Theorem axpasch 25821

Description: The inner Pasch axiom. Take a triangle A C E, a point D on A C, and a point B extending C E. Then A E and D B intersect at some point x. Axiom A7 of [Schwabhauser] p. 12. (Contributed by Scott Fenton, 3-Jun-2013.)

Assertion

Ref	Expression
axpasch	|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( ( D Btwn <. A , C >. /\ E Btwn <. B , C >. ) -> E. x e. ( EE ` N ) ( x Btwn <. D , B >. /\ x Btwn <. E , A >. ) ) )

Distinct variable groups:   x, A   x, B   x, C   x, D   x, E   x, N

Proof of Theorem axpasch

Dummy variables i q r s t k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		axpaschlem 25820	. . . . . . . . . 10 |- ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) )
2	1	3ad2ant3 1084	. . . . . . . . 9 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) -> E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) )
3		simp1 1061	. . . . . . . . . . . . . . . . . . 19 |- ( ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) -> q = ( ( 1 - r ) x. ( 1 - t ) ) )
4	3	oveq1d 6665	. . . . . . . . . . . . . . . . . 18 |- ( ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) -> ( q x. ( A ` i ) ) = ( ( ( 1 - r ) x. ( 1 - t ) ) x. ( A ` i ) ) )
5	4	eqcomd 2628	. . . . . . . . . . . . . . . . 17 |- ( ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) -> ( ( ( 1 - r ) x. ( 1 - t ) ) x. ( A ` i ) ) = ( q x. ( A ` i ) ) )
6		simp2 1062	. . . . . . . . . . . . . . . . . 18 |- ( ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) -> r = ( ( 1 - q ) x. ( 1 - s ) ) )
7	6	oveq1d 6665	. . . . . . . . . . . . . . . . 17 |- ( ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) -> ( r x. ( B ` i ) ) = ( ( ( 1 - q ) x. ( 1 - s ) ) x. ( B ` i ) ) )
8	5, 7	oveq12d 6668	. . . . . . . . . . . . . . . 16 |- ( ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) -> ( ( ( ( 1 - r ) x. ( 1 - t ) ) x. ( A ` i ) ) + ( r x. ( B ` i ) ) ) = ( ( q x. ( A ` i ) ) + ( ( ( 1 - q ) x. ( 1 - s ) ) x. ( B ` i ) ) ) )
9		simp3 1063	. . . . . . . . . . . . . . . . 17 |- ( ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) -> ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) )
10	9	oveq1d 6665	. . . . . . . . . . . . . . . 16 |- ( ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) -> ( ( ( 1 - r ) x. t ) x. ( C ` i ) ) = ( ( ( 1 - q ) x. s ) x. ( C ` i ) ) )
11	8, 10	oveq12d 6668	. . . . . . . . . . . . . . 15 |- ( ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) -> ( ( ( ( ( 1 - r ) x. ( 1 - t ) ) x. ( A ` i ) ) + ( r x. ( B ` i ) ) ) + ( ( ( 1 - r ) x. t ) x. ( C ` i ) ) ) = ( ( ( q x. ( A ` i ) ) + ( ( ( 1 - q ) x. ( 1 - s ) ) x. ( B ` i ) ) ) + ( ( ( 1 - q ) x. s ) x. ( C ` i ) ) ) )
12	11	3ad2ant3 1084	. . . . . . . . . . . . . 14 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) -> ( ( ( ( ( 1 - r ) x. ( 1 - t ) ) x. ( A ` i ) ) + ( r x. ( B ` i ) ) ) + ( ( ( 1 - r ) x. t ) x. ( C ` i ) ) ) = ( ( ( q x. ( A ` i ) ) + ( ( ( 1 - q ) x. ( 1 - s ) ) x. ( B ` i ) ) ) + ( ( ( 1 - q ) x. s ) x. ( C ` i ) ) ) )
13	12	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( ( ( 1 - r ) x. ( 1 - t ) ) x. ( A ` i ) ) + ( r x. ( B ` i ) ) ) + ( ( ( 1 - r ) x. t ) x. ( C ` i ) ) ) = ( ( ( q x. ( A ` i ) ) + ( ( ( 1 - q ) x. ( 1 - s ) ) x. ( B ` i ) ) ) + ( ( ( 1 - q ) x. s ) x. ( C ` i ) ) ) )
14		1re 10039	. . . . . . . . . . . . . . . . . . 19 |- 1 e. RR
15		simpl2l 1114	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> r e. ( 0 [,] 1 ) )
16		0re 10040	. . . . . . . . . . . . . . . . . . . . . 22 |- 0 e. RR
17	16, 14	elicc2i 12239	. . . . . . . . . . . . . . . . . . . . 21 |- ( r e. ( 0 [,] 1 ) <-> ( r e. RR /\ 0 <_ r /\ r <_ 1 ) )
18	17	simp1bi 1076	. . . . . . . . . . . . . . . . . . . 20 |- ( r e. ( 0 [,] 1 ) -> r e. RR )
19	15, 18	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> r e. RR )
20		resubcl 10345	. . . . . . . . . . . . . . . . . . 19 |- ( ( 1 e. RR /\ r e. RR ) -> ( 1 - r ) e. RR )
21	14, 19, 20	sylancr 695	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( 1 - r ) e. RR )
22	21	recnd 10068	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( 1 - r ) e. CC )
23		simp13l 1176	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) -> t e. ( 0 [,] 1 ) )
24	23	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> t e. ( 0 [,] 1 ) )
