Theorem fscgr 30847

Description: Congruence law for the general five segment configuration. Theorem 4.16 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)

Assertion

Ref	Expression
fscgr	|- ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( <. <. A , B >. , <. C , D >. >. FiveSeg <. <. E , F >. , <. G , H >. >. /\ A =/= B ) -> <. C , D >.Cgr <. G , H >. ) )

Proof of Theorem fscgr

Step	Hyp	Ref	Expression
1		brfs 30846	. . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. <. A , B >. , <. C , D >. >. FiveSeg <. <. E , F >. , <. G , H >. >. <-> ( A Colinear <. B , C >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) )
2	1	anbi1d 711	. 2 |- ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( <. <. A , B >. , <. C , D >. >. FiveSeg <. <. E , F >. , <. G , H >. >. /\ A =/= B ) <-> ( ( A Colinear <. B , C >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) /\ A =/= B ) ) )
3		simp11 1038	. . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> N e. NN )
4		simp12 1039	. . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
5		simp13 1040	. . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
6		simp21 1041	. . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
7		brcolinear 30826	. . . . . 6 |- ( ( 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 >. ) ) )
8	3, 4, 5, 6, 7	syl13anc 1270	. . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( A Colinear <. B , C >. <-> ( A Btwn <. B , C >. \/ B Btwn <. C , A >. \/ C Btwn <. A , B >. ) ) )
9		simp23 1043	. . . . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> E e. ( EE ` N ) )
10		simp31 1044	. . . . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> F e. ( EE ` N ) )
11		simp32 1045	. . . . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> G e. ( EE ` N ) )
12		cgr3permute2 30816	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ G e. ( EE ` N ) ) ) -> ( <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. <-> <. B , <. A , C >. >.Cgr3 <. F , <. E , G >. >. ) )
13	3, 4, 5, 6, 9, 10, 11, 12	syl133anc 1291	. . . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. <-> <. B , <. A , C >. >.Cgr3 <. F , <. E , G >. >. ) )
14		ancom 452	. . . . . . . . . . . . 13 |- ( ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) <-> ( <. B , D >.Cgr <. F , H >. /\ <. A , D >.Cgr <. E , H >. ) )
15	14	a1i 11	. . . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) <-> ( <. B , D >.Cgr <. F , H >. /\ <. A , D >.Cgr <. E , H >. ) ) )
16	13, 15	3anbi23d 1342	. . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( A Btwn <. B , C >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) <-> ( A Btwn <. B , C >. /\ <. B , <. A , C >. >.Cgr3 <. F , <. E , G >. >. /\ ( <. B , D >.Cgr <. F , H >. /\ <. A , D >.Cgr <. E , H >. ) ) ) )
17		simp22 1042	. . . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> D e. ( EE ` N ) )
18		simp33 1046	. . . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> H e. ( EE ` N ) )
19		brofs2 30844	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ B e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ F e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. <. B , A >. , <. C , D >. >. OuterFiveSeg <. <. F , E >. , <. G , H >. >. <-> ( A Btwn <. B , C >. /\ <. B , <. A , C >. >.Cgr3 <. F , <. E , G >. >. /\ ( <. B , D >.Cgr <. F , H >. /\ <. A , D >.Cgr <. E , H >. ) ) ) )
20	3, 5, 4, 6, 17, 10, 9, 11, 18, 19	syl333anc 1300	. . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. <. B , A >. , <. C , D >. >. OuterFiveSeg <. <. F , E >. , <. G , H >. >. <-> ( A Btwn <. B , C >. /\ <. B , <. A , C >. >.Cgr3 <. F , <. E , G >. >. /\ ( <. B , D >.Cgr <. F , H >. /\ <. A , D >.Cgr <. E , H >. ) ) ) )
21	16, 20	bitr4d 260	. . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( A Btwn <. B , C >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) <-> <. <. B , A >. , <. C , D >. >. OuterFiveSeg <. <. F , E >. , <. G , H >. >. ) )
22		necom 2677	. . . . . . . . . . 11 |- ( A =/= B <-> B =/= A )
23	22	a1i 11	. . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( A =/= B <-> B =/= A ) )
