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Theorem fscgr 32187
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
StepHypRef Expression
1 brfs 32186 . . 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 >. ) ) ) )
21anbi1d 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 ) ) /\ ( 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 1091 . . . . . 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 1092 . . . . . 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 1093 . . . . . 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 1094 . . . . . 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 32166 . . . . . 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 >. ) ) )
83, 4, 5, 6, 7syl13anc 1328 . . . . 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 1096 . . . . . . . . . . . . 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 1097 . . . . . . . . . . . . 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 1098 . . . . . . . . . . . . 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 32156 . . . . . . . . . . . . 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 >. >. ) )
133, 4, 5, 6, 9, 10, 11, 12syl133anc 1349 . . . . . . . . . . . 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 466 . . . . . . . . . . . . 13 |- ( ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) <-> ( <. B , D >.Cgr <. F , H >. /\ <. A , D >.Cgr <. E , H >. ) )
1514a1i 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 >. ) ) )
1613, 153anbi23d 1402 . . . . . . . . . . 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 1095 . . . . . . . . . . . 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 1099 . . . . . . . . . . . 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 32184 . . . . . . . . . . . 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 >. ) ) ) )
203, 5, 4, 6, 17, 10, 9, 11, 18, 19syl333anc 1358 . . . . . . . . . . 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 >. ) ) ) )
2116, 20bitr4d 271 . . . . . . . . . 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 2847 . . . . . . . . . . 11 |- ( A =/= B <-> B =/= A )
2322a1i 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 ) )
2421, 23anbi12d 747 . . . . . . . . 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 32113 . . . . . . . . . 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 >. ) )
263, 5, 4, 6, 17, 10, 9, 11, 18, 25syl333anc 1358 . . . . . . . . 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 >. ) )
2724, 26sylbid 230 . . . . . . . 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 >. ) )
2827expd 452 . . . . . . 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 >. ) ) )
29283expd 1284 . . . . . 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 32121 . . . . . . . . . . . . 13 |- ( ( N e. NN /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) /\ A e. ( EE ` N ) ) ) -> ( B Btwn <. C , A >. <-> B Btwn <. A , C >. ) )
313, 5, 6, 4, 30syl13anc 1328 . . . . . . . . . . . 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 >. ) )
32313anbi1d 1403 . . . . . . . . . . 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 32184 . . . . . . . . . . 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 >. ) ) ) )
3432, 33bitr4d 271 . . . . . . . . . 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 >. >. ) )
3534anbi1d 741 . . . . . . . . 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 32113 . . . . . . . . 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 >. ) )
3735, 36sylbid 230 . . . . . . . 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 >. ) )
3837expd 452 . . . . . . 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 >. ) ) )
39383expd 1284 . . . . . 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 32155 . . . . . . . . . . . 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 >. >. ) )
413, 4, 5, 6, 9, 10, 11, 40syl133anc 1349 . . . . . . . . . . 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 >. >. ) )
42413anbi2d 1404 . . . . . . . . . 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 32185 . . . . . . . . . . 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 >. ) ) ) )
443, 4, 6, 5, 17, 9, 11, 10, 18, 43syl333anc 1358 . . . . . . . . . 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 >. ) ) ) )
4542, 44bitr4d 271 . . . . . . . . 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 32151 . . . . . . . . . 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 >. ) )
473, 4, 6, 5, 17, 9, 11, 10, 18, 46syl333anc 1358 . . . . . . . . 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 >. ) )
4845, 47sylbid 230 . . . . . . . 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 >. ) )
4948a1dd 50 . . . . . . 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 >. ) ) )
50493expd 1284 . . . . . 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 >. ) ) ) ) )
5129, 39, 503jaod 1392 . . . . 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 >. ) ) ) ) )
528, 51sylbid 230 . . . 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 >. ) ) ) ) )
53523impd 1281 . . 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 >. ) ) )
5453impd 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 ) ) /\ ( 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 >. ) )
552, 54sylbid 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 ) ) /\ ( 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 >. ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   /\ w3a 1037   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   InnerFiveSeg cifs 32142  Cgr3ccgr3 32143   Colinear ccolin 32144   FiveSeg cfs 32145
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-fs 32149
This theorem is referenced by:  linecgr  32188
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