Theorem ax5seglem6 25814

Description: Lemma for ax5seg 25818. Given two line segments that are divided into pieces, if the pieces are congruent, then the scaling constant is the same. (Contributed by Scott Fenton, 12-Jun-2013.)

Assertion

Ref

Expression

ax5seglem6

|- ( ( ( N e. NN /\ ( ( 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 /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> T = S )

Distinct variable groups:   A, i   B, i   C, i   D, i   i, E   i, F   i, N   S, i   T, i

Proof of Theorem ax5seglem6

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp22l 1180	. . . 4 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> T e. ( 0 [,] 1 ) )
2		0re 10040	. . . . . 6 |- 0 e. RR
3		1re 10039	. . . . . 6 |- 1 e. RR
4	2, 3	elicc2i 12239	. . . . 5 |- ( T e. ( 0 [,] 1 ) <-> ( T e. RR /\ 0 <_ T /\ T <_ 1 ) )
5	4	simp1bi 1076	. . . 4 |- ( T e. ( 0 [,] 1 ) -> T e. RR )
6		resqcl 12931	. . . . 5 |- ( T e. RR -> ( T ^ 2 ) e. RR )
7	6	recnd 10068	. . . 4 |- ( T e. RR -> ( T ^ 2 ) e. CC )
8	1, 5, 7	3syl 18	. . 3 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> ( T ^ 2 ) e. CC )
9		simp22r 1181	. . . 4 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> S e. ( 0 [,] 1 ) )
10	2, 3	elicc2i 12239	. . . . 5 |- ( S e. ( 0 [,] 1 ) <-> ( S e. RR /\ 0 <_ S /\ S <_ 1 ) )
11	10	simp1bi 1076	. . . 4 |- ( S e. ( 0 [,] 1 ) -> S e. RR )
12		resqcl 12931	. . . . 5 |- ( S e. RR -> ( S ^ 2 ) e. RR )
13	12	recnd 10068	. . . 4 |- ( S e. RR -> ( S ^ 2 ) e. CC )
14	9, 11, 13	3syl 18	. . 3 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> ( S ^ 2 ) e. CC )
15		fzfid 12772	. . . 4 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> ( 1 ... N ) e. Fin )
16		simprl1 1106	. . . . . . . 8 |- ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) -> A e. ( EE ` N ) )
17	16	3ad2ant1 1082	. . . . . . 7 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> A e. ( EE ` N ) )
18		fveecn 25782	. . . . . . 7 |- ( ( A e. ( EE ` N ) /\ j e. ( 1 ... N ) ) -> ( A ` j ) e. CC )
19	17, 18	sylan 488	. . . . . 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 /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) /\ j e. ( 1 ... N ) ) -> ( A ` j ) e. CC )
20		simprl3 1108	. . . . . . . 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 ) ) ) ) -> C e. ( EE ` N ) )
21	20	3ad2ant1 1082	. . . . . . 7 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> C e. ( EE ` N ) )
22		fveecn 25782	. . . . . . 7 |- ( ( C e. ( EE ` N ) /\ j e. ( 1 ... N ) ) -> ( C ` j ) e. CC )
23	21, 22	sylan 488	. . . . . 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 /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) /\ j e. ( 1 ... N ) ) -> ( C ` j ) e. CC )
24	19, 23	subcld 10392	. . . . 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 /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) /\ j e. ( 1 ... N ) ) -> ( ( A ` j ) - ( C ` j ) ) e. CC )
25	24	sqcld 13006	. . . 4 |- ( ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) /\ j e. ( 1 ... N ) ) -> ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) e. CC )
26	15, 25	fsumcl 14464	. . 3 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) e. CC )
27		simp1l 1085	. . . 4 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> N e. NN )
28		simp1rl 1126	. . . 4 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) )
29		simp21 1094	. . . 4 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> A =/= B )
30		simp23l 1182	. . . 4 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) )
31		ax5seglem5 25813	. . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( A =/= B /\ T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) =/= 0 )
32	27, 28, 29, 1, 30, 31	syl23anc 1333	. . 3 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) =/= 0 )
33		simp3l 1089	. . . . . 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 /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> <. A , B >.Cgr <. D , E >. )
34		simprl2 1107	. . . . . . . 8 |- ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) -> B e. ( EE ` N ) )
35		simprr1 1109	. . . . . . . 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 ) ) ) ) -> D e. ( EE ` N ) )
36		simprr2 1110	. . . . . . . 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 ) )
37		brcgr 25780	. . . . . . . 8 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( <. A , B >.Cgr <. D , E >. <-> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( B ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( D ` j ) - ( E ` j ) ) ^ 2 ) ) )
38	16, 34, 35, 36, 37	syl22anc 1327	. . . . . . 7 |- ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) -> ( <. A , B >.Cgr <. D , E >. <-> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( B ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( D ` j ) - ( E ` j ) ) ^ 2 ) ) )