25	16, 14	elicc2i 12239	. . . . . . . . . . . . . . . . . . . . . 22 |- ( t e. ( 0 [,] 1 ) <-> ( t e. RR /\ 0 <_ t /\ t <_ 1 ) )
26	25	simp1bi 1076	. . . . . . . . . . . . . . . . . . . . 21 |- ( t e. ( 0 [,] 1 ) -> t e. RR )
27	24, 26	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> t e. RR )
28		resubcl 10345	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 1 e. RR /\ t e. RR ) -> ( 1 - t ) e. RR )
29	14, 27, 28	sylancr 695	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( 1 - t ) e. RR )
30		simp121 1193	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) -> A e. ( EE ` N ) )
31		fveere 25781	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. ( EE ` N ) /\ i e. ( 1 ... N ) ) -> ( A ` i ) e. RR )
32	30, 31	sylan 488	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( A ` i ) e. RR )
33	29, 32	remulcld 10070	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( 1 - t ) x. ( A ` i ) ) e. RR )
34	33	recnd 10068	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( 1 - t ) x. ( A ` i ) ) e. CC )
35		simp123 1195	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) -> C e. ( EE ` N ) )
36		fveere 25781	. . . . . . . . . . . . . . . . . . . 20 |- ( ( C e. ( EE ` N ) /\ i e. ( 1 ... N ) ) -> ( C ` i ) e. RR )
37	35, 36	sylan 488	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( C ` i ) e. RR )
38	27, 37	remulcld 10070	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( t x. ( C ` i ) ) e. RR )
39	38	recnd 10068	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( t x. ( C ` i ) ) e. CC )
40	22, 34, 39	adddid 10064	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) = ( ( ( 1 - r ) x. ( ( 1 - t ) x. ( A ` i ) ) ) + ( ( 1 - r ) x. ( t x. ( C ` i ) ) ) ) )
41	29	recnd 10068	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( 1 - t ) e. CC )
42	32	recnd 10068	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( A ` i ) e. CC )
43	22, 41, 42	mulassd 10063	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( 1 - r ) x. ( 1 - t ) ) x. ( A ` i ) ) = ( ( 1 - r ) x. ( ( 1 - t ) x. ( A ` i ) ) ) )
44	27	recnd 10068	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> t e. CC )
45		fveecn 25782	. . . . . . . . . . . . . . . . . . 19 |- ( ( C e. ( EE ` N ) /\ i e. ( 1 ... N ) ) -> ( C ` i ) e. CC )
46	35, 45	sylan 488	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( C ` i ) e. CC )
47	22, 44, 46	mulassd 10063	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( 1 - r ) x. t ) x. ( C ` i ) ) = ( ( 1 - r ) x. ( t x. ( C ` i ) ) ) )
48	43, 47	oveq12d 6668	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( ( 1 - r ) x. ( 1 - t ) ) x. ( A ` i ) ) + ( ( ( 1 - r ) x. t ) x. ( C ` i ) ) ) = ( ( ( 1 - r ) x. ( ( 1 - t ) x. ( A ` i ) ) ) + ( ( 1 - r ) x. ( t x. ( C ` i ) ) ) ) )
49	40, 48	eqtr4d 2659	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) = ( ( ( ( 1 - r ) x. ( 1 - t ) ) x. ( A ` i ) ) + ( ( ( 1 - r ) x. t ) x. ( C ` i ) ) ) )
50	49	oveq1d 6665	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( ( ( 1 - r ) x. ( 1 - t ) ) x. ( A ` i ) ) + ( ( ( 1 - r ) x. t ) x. ( C ` i ) ) ) + ( r x. ( B ` i ) ) ) )
51	21, 29	remulcld 10070	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( 1 - r ) x. ( 1 - t ) ) e. RR )
52	51, 32	remulcld 10070	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( 1 - r ) x. ( 1 - t ) ) x. ( A ` i ) ) e. RR )
53	52	recnd 10068	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( 1 - r ) x. ( 1 - t ) ) x. ( A ` i ) ) e. CC )
54	21, 27	remulcld 10070	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( 1 - r ) x. t ) e. RR )
55	54, 37	remulcld 10070	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( 1 - r ) x. t ) x. ( C ` i ) ) e. RR )
56	55	recnd 10068	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( 1 - r ) x. t ) x. ( C ` i ) ) e. CC )
57		simp122 1194	. . . . . . . . . . . . . . . . . 18 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) -> B e. ( EE ` N ) )
58		fveere 25781	. . . . . . . . . . . . . . . . . 18 |- ( ( B e. ( EE ` N ) /\ i e. ( 1 ... N ) ) -> ( B ` i ) e. RR )
59	57, 58	sylan 488	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( B ` i ) e. RR )
60	19, 59	remulcld 10070	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( r x. ( B ` i ) ) e. RR )
