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Theorem ax5seg 25818
Description: The five segment axiom. Take two triangles A D C and E H G, a point B on A C, and a point F on E G. If all corresponding line segments except for C D and G H are congruent, then so are C D and G H. Axiom A5 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 12-Jun-2013.)
Assertion
Ref Expression
ax5seg |- ( ( ( 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 Btwn <. A , C >. /\ F Btwn <. E , G >. ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) -> <. C , D >.Cgr <. G , H >. ) )
Proof of Theorem ax5seg
Dummy variables i j t s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzfid 12772 . . . . . . . . . 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 ) ) ) -> ( 1 ... N ) e. Fin )
2 simpl21 1139 . . . . . . . . . . . . 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 ) ) ) /\ j e. ( 1 ... N ) ) -> C e. ( EE ` N ) )
3 fveere 25781 . . . . . . . . . . . . 13 |- ( ( C e. ( EE ` N ) /\ j e. ( 1 ... N ) ) -> ( C ` j ) e. RR )
42, 3sylancom 701 . . . . . . . . . . . 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 ) ) ) /\ j e. ( 1 ... N ) ) -> ( C ` j ) e. RR )
5 simpl22 1140 . . . . . . . . . . . . 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 ) ) ) /\ j e. ( 1 ... N ) ) -> D e. ( EE ` N ) )
6 fveere 25781 . . . . . . . . . . . . 13 |- ( ( D e. ( EE ` N ) /\ j e. ( 1 ... N ) ) -> ( D ` j ) e. RR )
75, 6sylancom 701 . . . . . . . . . . . 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 ) ) ) /\ j e. ( 1 ... N ) ) -> ( D ` j ) e. RR )
84, 7resubcld 10458 . . . . . . . . . . 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 ) ) ) /\ j e. ( 1 ... N ) ) -> ( ( C ` j ) - ( D ` j ) ) e. RR )
98resqcld 13035 . . . . . . . . . 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 ) ) ) /\ j e. ( 1 ... N ) ) -> ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) e. RR )
101, 9fsumrecl 14465 . . . . . . . . 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 ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) e. RR )
1110recnd 10068 . . . . . . . 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 ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) e. CC )
1211adantr 481 . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) e. CC )
13 simpl32 1143 . . . . . . . . . . . . 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 ) ) ) /\ j e. ( 1 ... N ) ) -> G e. ( EE ` N ) )
14 fveere 25781 . . . . . . . . . . . . 13 |- ( ( G e. ( EE ` N ) /\ j e. ( 1 ... N ) ) -> ( G ` j ) e. RR )
1513, 14sylancom 701 . . . . . . . . . . . 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 ) ) ) /\ j e. ( 1 ... N ) ) -> ( G ` j ) e. RR )
16 simpl33 1144 . . . . . . . . . . . . 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 ) ) ) /\ j e. ( 1 ... N ) ) -> H e. ( EE ` N ) )
17 fveere 25781 . . . . . . . . . . . . 13 |- ( ( H e. ( EE ` N ) /\ j e. ( 1 ... N ) ) -> ( H ` j ) e. RR )
1816, 17sylancom 701 . . . . . . . . . . . 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 ) ) ) /\ j e. ( 1 ... N ) ) -> ( H ` j ) e. RR )
1915, 18resubcld 10458 . . . . . . . . . . 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 ) ) ) /\ j e. ( 1 ... N ) ) -> ( ( G ` j ) - ( H ` j ) ) e. RR )
2019resqcld 13035 . . . . . . . . . 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 ) ) ) /\ j e. ( 1 ... N ) ) -> ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) e. RR )
211, 20fsumrecl 14465 . . . . . . . . 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 ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) e. RR )
2221recnd 10068 . . . . . . . 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 ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) e. CC )
2322adantr 481 . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) e. CC )
24 0re 10040 . . . . . . . . . . . . 13 |- 0 e. RR
25 1re 10039 . . . . . . . . . . . . 13 |- 1 e. RR
2624, 25elicc2i 12239 . . . . . . . . . . . 12 |- ( t e. ( 0 [,] 1 ) <-> ( t e. RR /\ 0 <_ t /\ t <_ 1 ) )
2726simp1bi 1076 . . . . . . . . . . 11 |- ( t e. ( 0 [,] 1 ) -> t e. RR )
2827recnd 10068 . . . . . . . . . 10 |- ( t e. ( 0 [,] 1 ) -> t e. CC )
2928ad2antrr 762 . . . . . . . . 9 |- ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) -> t e. CC )
30293ad2ant1 1082 . . . . . . . 8 |- ( ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) -> t e. CC )
3130adantl 482 . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> t e. CC )
32 simpl11 1136 . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> N e. NN )
33 simp12 1092 . . . . . . . . . 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 e. ( EE ` N ) )
34 simp13 1093 . . . . . . . . . 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 e. ( EE ` N ) )
35 simp21 1094 . . . . . . . . . 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 e. ( EE ` N ) )
3633, 34, 353jca 1242 . . . . . . . . 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 e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) )
3736adantr 481 . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) )
