Theorem axeuclidlem 24137

Description: Lemma for axeuclid 24138. Handle the algebraic aspects of the theorem. (Contributed by Scott Fenton, 9-Sep-2013.)

Assertion

Ref

Expression

axeuclidlem

|- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) /\ A. i e. ( 1 ... N ) ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( T ` i ) ) ) = ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) -> E. x e. ( EE ` N ) E. y e. ( EE ` N ) E. r e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) E. u e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) )

Distinct variable groups:   A, i, r, s, u, x, y   B, i, r, s, u, x, y   C, i, r, s, u, x, y   i, N, r, s, u, x, y   P, i, r, s, u, x, y   Q, i, r, s, u, x, y   T, i, r, s, u, x, y

Proof of Theorem axeuclidlem

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp21 1030	. . 3 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) /\ A. i e. ( 1 ... N ) ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( T ` i ) ) ) = ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) -> P e. ( 0 [,] 1 ) )
2		simp22 1031	. . 3 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) /\ A. i e. ( 1 ... N ) ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( T ` i ) ) ) = ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) -> Q e. ( 0 [,] 1 ) )
3		fveere 24076	. . . . . . . . . . . . 13 |- ( ( A e. ( EE ` N ) /\ k e. ( 1 ... N ) ) -> ( A ` k ) e. RR )
4	3	expcom 435	. . . . . . . . . . . 12 |- ( k e. ( 1 ... N ) -> ( A e. ( EE ` N ) -> ( A ` k ) e. RR ) )
5		fveere 24076	. . . . . . . . . . . . 13 |- ( ( B e. ( EE ` N ) /\ k e. ( 1 ... N ) ) -> ( B ` k ) e. RR )
6	5	expcom 435	. . . . . . . . . . . 12 |- ( k e. ( 1 ... N ) -> ( B e. ( EE ` N ) -> ( B ` k ) e. RR ) )
7	4, 6	anim12d 563	. . . . . . . . . . 11 |- ( k e. ( 1 ... N ) -> ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( ( A ` k ) e. RR /\ ( B ` k ) e. RR ) ) )
8		fveere 24076	. . . . . . . . . . . . 13 |- ( ( C e. ( EE ` N ) /\ k e. ( 1 ... N ) ) -> ( C ` k ) e. RR )
9	8	expcom 435	. . . . . . . . . . . 12 |- ( k e. ( 1 ... N ) -> ( C e. ( EE ` N ) -> ( C ` k ) e. RR ) )
10		fveere 24076	. . . . . . . . . . . . 13 |- ( ( T e. ( EE ` N ) /\ k e. ( 1 ... N ) ) -> ( T ` k ) e. RR )
11	10	expcom 435	. . . . . . . . . . . 12 |- ( k e. ( 1 ... N ) -> ( T e. ( EE ` N ) -> ( T ` k ) e. RR ) )
12	9, 11	anim12d 563	. . . . . . . . . . 11 |- ( k e. ( 1 ... N ) -> ( ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) -> ( ( C ` k ) e. RR /\ ( T ` k ) e. RR ) ) )
13	7, 12	anim12d 563	. . . . . . . . . 10 |- ( k e. ( 1 ... N ) -> ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) -> ( ( ( A ` k ) e. RR /\ ( B ` k ) e. RR ) /\ ( ( C ` k ) e. RR /\ ( T ` k ) e. RR ) ) ) )
14	13	impcom 430	. . . . . . . . 9 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ k e. ( 1 ... N ) ) -> ( ( ( A ` k ) e. RR /\ ( B ` k ) e. RR ) /\ ( ( C ` k ) e. RR /\ ( T ` k ) e. RR ) ) )
15		unitssre 11676	. . . . . . . . . . . . . . . 16 |- ( 0 [,] 1 ) C_ RR
16	15	sseli 3485	. . . . . . . . . . . . . . 15 |- ( P e. ( 0 [,] 1 ) -> P e. RR )
17	16	3ad2ant1 1018	. . . . . . . . . . . . . 14 |- ( ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) -> P e. RR )
18	17	adantl 466	. . . . . . . . . . . . 13 |- ( ( ( ( ( A ` k ) e. RR /\ ( B ` k ) e. RR ) /\ ( ( C ` k ) e. RR /\ ( T ` k ) e. RR ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> P e. RR )
19		peano2rem 9891	. . . . . . . . . . . . 13 |- ( P e. RR -> ( P - 1 ) e. RR )