61	60	recnd 10068	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( r x. ( B ` i ) ) e. CC )
62	53, 56, 61	add32d 10263	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( ( ( 1 - r ) x. ( 1 - t ) ) x. ( A ` i ) ) + ( ( ( 1 - r ) x. t ) x. ( C ` i ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( ( ( 1 - r ) x. ( 1 - t ) ) x. ( A ` i ) ) + ( r x. ( B ` i ) ) ) + ( ( ( 1 - r ) x. t ) x. ( C ` i ) ) ) )
63	50, 62	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( ( ( 1 - r ) x. ( 1 - t ) ) x. ( A ` i ) ) + ( r x. ( B ` i ) ) ) + ( ( ( 1 - r ) x. t ) x. ( C ` i ) ) ) )
64		simpl2r 1115	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> q e. ( 0 [,] 1 ) )
65	16, 14	elicc2i 12239	. . . . . . . . . . . . . . . . . . . 20 |- ( q e. ( 0 [,] 1 ) <-> ( q e. RR /\ 0 <_ q /\ q <_ 1 ) )
66	65	simp1bi 1076	. . . . . . . . . . . . . . . . . . 19 |- ( q e. ( 0 [,] 1 ) -> q e. RR )
67	64, 66	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> q e. RR )
68		resubcl 10345	. . . . . . . . . . . . . . . . . 18 |- ( ( 1 e. RR /\ q e. RR ) -> ( 1 - q ) e. RR )
69	14, 67, 68	sylancr 695	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( 1 - q ) e. RR )
70		simp13r 1177	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) -> s e. ( 0 [,] 1 ) )
71	70	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> s e. ( 0 [,] 1 ) )
72	16, 14	elicc2i 12239	. . . . . . . . . . . . . . . . . . . . 21 |- ( s e. ( 0 [,] 1 ) <-> ( s e. RR /\ 0 <_ s /\ s <_ 1 ) )
73	72	simp1bi 1076	. . . . . . . . . . . . . . . . . . . 20 |- ( s e. ( 0 [,] 1 ) -> s e. RR )
74	71, 73	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> s e. RR )
75		resubcl 10345	. . . . . . . . . . . . . . . . . . 19 |- ( ( 1 e. RR /\ s e. RR ) -> ( 1 - s ) e. RR )
76	14, 74, 75	sylancr 695	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( 1 - s ) e. RR )
77	76, 59	remulcld 10070	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( 1 - s ) x. ( B ` i ) ) e. RR )
78	69, 77	remulcld 10070	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( 1 - q ) x. ( ( 1 - s ) x. ( B ` i ) ) ) e. RR )
79	78	recnd 10068	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( 1 - q ) x. ( ( 1 - s ) x. ( B ` i ) ) ) e. CC )
80	74, 37	remulcld 10070	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( s x. ( C ` i ) ) e. RR )
81	69, 80	remulcld 10070	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( 1 - q ) x. ( s x. ( C ` i ) ) ) e. RR )
82	81	recnd 10068	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( 1 - q ) x. ( s x. ( C ` i ) ) ) e. CC )
83	67, 32	remulcld 10070	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( q x. ( A ` i ) ) e. RR )
84	83	recnd 10068	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( q x. ( A ` i ) ) e. CC )
85	79, 82, 84	add32d 10263	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( ( 1 - q ) x. ( ( 1 - s ) x. ( B ` i ) ) ) + ( ( 1 - q ) x. ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) = ( ( ( ( 1 - q ) x. ( ( 1 - s ) x. ( B ` i ) ) ) + ( q x. ( A ` i ) ) ) + ( ( 1 - q ) x. ( s x. ( C ` i ) ) ) ) )
86	69	recnd 10068	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( 1 - q ) e. CC )
87	77	recnd 10068	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( 1 - s ) x. ( B ` i ) ) e. CC )
88	80	recnd 10068	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( s x. ( C ` i ) ) e. CC )
89	86, 87, 88	adddid 10064	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) = ( ( ( 1 - q ) x. ( ( 1 - s ) x. ( B ` i ) ) ) + ( ( 1 - q ) x. ( s x. ( C ` i ) ) ) ) )
90	89	oveq1d 6665	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) = ( ( ( ( 1 - q ) x. ( ( 1 - s ) x. ( B ` i ) ) ) + ( ( 1 - q ) x. ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) )
91	76	recnd 10068	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( 1 - s ) e. CC )
92	59	recnd 10068	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( B ` i ) e. CC )
93	86, 91, 92	mulassd 10063	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( 1 - q ) x. ( 1 - s ) ) x. ( B ` i ) ) = ( ( 1 - q ) x. ( ( 1 - s ) x. ( B ` i ) ) ) )
94	93	oveq2d 6666	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( q x. ( A ` i ) ) + ( ( ( 1 - q ) x. ( 1 - s ) ) x. ( B ` i ) ) ) = ( ( q x. ( A ` i ) ) + ( ( 1 - q ) x. ( ( 1 - s ) x. ( B ` i ) ) ) ) )