24	21, 23	anbi12d 717	. . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( ( A Btwn <. B , C >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) /\ A =/= B ) <-> ( <. <. B , A >. , <. C , D >. >. OuterFiveSeg <. <. F , E >. , <. G , H >. >. /\ B =/= A ) ) )
25		5segofs 30773	. . . . . . . . . 10 |- ( ( ( N e. NN /\ B e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ F e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( <. <. B , A >. , <. C , D >. >. OuterFiveSeg <. <. F , E >. , <. G , H >. >. /\ B =/= A ) -> <. C , D >.Cgr <. G , H >. ) )
26	3, 5, 4, 6, 17, 10, 9, 11, 18, 25	syl333anc 1300	. . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( <. <. B , A >. , <. C , D >. >. OuterFiveSeg <. <. F , E >. , <. G , H >. >. /\ B =/= A ) -> <. C , D >.Cgr <. G , H >. ) )
27	24, 26	sylbid 219	. . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( ( A Btwn <. B , C >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) /\ A =/= B ) -> <. C , D >.Cgr <. G , H >. ) )
28	27	expd 438	. . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( A Btwn <. B , C >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) -> ( A =/= B -> <. C , D >.Cgr <. G , H >. ) ) )
29	28	3expd 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 ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( A Btwn <. B , C >. -> ( <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. -> ( ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) -> ( A =/= B -> <. C , D >.Cgr <. G , H >. ) ) ) ) )
30		btwncom 30781	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) /\ A e. ( EE ` N ) ) ) -> ( B Btwn <. C , A >. <-> B Btwn <. A , C >. ) )
31	3, 5, 6, 4, 30	syl13anc 1270	. . . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( B Btwn <. C , A >. <-> B Btwn <. A , C >. ) )
32	31	3anbi1d 1343	. . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( B Btwn <. C , A >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) <-> ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) )
33		brofs2 30844	. . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. <. A , B >. , <. C , D >. >. OuterFiveSeg <. <. E , F >. , <. G , H >. >. <-> ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) )
34	32, 33	bitr4d 260	. . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( B Btwn <. C , A >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) <-> <. <. A , B >. , <. C , D >. >. OuterFiveSeg <. <. E , F >. , <. G , H >. >. ) )
35	34	anbi1d 711	. . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( ( B Btwn <. C , A >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) /\ A =/= B ) <-> ( <. <. A , B >. , <. C , D >. >. OuterFiveSeg <. <. E , F >. , <. G , H >. >. /\ A =/= B ) ) )
36		5segofs 30773	. . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( <. <. A , B >. , <. C , D >. >. OuterFiveSeg <. <. E , F >. , <. G , H >. >. /\ A =/= B ) -> <. C , D >.Cgr <. G , H >. ) )
37	35, 36	sylbid 219	. . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( ( B Btwn <. C , A >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) /\ A =/= B ) -> <. C , D >.Cgr <. G , H >. ) )
38	37	expd 438	. . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( B Btwn <. C , A >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) -> ( A =/= B -> <. C , D >.Cgr <. G , H >. ) ) )
39	38	3expd 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 ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( B Btwn <. C , A >. -> ( <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. -> ( ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) -> ( A =/= B -> <. C , D >.Cgr <. G , H >. ) ) ) ) )
40		cgr3permute1 30815	. . . . . . . . . . . 12 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ G e. ( EE ` N ) ) ) -> ( <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. <-> <. A , <. C , B >. >.Cgr3 <. E , <. G , F >. >. ) )
41	3, 4, 5, 6, 9, 10, 11, 40	syl133anc 1291	. . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. <-> <. A , <. C , B >. >.Cgr3 <. E , <. G , F >. >. ) )
