 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 /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) )
32		simprrl 804	. . . . . . . 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 =/= B /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) )
33		simprrr 805	. . . . . . . 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 =/= B /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) )
34		cgrtriv 32109	. . . . . . . . . 10 |- ( ( N e. NN /\ A e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> <. A , A >.Cgr <. D , D >. )
35	21, 22, 26, 34	syl3anc 1326	. . . . . . . . 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 /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> <. A , A >.Cgr <. D , D >. )
36	33	simpld 475	. . . . . . . . . 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 /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> <. A , B >.Cgr <. D , E >. )
37		cgrcomlr 32105	. . . . . . . . . . 11 |- ( ( 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 >. ) )
38	21, 22, 23, 26, 28, 37	syl122anc 1335	. . . . . . . . . 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 /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> ( <. A , B >.Cgr <. D , E >. <-> <. B , A >.Cgr <. E , D >. ) )
39	36, 38	mpbid 222	. . . . . . . . 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 /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> <. B , A >.Cgr <. E , D >. )
40	35, 39	jca 554	. . . . . . . 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 =/= B /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> ( <. A , A >.Cgr <. D , D >. /\ <. B , A >.Cgr <. E , D >. ) )
41		brofs 32112	. . . . . . . . 9 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. <. A , B >. , <. C , A >. >. OuterFiveSeg <. <. D , E >. , <. F , D >. >. <-> ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) /\ ( <. A , A >.Cgr <. D , D >. /\ <. B , A >.Cgr <. E , D >. ) ) ) )
42	21, 22, 23, 25, 22, 26, 28, 29, 26, 41	syl333anc 1358	. . . . . . . 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 =/= B /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> ( <. <. A , B >. , <. C , A >. >. OuterFiveSeg <. <. D , E >. , <. F , D >. >. <-> ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) /\ ( <. A , A >.Cgr <. D , D >. /\ <. B , A >.Cgr <. E , D >. ) ) ) )
43	32, 33, 40, 42	mpbir3and 1245	. . . . . . 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 ) ) ) /\ ( A =/= B /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> <. <. A , B >. , <. C , A >. >. OuterFiveSeg <. <. D , E >. , <. F , D >. >. )
44		simprl 794	. . . . . . 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 ) ) ) /\ ( A =/= B /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> A =/= B )
45	43, 44	jca 554	. . . . . 6 |- ( ( ( 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 /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> ( <. <. A , B >. , <. C , A >. >. OuterFiveSeg <. <. D , E >. , <. F , D >. >. /\ A =/= B ) )
46		5segofs 32113	. . . . . 6 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( <. <. A , B >. , <. C , A >. >. OuterFiveSeg <. <. D , E >. , <. F , D >. >. /\ A =/= B ) -> <. C , A >.Cgr <. F , D >. ) )
47	31, 45, 46	sylc 65	. . . . 5 |- ( ( ( 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 /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> <. C , A >.Cgr <. F , D >. )
48		cgrcomlr 32105	. . . . . 6 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. C , A >.Cgr <. F , D >. <-> <. A , C >.Cgr <. D , F >. ) )
49	21, 25, 22, 29, 26, 48	syl122anc 1335	. . . . 5 |- ( ( ( 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 /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> ( <. C , A >.Cgr <. F , D >. <-> <. A , C >.Cgr <. D , F >. ) )
50	47, 49	mpbid 222	. . . 4 |- ( ( ( 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 /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) ) ) -> <. A , C >.Cgr <. D , F >. )
51	50	exp32 631	. . 3 |- ( ( 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 -> ( ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> <. A , C >.Cgr <. D , F >. ) ) )
52	51	com12 32	. 2 |- ( A =/= B -> ( ( 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 >. ) ) )
53	20, 52	pm2.61ine 2877	1 |- ( ( 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 >. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770   OuterFiveSeg cofs 32089

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

This theorem is referenced by:  cgrextendand  32116  segconeq  32117  lineext  32183  brofs2  32184