95	84, 79, 94	comraddd 10250	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( q x. ( A ` i ) ) + ( ( ( 1 - q ) x. ( 1 - s ) ) x. ( B ` i ) ) ) = ( ( ( 1 - q ) x. ( ( 1 - s ) x. ( B ` i ) ) ) + ( q x. ( A ` i ) ) ) )
96	74	recnd 10068	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> s e. CC )
97	86, 96, 46	mulassd 10063	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( 1 - q ) x. s ) x. ( C ` i ) ) = ( ( 1 - q ) x. ( s x. ( C ` i ) ) ) )
98	95, 97	oveq12d 6668	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( q x. ( A ` i ) ) + ( ( ( 1 - q ) x. ( 1 - s ) ) x. ( B ` i ) ) ) + ( ( ( 1 - q ) x. s ) x. ( C ` i ) ) ) = ( ( ( ( 1 - q ) x. ( ( 1 - s ) x. ( B ` i ) ) ) + ( q x. ( A ` i ) ) ) + ( ( 1 - q ) x. ( s x. ( C ` i ) ) ) ) )
99	85, 90, 98	3eqtr4d 2666	. . . . . . . . . . . . 13 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) = ( ( ( q x. ( A ` i ) ) + ( ( ( 1 - q ) x. ( 1 - s ) ) x. ( B ` i ) ) ) + ( ( ( 1 - q ) x. s ) x. ( C ` i ) ) ) )
100	13, 63, 99	3eqtr4d 2666	. . . . . . . . . . . 12 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) )
101	100	ralrimiva 2966	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) /\ ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) ) -> A. i e. ( 1 ... N ) ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) )
102	101	3expia 1267	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) -> ( ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) -> A. i e. ( 1 ... N ) ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) )
103	102	reximdvva 3019	. . . . . . . . 9 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) -> ( E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) ( q = ( ( 1 - r ) x. ( 1 - t ) ) /\ r = ( ( 1 - q ) x. ( 1 - s ) ) /\ ( ( 1 - r ) x. t ) = ( ( 1 - q ) x. s ) ) -> E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) )
104	2, 103	mpd 15	. . . . . . . 8 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) -> E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) )
105		simplrl 800	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ k e. ( 1 ... N ) ) -> r e. ( 0 [,] 1 ) )
106	105, 18	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ k e. ( 1 ... N ) ) -> r e. RR )
107	14, 106, 20	sylancr 695	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ k e. ( 1 ... N ) ) -> ( 1 - r ) e. RR )
108		simpl3l 1116	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) -> t e. ( 0 [,] 1 ) )
109	108	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ k e. ( 1 ... N ) ) -> t e. ( 0 [,] 1 ) )
110	109, 26	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ k e. ( 1 ... N ) ) -> t e. RR )
111	14, 110, 28	sylancr 695	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ k e. ( 1 ... N ) ) -> ( 1 - t ) e. RR )
112		simpl21 1139	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) -> A e. ( EE ` N ) )
113		fveere 25781	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. ( EE ` N ) /\ k e. ( 1 ... N ) ) -> ( A ` k ) e. RR )
114	112, 113	sylan 488	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ k e. ( 1 ... N ) ) -> ( A ` k ) e. RR )
115	111, 114	remulcld 10070	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ k e. ( 1 ... N ) ) -> ( ( 1 - t ) x. ( A ` k ) ) e. RR )
116		simpl23 1141	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) -> C e. ( EE ` N ) )
117		fveere 25781	. . . . . . . . . . . . . . . . . . 19 |- ( ( C e. ( EE ` N ) /\ k e. ( 1 ... N ) ) -> ( C ` k ) e. RR )
118	116, 117	sylan 488	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ k e. ( 1 ... N ) ) -> ( C ` k ) e. RR )
119	110, 118	remulcld 10070	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ k e. ( 1 ... N ) ) -> ( t x. ( C ` k ) ) e. RR )
120	115, 119	readdcld 10069	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ k e. ( 1 ... N ) ) -> ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) e. RR )
121	107, 120	remulcld 10070	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ k e. ( 1 ... N ) ) -> ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) e. RR )
122		simpl22 1140	. . . . . . . . . . . . . . . . 17 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) -> B e. ( EE ` N ) )
123		fveere 25781	. . . . . . . . . . . . . . . . 17 |- ( ( B e. ( EE ` N ) /\ k e. ( 1 ... N ) ) -> ( B ` k ) e. RR )