38 simprrl 804 . . . . . . . . . 10 |- ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) -> A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) )
39383ad2ant1 1082 . . . . . . . . 9 |- ( ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) -> A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) )
4039adantl 482 . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) )
41 simp1rl 1126 . . . . . . . . 9 |- ( ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) -> A =/= B )
4241adantl 482 . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> A =/= B )
43 ax5seglem4 25812 . . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A =/= B ) -> t =/= 0 )
4432, 37, 40, 42, 43syl211anc 1332 . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> t =/= 0 )
45 simpr3r 1123 . . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> <. B , D >.Cgr <. F , H >. )
46 simpl13 1138 . . . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> B e. ( EE ` N ) )
47 simpl22 1140 . . . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> D e. ( EE ` N ) )
48 simpl31 1142 . . . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> F e. ( EE ` N ) )
49 simpl33 1144 . . . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> H e. ( EE ` N ) )
50 brcgr 25780 . . . . . . . . . . . 12 |- ( ( ( B e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. B , D >.Cgr <. F , H >. <-> sum_ j e. ( 1 ... N ) ( ( ( B ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( F ` j ) - ( H ` j ) ) ^ 2 ) ) )
5146, 47, 48, 49, 50syl22anc 1327 . . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( <. B , D >.Cgr <. F , H >. <-> sum_ j e. ( 1 ... N ) ( ( ( B ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( F ` j ) - ( H ` j ) ) ^ 2 ) ) )
5245, 51mpbid 222 . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( B ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( F ` j ) - ( H ` j ) ) ^ 2 ) )
53 simp23 1096 . . . . . . . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> E e. ( EE ` N ) )
54 simp31 1097 . . . . . . . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> F e. ( EE ` N ) )
55 simp32 1098 . . . . . . . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> G e. ( EE ` N ) )
5653, 54, 553jca 1242 . . . . . . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ G e. ( EE ` N ) ) )
5736, 56jca 554 . . . . . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ G e. ( EE ` N ) ) ) )
5857adantr 481 . . . . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ G e. ( EE ` N ) ) ) )
59 simpr1l 1118 . . . . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) )
60 simprrr 805 . . . . . . . . . . . . . . . 16 |- ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) -> A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) )
61603ad2ant1 1082 . . . . . . . . . . . . . . 15 |- ( ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) -> A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) )
6261adantl 482 . . . . . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) )
6340, 62jca 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) )
64 simpr2l 1120 . . . . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> <. A , B >.Cgr <. E , F >. )
65 simpr2r 1121 . . . . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> <. B , C >.Cgr <. F , G >. )
66 ax5seglem6 25814 . . . . . . . . . . . . 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 /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) ) -> t = s )
6732, 58, 42, 59, 63, 64, 65, 66syl232anc 1353 . . . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> t = s )
6867oveq2d 6666 . . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( 1 - t ) = ( 1 - s ) )
6956adantr 481 . . . . . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ G e. ( EE ` N ) ) )
70 ax5seglem3 25811 . . . . . . . . . . . . . 14 |- ( ( ( 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 ) ) ) /\ ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( G ` j ) ) ^ 2 ) )
7132, 37, 69, 59, 63, 64, 65, 70syl322anc 1354 . . . . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( G ` j ) ) ^ 2 ) )
7267, 71oveq12d 6668 . . . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( t x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) = ( s x. sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( G ` j ) ) ^ 2 ) ) )
73 simpr3l 1122 . . . . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> <. A , D >.Cgr <. E , H >. )
74 simpl12 1137 . . . . . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> A e. ( EE ` N ) )
75 simpl23 1141 . . . . . . . . . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> E e. ( EE ` N ) )
76 brcgr 25780 . . . . . . . . . . . . . 14 |- ( ( ( A e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. A , D >.Cgr <. E , H >. <-> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( H ` j ) ) ^ 2 ) ) )
7774, 47, 75, 49, 76syl22anc 1327 . . . . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( <. A , D >.Cgr <. E , H >. <-> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( H ` j ) ) ^ 2 ) ) )
7873, 77mpbid 222 . . . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( H ` j ) ) ^ 2 ) )
