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Theorem btwnxfr 32163
Description: A condition for extending betweenness to a new set of points based on congruence with another set of points. Theorem 4.6 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 4-Oct-2013.)
Assertion
Ref Expression
btwnxfr |- ( ( 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 , C >. >.Cgr3 <. D , <. E , F >. >. ) -> E Btwn <. D , F >. ) )
Proof of Theorem btwnxfr
Dummy variable e is distinct from all other variables.
StepHypRef Expression
1 brcgr3 32153 . . . . . 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 , C >. >.Cgr3 <. D , <. E , F >. >. <-> ( <. A , B >.Cgr <. D , E >. /\ <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) )
2 simp2 1062 . . . . . 6 |- ( ( <. A , B >.Cgr <. D , E >. /\ <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) -> <. A , C >.Cgr <. D , F >. )
31, 2syl6bi 243 . . . . 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 , C >. >.Cgr3 <. D , <. E , F >. >. -> <. A , C >.Cgr <. D , F >. ) )
4 simp1 1061 . . . . . 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 ) ) ) -> N e. NN )
5 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 ) ) ) -> A e. ( EE ` N ) )
6 simp22 1095 . . . . . 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 ) ) ) -> B e. ( EE ` N ) )
7 simp23 1096 . . . . . 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 ) ) ) -> C e. ( EE ` N ) )
8 simp31 1097 . . . . . 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 ) ) ) -> D e. ( EE ` N ) )
9 simp33 1099 . . . . . 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 ) ) ) -> F e. ( EE ` N ) )
10 cgrxfr 32162 . . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , C >. /\ <. A , C >.Cgr <. D , F >. ) -> E. e e. ( EE ` N ) ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) ) )
114, 5, 6, 7, 8, 9, 10syl132anc 1344 . . . . 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 ) ) ) -> ( ( B Btwn <. A , C >. /\ <. A , C >.Cgr <. D , F >. ) -> E. e e. ( EE ` N ) ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) ) )
123, 11sylan2d 499 . . . 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 ) ) ) -> ( ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) -> E. e e. ( EE ` N ) ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) ) )
1312imp 445 . . 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 ) ) ) /\ ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) ) -> E. e e. ( EE ` N ) ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) )
14 simprrl 804 . . . . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) /\ ( ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) /\ ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) ) ) -> e Btwn <. D , F >. )
1514, 14jca 554 . . . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) /\ ( ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) /\ ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) ) ) -> ( e Btwn <. D , F >. /\ e Btwn <. D , F >. ) )
16 simpl1 1064 . . . . . . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) -> N e. NN )
17 simpl31 1142 . . . . . . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) -> D e. ( EE ` N ) )
18 simpl33 1144 . . . . . . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) -> F e. ( EE ` N ) )
1916, 17, 18cgrrflxd 32095 . . . . . . . . . . . . . 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 e. ( EE ` N ) ) -> <. D , F >.Cgr <. D , F >. )
20 simpr 477 . . . . . . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) -> e e. ( EE ` N ) )
2116, 20, 18cgrrflxd 32095 . . . . . . . . . . . . . 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 e. ( EE ` N ) ) -> <. e , F >.Cgr <. e , F >. )
2219, 21jca 554 . . . . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) -> ( <. D , F >.Cgr <. D , F >. /\ <. e , F >.Cgr <. e , F >. ) )
2322adantr 481 . . . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) /\ ( ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) /\ ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) ) ) -> ( <. D , F >.Cgr <. D , F >. /\ <. e , F >.Cgr <. e , F >. ) )
24 simpr 477 . . . . . . . . . . . . . 14 |- ( ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) -> <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. )
25 simpr 477 . . . . . . . . . . . . . 14 |- ( ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) -> <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. )
26 simpl2 1065 . . . . . . . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) -> ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) )
27 simpl3 1066 . . . . . . . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) -> ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) )