124	122, 123	sylan 488	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ k e. ( 1 ... N ) ) -> ( B ` k ) e. RR )
125	106, 124	remulcld 10070	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ k e. ( 1 ... N ) ) -> ( r x. ( B ` k ) ) e. RR )
126	121, 125	readdcld 10069	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) /\ k e. ( 1 ... N ) ) -> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) e. RR )
127	126	ralrimiva 2966	. . . . . . . . . . . . 13 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ ( r e. ( 0 [,] 1 ) /\ q e. ( 0 [,] 1 ) ) ) -> A. k e. ( 1 ... N ) ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) e. RR )
128	127	anassrs 680	. . . . . . . . . . . 12 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ r e. ( 0 [,] 1 ) ) /\ q e. ( 0 [,] 1 ) ) -> A. k e. ( 1 ... N ) ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) e. RR )
129		simpll1 1100	. . . . . . . . . . . . 13 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ r e. ( 0 [,] 1 ) ) /\ q e. ( 0 [,] 1 ) ) -> N e. NN )
130		mptelee 25775	. . . . . . . . . . . . 13 |- ( N e. NN -> ( ( k e. ( 1 ... N ) |-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) ) e. ( EE ` N ) <-> A. k e. ( 1 ... N ) ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) e. RR ) )
131	129, 130	syl 17	. . . . . . . . . . . 12 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ r e. ( 0 [,] 1 ) ) /\ q e. ( 0 [,] 1 ) ) -> ( ( k e. ( 1 ... N ) |-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) ) e. ( EE ` N ) <-> A. k e. ( 1 ... N ) ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) e. RR ) )
132	128, 131	mpbird 247	. . . . . . . . . . 11 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ r e. ( 0 [,] 1 ) ) /\ q e. ( 0 [,] 1 ) ) -> ( k e. ( 1 ... N ) |-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) ) e. ( EE ` N ) )
133		fveq1 6190	. . . . . . . . . . . . . . . . . 18 |- ( x = ( k e. ( 1 ... N ) |-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) ) -> ( x ` i ) = ( ( k e. ( 1 ... N ) |-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) ) ` i ) )
134		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( k = i -> ( A ` k ) = ( A ` i ) )
135	134	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = i -> ( ( 1 - t ) x. ( A ` k ) ) = ( ( 1 - t ) x. ( A ` i ) ) )
136		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( k = i -> ( C ` k ) = ( C ` i ) )
137	136	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = i -> ( t x. ( C ` k ) ) = ( t x. ( C ` i ) ) )
138	135, 137	oveq12d 6668	. . . . . . . . . . . . . . . . . . . . 21 |- ( k = i -> ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) )
139	138	oveq2d 6666	. . . . . . . . . . . . . . . . . . . 20 |- ( k = i -> ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) = ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) )
140		fveq2 6191	. . . . . . . . . . . . . . . . . . . . 21 |- ( k = i -> ( B ` k ) = ( B ` i ) )
141	140	oveq2d 6666	. . . . . . . . . . . . . . . . . . . 20 |- ( k = i -> ( r x. ( B ` k ) ) = ( r x. ( B ` i ) ) )
142	139, 141	oveq12d 6668	. . . . . . . . . . . . . . . . . . 19 |- ( k = i -> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) )
143		eqid 2622	. . . . . . . . . . . . . . . . . . 19 |- ( k e. ( 1 ... N ) |-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) ) = ( k e. ( 1 ... N ) |-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) )
144		ovex 6678	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) e. _V
145	142, 143, 144	fvmpt 6282	. . . . . . . . . . . . . . . . . 18 |- ( i e. ( 1 ... N ) -> ( ( k e. ( 1 ... N ) |-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) ) ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) )
146	133, 145	sylan9eq 2676	. . . . . . . . . . . . . . . . 17 |- ( ( x = ( k e. ( 1 ... N ) |-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) ) /\ i e. ( 1 ... N ) ) -> ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) )
147	146	eqeq1d 2624	. . . . . . . . . . . . . . . 16 |- ( ( x = ( k e. ( 1 ... N ) |-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) <-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) ) )
148	146	eqeq1d 2624	. . . . . . . . . . . . . . . 16 |- ( ( x = ( k e. ( 1 ... N ) |-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) <-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) )