39	38	3ad2ant1 1082	. . . . . 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 /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> ( <. A , B >.Cgr <. D , E >. <-> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( B ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( D ` j ) - ( E ` j ) ) ^ 2 ) ) )
40	33, 39	mpbid 222	. . . . 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 /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( B ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( D ` j ) - ( E ` j ) ) ^ 2 ) )
41		ax5seglem1 25808	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( B ` j ) ) ^ 2 ) = ( ( T ^ 2 ) x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) )
42	27, 17, 21, 1, 30, 41	syl122anc 1335	. . . . 5 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( B ` j ) ) ^ 2 ) = ( ( T ^ 2 ) x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) )
43	35	3ad2ant1 1082	. . . . . 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 /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> D e. ( EE ` N ) )
44		simprr3 1111	. . . . . . 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 ) ) ) ) -> F e. ( EE ` N ) )
45	44	3ad2ant1 1082	. . . . . 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 /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> F e. ( EE ` N ) )
46		simp23r 1183	. . . . . 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 /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> A. i e. ( 1 ... N ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) )
47		ax5seglem1 25808	. . . . . 6 |- ( ( N e. NN /\ ( D e. ( EE ` N ) /\ F e. ( EE ` N ) ) /\ ( S e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( D ` j ) - ( E ` j ) ) ^ 2 ) = ( ( S ^ 2 ) x. sum_ j e. ( 1 ... N ) ( ( ( D ` j ) - ( F ` j ) ) ^ 2 ) ) )
48	27, 43, 45, 9, 46, 47	syl122anc 1335	. . . . 5 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> sum_ j e. ( 1 ... N ) ( ( ( D ` j ) - ( E ` j ) ) ^ 2 ) = ( ( S ^ 2 ) x. sum_ j e. ( 1 ... N ) ( ( ( D ` j ) - ( F ` j ) ) ^ 2 ) ) )
49	40, 42, 48	3eqtr3d 2664	. . . 4 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> ( ( T ^ 2 ) x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) = ( ( S ^ 2 ) x. sum_ j e. ( 1 ... N ) ( ( ( D ` j ) - ( F ` j ) ) ^ 2 ) ) )
50		simp1rr 1127	. . . . . 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 /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) )
51		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 ) ) ) ) /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) )
52		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 ) ) ) ) /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) )
53		simp3r 1090	. . . . . 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 /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> <. B , C >.Cgr <. E , F >. )
54		ax5seglem3 25811	. . . . . 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 ) ) ) /\ ( ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( D ` j ) - ( F ` j ) ) ^ 2 ) )
55	27, 28, 50, 51, 52, 33, 53, 54	syl322anc 1354	. . . . 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 /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( D ` j ) - ( F ` j ) ) ^ 2 ) )
56	55	oveq2d 6666	. . . 4 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> ( ( S ^ 2 ) x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) = ( ( S ^ 2 ) x. sum_ j e. ( 1 ... N ) ( ( ( D ` j ) - ( F ` j ) ) ^ 2 ) ) )
57	49, 56	eqtr4d 2659	. . 3 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> ( ( T ^ 2 ) x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) = ( ( S ^ 2 ) x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) )
58	8, 14, 26, 32, 57	mulcan2ad 10663	. 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 ) ) ) ) /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> ( T ^ 2 ) = ( S ^ 2 ) )
59	4	simp2bi 1077	. . . . 5 |- ( T e. ( 0 [,] 1 ) -> 0 <_ T )
60	5, 59	jca 554	. . . 4 |- ( T e. ( 0 [,] 1 ) -> ( T e. RR /\ 0 <_ T ) )
61	1, 60	syl 17	. . 3 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> ( T e. RR /\ 0 <_ T ) )
62	10	simp2bi 1077	. . . . 5 |- ( S e. ( 0 [,] 1 ) -> 0 <_ S )
63	11, 62	jca 554	. . . 4 |- ( S e. ( 0 [,] 1 ) -> ( S e. RR /\ 0 <_ S ) )
64	9, 63	syl 17	. . 3 |- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> ( S e. RR /\ 0 <_ S ) )
65		sq11 12936	. . 3 |- ( ( ( T e. RR /\ 0 <_ T ) /\ ( S e. RR /\ 0 <_ S ) ) -> ( ( T ^ 2 ) = ( S ^ 2 ) <-> T = S ) )
66	61, 64, 65	syl2anc 693	. 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 ) ) ) ) /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> ( ( T ^ 2 ) = ( S ^ 2 ) <-> T = S ) )
67	58, 66	mpbid 222	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 ) ) ) ) /\ ( 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 ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> T = S )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   <.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  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-cgr 25773

This theorem is referenced by:  ax5seg  25818