2817, 20, 183jca 1242 . . . . . . . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) -> ( D e. ( EE ` N ) /\ e e. ( EE ` N ) /\ F e. ( EE ` N ) ) )
29 cgr3tr4 32159 . . . . . . . . . . . . . . . 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 ) ) /\ ( D e. ( EE ` N ) /\ e e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) -> ( ( <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) -> <. D , <. E , F >. >.Cgr3 <. D , <. e , F >. >. ) )
3016, 26, 27, 28, 29syl13anc 1328 . . . . . . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) -> ( ( <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) -> <. D , <. E , F >. >.Cgr3 <. D , <. e , F >. >. ) )
31 cgr3com 32160 . . . . . . . . . . . . . . . . 17 |- ( ( N e. NN /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ e e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) -> ( <. D , <. E , F >. >.Cgr3 <. D , <. e , F >. >. <-> <. D , <. e , F >. >.Cgr3 <. D , <. E , F >. >. ) )
3216, 27, 17, 20, 18, 31syl113anc 1338 . . . . . . . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) -> ( <. D , <. E , F >. >.Cgr3 <. D , <. e , F >. >. <-> <. D , <. e , F >. >.Cgr3 <. D , <. E , F >. >. ) )
33 simpl32 1143 . . . . . . . . . . . . . . . . . 18 |- ( ( ( 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 ) ) -> E e. ( EE ` N ) )
34 brcgr3 32153 . . . . . . . . . . . . . . . . . 18 |- ( ( N e. NN /\ ( D e. ( EE ` N ) /\ e e. ( EE ` N ) /\ F e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) -> ( <. D , <. e , F >. >.Cgr3 <. D , <. E , F >. >. <-> ( <. D , e >.Cgr <. D , E >. /\ <. D , F >.Cgr <. D , F >. /\ <. e , F >.Cgr <. E , F >. ) ) )
3516, 17, 20, 18, 17, 33, 18, 34syl133anc 1349 . . . . . . . . . . . . . . . . 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 ) ) -> ( <. D , <. e , F >. >.Cgr3 <. D , <. E , F >. >. <-> ( <. D , e >.Cgr <. D , E >. /\ <. D , F >.Cgr <. D , F >. /\ <. e , F >.Cgr <. E , F >. ) ) )
36 simpr1 1067 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( 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 ) ) /\ ( <. D , e >.Cgr <. D , E >. /\ <. D , F >.Cgr <. D , F >. /\ <. e , F >.Cgr <. E , F >. ) ) -> <. D , e >.Cgr <. D , E >. )
37 simpr3 1069 . . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( 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 ) ) /\ ( <. D , e >.Cgr <. D , E >. /\ <. D , F >.Cgr <. D , F >. /\ <. e , F >.Cgr <. E , F >. ) ) -> <. e , F >.Cgr <. E , F >. )
3816, 20, 18, 33, 18, 37cgrcomlrand 32108 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( 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 ) ) /\ ( <. D , e >.Cgr <. D , E >. /\ <. D , F >.Cgr <. D , F >. /\ <. e , F >.Cgr <. E , F >. ) ) -> <. F , e >.Cgr <. F , E >. )
3936, 38jca 554 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ( 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 ) ) /\ ( <. D , e >.Cgr <. D , E >. /\ <. D , F >.Cgr <. D , F >. /\ <. e , F >.Cgr <. E , F >. ) ) -> ( <. D , e >.Cgr <. D , E >. /\ <. F , e >.Cgr <. F , E >. ) )
4039ex 450 . . . . . . . . . . . . . . . . 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 ) ) -> ( ( <. D , e >.Cgr <. D , E >. /\ <. D , F >.Cgr <. D , F >. /\ <. e , F >.Cgr <. E , F >. ) -> ( <. D , e >.Cgr <. D , E >. /\ <. F , e >.Cgr <. F , E >. ) ) )
4135, 40sylbid 230 . . . . . . . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) -> ( <. D , <. e , F >. >.Cgr3 <. D , <. E , F >. >. -> ( <. D , e >.Cgr <. D , E >. /\ <. F , e >.Cgr <. F , E >. ) ) )
4232, 41sylbid 230 . . . . . . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) -> ( <. D , <. E , F >. >.Cgr3 <. D , <. e , F >. >. -> ( <. D , e >.Cgr <. D , E >. /\ <. F , e >.Cgr <. F , E >. ) ) )
4330, 42syld 47 . . . . . . . . . . . . . 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 e. ( EE ` N ) ) -> ( ( <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) -> ( <. D , e >.Cgr <. D , E >. /\ <. F , e >.Cgr <. F , E >. ) ) )
4424, 25, 43syl2ani 688 . . . . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) -> ( ( ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) /\ ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) ) -> ( <. D , e >.Cgr <. D , E >. /\ <. F , e >.Cgr <. F , E >. ) ) )
4544imp 445 . . . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) /\ ( ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) /\ ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) ) ) -> ( <. D , e >.Cgr <. D , E >. /\ <. F , e >.Cgr <. F , E >. ) )
4615, 23, 453jca 1242 . . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) /\ ( ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) /\ ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) ) ) -> ( ( e Btwn <. D , F >. /\ e Btwn <. D , F >. ) /\ ( <. D , F >.Cgr <. D , F >. /\ <. e , F >.Cgr <. e , F >. ) /\ ( <. D , e >.Cgr <. D , E >. /\ <. F , e >.Cgr <. F , E >. ) ) )