149	147, 148	anbi12d 747	. . . . . . . . . . . . . . 15 |- ( ( x = ( k e. ( 1 ... N ) |-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) <-> ( ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) ) )
150		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) )
151	150	biantrur 527	. . . . . . . . . . . . . . 15 |- ( ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) <-> ( ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) )
152	149, 151	syl6bbr 278	. . . . . . . . . . . . . 14 |- ( ( x = ( k e. ( 1 ... N ) |-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) <-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) )
153	152	ralbidva 2985	. . . . . . . . . . . . 13 |- ( x = ( k e. ( 1 ... N ) |-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) ) -> ( A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) <-> A. i e. ( 1 ... N ) ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) )
154	153	rspcev 3309	. . . . . . . . . . . 12 |- ( ( ( k e. ( 1 ... N ) |-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) ) e. ( EE ` N ) /\ A. i e. ( 1 ... N ) ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) -> E. x e. ( EE ` N ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) )
155	154	ex 450	. . . . . . . . . . 11 |- ( ( k e. ( 1 ... N ) |-> ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` k ) ) + ( t x. ( C ` k ) ) ) ) + ( r x. ( B ` k ) ) ) ) e. ( EE ` N ) -> ( A. i e. ( 1 ... N ) ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) -> E. x e. ( EE ` N ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) ) )
156	132, 155	syl 17	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ r e. ( 0 [,] 1 ) ) /\ q e. ( 0 [,] 1 ) ) -> ( A. i e. ( 1 ... N ) ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) -> E. x e. ( EE ` N ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) ) )
157	156	reximdva 3017	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) /\ r e. ( 0 [,] 1 ) ) -> ( E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) -> E. q e. ( 0 [,] 1 ) E. x e. ( EE ` N ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) ) )
158	157	reximdva 3017	. . . . . . . 8 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) -> ( E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) -> E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) E. x e. ( EE ` N ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) ) )
159	104, 158	mpd 15	. . . . . . 7 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) -> E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) E. x e. ( EE ` N ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) )
160		rexcom 3099	. . . . . . . . 9 |- ( E. q e. ( 0 [,] 1 ) E. x e. ( EE ` N ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) <-> E. x e. ( EE ` N ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) )
161	160	rexbii 3041	. . . . . . . 8 |- ( E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) E. x e. ( EE ` N ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) <-> E. r e. ( 0 [,] 1 ) E. x e. ( EE ` N ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) )
162		rexcom 3099	. . . . . . . 8 |- ( E. r e. ( 0 [,] 1 ) E. x e. ( EE ` N ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) <-> E. x e. ( EE ` N ) E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) )
163	161, 162	bitri 264	. . . . . . 7 |- ( E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) E. x e. ( EE ` N ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) <-> E. x e. ( EE ` N ) E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) )
164	159, 163	sylib 208	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) -> E. x e. ( EE ` N ) E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) )
165		oveq2 6658	. . . . . . . . . . . . 13 |- ( ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) -> ( ( 1 - r ) x. ( D ` i ) ) = ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) )
166	165	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) -> ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) )
167	166	eqeq2d 2632	. . . . . . . . . . 11 |- ( ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) -> ( ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) <-> ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) ) )