42	41	3anbi2d 1344	. . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( C Btwn <. A , B >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) <-> ( C Btwn <. A , B >. /\ <. A , <. C , B >. >.Cgr3 <. E , <. G , F >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) )
43		brifs2 30845	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( G e. ( EE ` N ) /\ F e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. <. A , C >. , <. B , D >. >. InnerFiveSeg <. <. E , G >. , <. F , H >. >. <-> ( C Btwn <. A , B >. /\ <. A , <. C , B >. >.Cgr3 <. E , <. G , F >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) )
44	3, 4, 6, 5, 17, 9, 11, 10, 18, 43	syl333anc 1300	. . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. <. A , C >. , <. B , D >. >. InnerFiveSeg <. <. E , G >. , <. F , H >. >. <-> ( C Btwn <. A , B >. /\ <. A , <. C , B >. >.Cgr3 <. E , <. G , F >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) )
45	42, 44	bitr4d 260	. . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( C Btwn <. A , B >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) <-> <. <. A , C >. , <. B , D >. >. InnerFiveSeg <. <. E , G >. , <. F , H >. >. ) )
46		ifscgr 30811	. . . . . . . . . 10 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( G e. ( EE ` N ) /\ F e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. <. A , C >. , <. B , D >. >. InnerFiveSeg <. <. E , G >. , <. F , H >. >. -> <. C , D >.Cgr <. G , H >. ) )
47	3, 4, 6, 5, 17, 9, 11, 10, 18, 46	syl333anc 1300	. . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. <. A , C >. , <. B , D >. >. InnerFiveSeg <. <. E , G >. , <. F , H >. >. -> <. C , D >.Cgr <. G , H >. ) )
48	45, 47	sylbid 219	. . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( C Btwn <. A , B >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) -> <. C , D >.Cgr <. G , H >. ) )
49	48	a1dd 47	. . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( C Btwn <. A , B >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) -> ( A =/= B -> <. C , D >.Cgr <. G , H >. ) ) )
50	49	3expd 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 ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( C Btwn <. A , B >. -> ( <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. -> ( ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) -> ( A =/= B -> <. C , D >.Cgr <. G , H >. ) ) ) ) )
51	29, 39, 50	3jaod 1332	. . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( A Btwn <. B , C >. \/ B Btwn <. C , A >. \/ C Btwn <. A , B >. ) -> ( <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. -> ( ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) -> ( A =/= B -> <. C , D >.Cgr <. G , H >. ) ) ) ) )
52	8, 51	sylbid 219	. . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( A Colinear <. B , C >. -> ( <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. -> ( ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) -> ( A =/= B -> <. C , D >.Cgr <. G , H >. ) ) ) ) )
53	52	3impd 1223	. . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( A Colinear <. B , C >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) -> ( A =/= B -> <. C , D >.Cgr <. G , H >. ) ) )
54	53	impd 433	. 2 |- ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( ( A Colinear <. B , C >. /\ <. A , <. B , C >. >.Cgr3 <. E , <. F , G >. >. /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) /\ A =/= B ) -> <. C , D >.Cgr <. G , H >. ) )
55	2, 54	sylbid 219	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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( <. <. A , B >. , <. C , D >. >. FiveSeg <. <. E , F >. , <. G , H >. >. /\ A =/= B ) -> <. C , D >.Cgr <. G , H >. ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   \/ w3o 984   /\ w3a 985   e. wcel 1887   =/= wne 2622   <.cop 3974   class class class wbr 4402   ` cfv 5582   NNcn 10609   EEcee 24918   Btwn cbtwn 24919  Cgrccgr 24920   OuterFiveSeg cofs 30749   InnerFiveSeg cifs 30802  Cgr3ccgr3 30803   Colinear ccolin 30804   FiveSeg cfs 30805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-ee 24921  df-btwn 24922  df-cgr 24923  df-ofs 30750  df-colinear 30806  df-ifs 30807  df-cgr3 30808  df-fs 30809

This theorem is referenced by:  linecgr  30848