4746ex 450 . . . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) -> ( ( ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) /\ ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) ) -> ( ( e Btwn <. D , F >. /\ e Btwn <. D , F >. ) /\ ( <. D , F >.Cgr <. D , F >. /\ <. e , F >.Cgr <. e , F >. ) /\ ( <. D , e >.Cgr <. D , E >. /\ <. F , e >.Cgr <. F , E >. ) ) ) )
48 brifs 32150 . . . . . . . . . . . 12 |- ( ( ( N e. NN /\ D e. ( EE ` N ) /\ e e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ e e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( e e. ( EE ` N ) /\ F e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( <. <. D , e >. , <. F , e >. >. InnerFiveSeg <. <. D , e >. , <. F , E >. >. <-> ( ( e Btwn <. D , F >. /\ e Btwn <. D , F >. ) /\ ( <. D , F >.Cgr <. D , F >. /\ <. e , F >.Cgr <. e , F >. ) /\ ( <. D , e >.Cgr <. D , E >. /\ <. F , e >.Cgr <. F , E >. ) ) ) )
49 ifscgr 32151 . . . . . . . . . . . 12 |- ( ( ( N e. NN /\ D e. ( EE ` N ) /\ e e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ e e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( e e. ( EE ` N ) /\ F e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( <. <. D , e >. , <. F , e >. >. InnerFiveSeg <. <. D , e >. , <. F , E >. >. -> <. e , e >.Cgr <. e , E >. ) )
5048, 49sylbird 250 . . . . . . . . . . 11 |- ( ( ( N e. NN /\ D e. ( EE ` N ) /\ e e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ e e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( e e. ( EE ` N ) /\ F e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( ( ( e Btwn <. D , F >. /\ e Btwn <. D , F >. ) /\ ( <. D , F >.Cgr <. D , F >. /\ <. e , F >.Cgr <. e , F >. ) /\ ( <. D , e >.Cgr <. D , E >. /\ <. F , e >.Cgr <. F , E >. ) ) -> <. e , e >.Cgr <. e , E >. ) )
5116, 17, 20, 18, 20, 17, 20, 18, 33, 50syl333anc 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 ) ) ) /\ e e. ( EE ` N ) ) -> ( ( ( e Btwn <. D , F >. /\ e Btwn <. D , F >. ) /\ ( <. D , F >.Cgr <. D , F >. /\ <. e , F >.Cgr <. e , F >. ) /\ ( <. D , e >.Cgr <. D , E >. /\ <. F , e >.Cgr <. F , E >. ) ) -> <. e , e >.Cgr <. e , E >. ) )
5247, 51syld 47 . . . . . . . . 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 ) ) -> ( ( ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) /\ ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) ) -> <. e , e >.Cgr <. e , E >. ) )
53 cgrid2 32110 . . . . . . . . . 10 |- ( ( N e. NN /\ ( e e. ( EE ` N ) /\ e e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( <. e , e >.Cgr <. e , E >. -> e = E ) )
5416, 20, 20, 33, 53syl13anc 1328 . . . . . . . . 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 ) ) -> ( <. e , e >.Cgr <. e , E >. -> e = E ) )
5552, 54syld 47 . . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) -> ( ( ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) /\ ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) ) -> e = E ) )
5655imp 445 . . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) /\ ( ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) /\ ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) ) ) -> e = E )
5756, 14eqbrtrrd 4677 . . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) /\ ( ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) /\ ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) ) ) -> E Btwn <. D , F >. )
5857expr 643 . . . . 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 ) ) ) /\ e e. ( EE ` N ) ) /\ ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) ) -> ( ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) -> E Btwn <. D , F >. ) )
5958an32s 846 . . . 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 ) ) ) /\ ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) ) /\ e e. ( EE ` N ) ) -> ( ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) -> E Btwn <. D , F >. ) )
6059rexlimdva 3031 . . 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 ) ) ) /\ ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) ) -> ( E. e e. ( EE ` N ) ( e Btwn <. D , F >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. e , F >. >. ) -> E Btwn <. D , F >. ) )
6113, 60mpd 15 . 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 ) ) ) /\ ( B Btwn <. A , C >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) ) -> E Btwn <. D , F >. )
6261ex 450 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 >. /\ <. A , <. B , C >. >.Cgr3 <. D , <. E , F >. >. ) -> E Btwn <. D , F >. ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770   InnerFiveSeg cifs 32142  Cgr3ccgr3 32143
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-ifs 32147  df-cgr3 32148
This theorem is referenced by:  colinearxfr  32182  brofs2  32184  brifs2  32185  endofsegid  32192  brsegle2  32216
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