168		oveq2 6658	. . . . . . . . . . . . 13 |- ( ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) -> ( ( 1 - q ) x. ( E ` i ) ) = ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) )
169	168	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) -> ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) )
170	169	eqeq2d 2632	. . . . . . . . . . 11 |- ( ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) -> ( ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) <-> ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) )
171	167, 170	bi2anan9 917	. . . . . . . . . 10 |- ( ( ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) -> ( ( ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) <-> ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) ) )
172	171	ralimi 2952	. . . . . . . . 9 |- ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) -> A. i e. ( 1 ... N ) ( ( ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) <-> ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) ) )
173		ralbi 3068	. . . . . . . . 9 |- ( A. i e. ( 1 ... N ) ( ( ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) <-> ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) ) -> ( A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) <-> A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) ) )
174	172, 173	syl 17	. . . . . . . 8 |- ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) -> ( A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) <-> A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) ) )
175	174	rexbidv 3052	. . . . . . 7 |- ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) -> ( E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) <-> E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) ) )
176	175	2rexbidv 3057	. . . . . 6 |- ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) -> ( E. x e. ( EE ` N ) E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) <-> E. x e. ( EE ` N ) E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) + ( q x. ( A ` i ) ) ) ) ) )
177	164, 176	syl5ibrcom 237	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) ) -> ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) -> E. x e. ( EE ` N ) E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) ) )
178	177	3expia 1267	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( t e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) -> E. x e. ( EE ` N ) E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) ) ) )
179	178	rexlimdvv 3037	. . 3 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) -> E. x e. ( EE ` N ) E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) ) )
180	179	3adant3 1081	. 2 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) -> E. x e. ( EE ` N ) E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) ) )
181		simp3l 1089	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> D e. ( EE ` N ) )
182		simp21 1094	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
183		simp23 1096	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
184		brbtwn 25779	. . . . 5 |- ( ( D e. ( EE ` N ) /\ A e. ( EE ` N ) /\ C e. ( EE ` N ) ) -> ( D Btwn <. A , C >. <-> E. t e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) )
185	181, 182, 183, 184	syl3anc 1326	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( D Btwn <. A , C >. <-> E. t e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) ) )
186		simp3r 1090	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> E e. ( EE ` N ) )
187		simp22 1095	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
188		brbtwn 25779	. . . . 5 |- ( ( E e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) -> ( E Btwn <. B , C >. <-> E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) )
189	186, 187, 183, 188	syl3anc 1326	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( E Btwn <. B , C >. <-> E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) )
190	185, 189	anbi12d 747	. . 3 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( ( D Btwn <. A , C >. /\ E Btwn <. B , C >. ) <-> ( E. t e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) ) )
191		r19.26 3064	. . . . 5 |- ( A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) <-> ( A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) )