7972, 78oveq12d 6668 . . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( ( t x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( D ` j ) ) ^ 2 ) ) = ( ( s x. sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( G ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( H ` j ) ) ^ 2 ) ) )
8068, 79oveq12d 6668 . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( ( 1 - t ) x. ( ( t x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( D ` j ) ) ^ 2 ) ) ) = ( ( 1 - s ) x. ( ( s x. sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( G ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( H ` j ) ) ^ 2 ) ) ) )
8152, 80oveq12d 6668 . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( sum_ j e. ( 1 ... N ) ( ( ( B ` j ) - ( D ` j ) ) ^ 2 ) + ( ( 1 - t ) x. ( ( t x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( D ` j ) ) ^ 2 ) ) ) ) = ( sum_ j e. ( 1 ... N ) ( ( ( F ` j ) - ( H ` j ) ) ^ 2 ) + ( ( 1 - s ) x. ( ( s x. sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( G ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( H ` j ) ) ^ 2 ) ) ) ) )
8233, 34jca 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) )
83 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 ) )
8482, 35, 83jca32 558 . . . . . . . . . . 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 e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) )
8584adantr 481 . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) )
86 simp1ll 1124 . . . . . . . . . . 11 |- ( ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) -> t e. ( 0 [,] 1 ) )
8786adantl 482 . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> t e. ( 0 [,] 1 ) )
88 ax5seglem9 25817 . . . . . . . . . 10 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) ) /\ ( t e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) ) -> ( t x. sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) ) = ( sum_ j e. ( 1 ... N ) ( ( ( B ` j ) - ( D ` j ) ) ^ 2 ) + ( ( 1 - t ) x. ( ( t x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( D ` j ) ) ^ 2 ) ) ) ) )
8932, 85, 87, 40, 88syl22anc 1327 . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( t x. sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) ) = ( sum_ j e. ( 1 ... N ) ( ( ( B ` j ) - ( D ` j ) ) ^ 2 ) + ( ( 1 - t ) x. ( ( t x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( D ` j ) ) ^ 2 ) ) ) ) )
90 3simpc 1060 . . . . . . . . . . . . 13 |- ( ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) -> ( G e. ( EE ` N ) /\ H e. ( EE ` N ) ) )
91903ad2ant3 1084 . . . . . . . . . . . 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 ) ) ) -> ( G e. ( EE ` N ) /\ H e. ( EE ` N ) ) )
9253, 54, 91jca31 557 . . . . . . . . . . 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 ) ) ) -> ( ( E e. ( EE ` N ) /\ F e. ( EE ` N ) ) /\ ( G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) )
9392adantr 481 . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( ( E e. ( EE ` N ) /\ F e. ( EE ` N ) ) /\ ( G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) )
94 simp1lr 1125 . . . . . . . . . . 11 |- ( ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) -> s e. ( 0 [,] 1 ) )
9594adantl 482 . . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> s e. ( 0 [,] 1 ) )
96 ax5seglem9 25817 . . . . . . . . . 10 |- ( ( ( N e. NN /\ ( ( E e. ( EE ` N ) /\ F e. ( EE ` N ) ) /\ ( G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) ) /\ ( s e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) -> ( s x. sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) = ( sum_ j e. ( 1 ... N ) ( ( ( F ` j ) - ( H ` j ) ) ^ 2 ) + ( ( 1 - s ) x. ( ( s x. sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( G ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( H ` j ) ) ^ 2 ) ) ) ) )
9732, 93, 95, 62, 96syl22anc 1327 . . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( s x. sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) = ( sum_ j e. ( 1 ... N ) ( ( ( F ` j ) - ( H ` j ) ) ^ 2 ) + ( ( 1 - s ) x. ( ( s x. sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( G ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( E ` j ) - ( H ` j ) ) ^ 2 ) ) ) ) )
9881, 89, 973eqtr4d 2666 . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( t x. sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) ) = ( s x. sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) )
9967oveq1d 6665 . . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( t x. sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) = ( s x. sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) )
10098, 99eqtr4d 2659 . . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> ( t x. sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) ) = ( t x. sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) )
10112, 23, 31, 44, 100mulcanad 10662 . . . . . 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 ) ) ) /\ ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) )
1021013exp2 1285 . . . . 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 ) ) ) -> ( ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) /\ ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) -> ( ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) -> ( ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) -> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) ) ) )