192	191	2rexbii 3042	. . . 4 |- ( E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) <-> E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) )
193		reeanv 3107	. . . 4 |- ( E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) <-> ( E. t e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) )
194	192, 193	bitri 264	. . 3 |- ( E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) <-> ( E. t e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) )
195	190, 194	syl6bbr 278	. 2 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( ( D Btwn <. A , C >. /\ E Btwn <. B , C >. ) <-> E. t e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( D ` i ) = ( ( ( 1 - t ) x. ( A ` i ) ) + ( t x. ( C ` i ) ) ) /\ ( E ` i ) = ( ( ( 1 - s ) x. ( B ` i ) ) + ( s x. ( C ` i ) ) ) ) ) )
196		simpr 477	. . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> x e. ( EE ` N ) )
197		simpl3l 1116	. . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> D e. ( EE ` N ) )
198		simpl22 1140	. . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> B e. ( EE ` N ) )
199		brbtwn 25779	. . . . . 6 |- ( ( x e. ( EE ` N ) /\ D e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( x Btwn <. D , B >. <-> E. r e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) ) )
200	196, 197, 198, 199	syl3anc 1326	. . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( x Btwn <. D , B >. <-> E. r e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) ) )
201		simpl3r 1117	. . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> E e. ( EE ` N ) )
202		simpl21 1139	. . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> A e. ( EE ` N ) )
203		brbtwn 25779	. . . . . 6 |- ( ( x e. ( EE ` N ) /\ E e. ( EE ` N ) /\ A e. ( EE ` N ) ) -> ( x Btwn <. E , A >. <-> E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) )
204	196, 201, 202, 203	syl3anc 1326	. . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( x Btwn <. E , A >. <-> E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) )
205	200, 204	anbi12d 747	. . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( ( x Btwn <. D , B >. /\ x Btwn <. E , A >. ) <-> ( E. r e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) ) )
206		r19.26 3064	. . . . . 6 |- ( A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) <-> ( A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) )
207	206	2rexbii 3042	. . . . 5 |- ( E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) <-> E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) )
208		reeanv 3107	. . . . 5 |- ( E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) ( A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) <-> ( E. r e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) )
209	207, 208	bitri 264	. . . 4 |- ( E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) <-> ( E. r e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) )
210	205, 209	syl6bbr 278	. . 3 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( ( x Btwn <. D , B >. /\ x Btwn <. E , A >. ) <-> E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) ) )
211	210	rexbidva 3049	. 2 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( E. x e. ( EE ` N ) ( x Btwn <. D , B >. /\ x Btwn <. E , A >. ) <-> E. x e. ( EE ` N ) E. r e. ( 0 [,] 1 ) E. q e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( x ` i ) = ( ( ( 1 - r ) x. ( D ` i ) ) + ( r x. ( B ` i ) ) ) /\ ( x ` i ) = ( ( ( 1 - q ) x. ( E ` i ) ) + ( q x. ( A ` i ) ) ) ) ) )
212	180, 195, 211	3imtr4d 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 ) ) ) -> ( ( D Btwn <. A , C >. /\ E Btwn <. B , C >. ) -> E. x e. ( EE ` N ) ( x Btwn <. D , B >. /\ x Btwn <. E , A >. ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   ` 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   [,]cicc 12178   ...cfz 12326   EEcee 25768   Btwn cbtwn 25769

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-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-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  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-z 11378  df-uz 11688  df-icc 12182  df-fz 12327  df-ee 25771  df-btwn 25772

This theorem is referenced by:  eengtrkg  25865  btwncomim  32120  btwnswapid  32124  btwnintr  32126  btwnexch3  32127  trisegint  32135  btwnconn1lem13  32206