103102expd 452 . . . 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 ) ) ) -> ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> ( ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) -> ( ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) -> ( ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) -> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) ) ) ) )
104103rexlimdvv 3037 . . 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 ) ) ) -> ( E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) -> ( ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) -> ( ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) -> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) ) ) )
1051043impd 1281 . 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 ) ) ) -> ( ( E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) -> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) )
106 brbtwn 25779 . . . . . . . 8 |- ( ( B e. ( EE ` N ) /\ A e. ( EE ` N ) /\ C e. ( EE ` N ) ) -> ( B Btwn <. A , C >. <-> E. t e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) )
10734, 33, 35, 106syl3anc 1326 . . . . . . 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 <. A , C >. <-> E. t e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) )
108 brbtwn 25779 . . . . . . . 8 |- ( ( F e. ( EE ` N ) /\ E e. ( EE ` N ) /\ G e. ( EE ` N ) ) -> ( F Btwn <. E , G >. <-> E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) )
10954, 53, 55, 108syl3anc 1326 . . . . . . 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 ) ) ) -> ( F Btwn <. E , G >. <-> E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) )
110107, 109anbi12d 747 . . . . . 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 <. A , C >. /\ F Btwn <. E , G >. ) <-> ( E. t e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) )
111 reeanv 3107 . . . . . 6 |- ( E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) <-> ( E. t e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) )
112110, 111syl6bbr 278 . . . . 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 ) ) ) -> ( ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) <-> E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) )
113112anbi2d 740 . . . 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 =/= B /\ ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) ) <-> ( A =/= B /\ E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) )
114 3anass 1042 . . . 4 |- ( ( A =/= B /\ B Btwn <. A , C >. /\ F Btwn <. E , G >. ) <-> ( A =/= B /\ ( B Btwn <. A , C >. /\ F Btwn <. E , G >. ) ) )
115 r19.42v 3092 . . . . . 6 |- ( E. s e. ( 0 [,] 1 ) ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) <-> ( A =/= B /\ E. s e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) )
116115rexbii 3041 . . . . 5 |- ( E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) <-> E. t e. ( 0 [,] 1 ) ( A =/= B /\ E. s e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) )
117 r19.42v 3092 . . . . 5 |- ( E. t e. ( 0 [,] 1 ) ( A =/= B /\ E. s e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) <-> ( A =/= B /\ E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) )
118116, 117bitri 264 . . . 4 |- ( E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) <-> ( A =/= B /\ E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) )
119113, 114, 1183bitr4g 303 . . 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 /\ B Btwn <. A , C >. /\ F Btwn <. E , G >. ) <-> E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) ) )
1201193anbi1d 1403 . 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 /\ B Btwn <. A , C >. /\ F Btwn <. E , G >. ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) <-> ( E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A =/= B /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( F ` i ) = ( ( ( 1 - s ) x. ( E ` i ) ) + ( s x. ( G ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) ) )
121 simp33 1099 . . 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 ) ) ) -> H e. ( EE ` N ) )
122 brcgr 25780 . . 3 |- ( ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. C , D >.Cgr <. G , H >. <-> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) )
12335, 83, 55, 121, 122syl22anc 1327 . 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 ) ) ) -> ( <. C , D >.Cgr <. G , H >. <-> sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( G ` j ) - ( H ` j ) ) ^ 2 ) ) )
124105, 120, 1233imtr4d 283 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 /\ B Btwn <. A , C >. /\ F Btwn <. E , G >. ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) -> <. C , D >.Cgr <. G , H >. ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   NNcn 11020   2c2 11070   [,]cicc 12178   ...cfz 12326   ^cexp 12860   sum_csu 14416   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770
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
This theorem is referenced by:  eengtrkg  25865  5segofs  32113
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