20	18, 19	syl 16	. . . . . . . . . . . 12 |- ( ( ( ( ( A ` k ) e. RR /\ ( B ` k ) e. RR ) /\ ( ( C ` k ) e. RR /\ ( T ` k ) e. RR ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( P - 1 ) e. RR )
21		simplll 759	. . . . . . . . . . . 12 |- ( ( ( ( ( A ` k ) e. RR /\ ( B ` k ) e. RR ) /\ ( ( C ` k ) e. RR /\ ( T ` k ) e. RR ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( A ` k ) e. RR )
22	20, 21	remulcld 9627	. . . . . . . . . . 11 |- ( ( ( ( ( A ` k ) e. RR /\ ( B ` k ) e. RR ) /\ ( ( C ` k ) e. RR /\ ( T ` k ) e. RR ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( P - 1 ) x. ( A ` k ) ) e. RR )
23		simpllr 760	. . . . . . . . . . 11 |- ( ( ( ( ( A ` k ) e. RR /\ ( B ` k ) e. RR ) /\ ( ( C ` k ) e. RR /\ ( T ` k ) e. RR ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( B ` k ) e. RR )
24	22, 23	readdcld 9626	. . . . . . . . . 10 |- ( ( ( ( ( A ` k ) e. RR /\ ( B ` k ) e. RR ) /\ ( ( C ` k ) e. RR /\ ( T ` k ) e. RR ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) e. RR )
25		simpr3 1005	. . . . . . . . . 10 |- ( ( ( ( ( A ` k ) e. RR /\ ( B ` k ) e. RR ) /\ ( ( C ` k ) e. RR /\ ( T ` k ) e. RR ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> P =/= 0 )
26	24, 18, 25	redivcld 10378	. . . . . . . . 9 |- ( ( ( ( ( A ` k ) e. RR /\ ( B ` k ) e. RR ) /\ ( ( C ` k ) e. RR /\ ( T ` k ) e. RR ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) / P ) e. RR )
27	14, 26	sylan 471	. . . . . . . 8 |- ( ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ k e. ( 1 ... N ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) / P ) e. RR )
28	27	an32s 804	. . . . . . 7 |- ( ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) /\ k e. ( 1 ... N ) ) -> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) / P ) e. RR )
29	28	ralrimiva 2857	. . . . . 6 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> A. k e. ( 1 ... N ) ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) / P ) e. RR )
30		eleenn 24071	. . . . . . . 8 |- ( A e. ( EE ` N ) -> N e. NN )
31	30	ad3antrrr 729	. . . . . . 7 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> N e. NN )
32		mptelee 24070	. . . . . . 7 |- ( N e. NN -> ( ( k e. ( 1 ... N ) |-> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) / P ) ) e. ( EE ` N ) <-> A. k e. ( 1 ... N ) ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) / P ) e. RR ) )
33	31, 32	syl 16	. . . . . 6 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( k e. ( 1 ... N ) |-> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) / P ) ) e. ( EE ` N ) <-> A. k e. ( 1 ... N ) ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) / P ) e. RR ) )
34	29, 33	mpbird 232	. . . . 5 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( k e. ( 1 ... N ) |-> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) / P ) ) e. ( EE ` N ) )
35	34	3adant3 1017	. . . 4 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) /\ A. i e. ( 1 ... N ) ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( T ` i ) ) ) = ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) -> ( k e. ( 1 ... N ) |-> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) / P ) ) e. ( EE ` N ) )
36		simplrl 761	. . . . . . . . . . 11 |- ( ( ( ( ( A ` k ) e. RR /\ ( B ` k ) e. RR ) /\ ( ( C ` k ) e. RR /\ ( T ` k ) e. RR ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( C ` k ) e. RR )
37	22, 36	readdcld 9626	. . . . . . . . . 10 |- ( ( ( ( ( A ` k ) e. RR /\ ( B ` k ) e. RR ) /\ ( ( C ` k ) e. RR /\ ( T ` k ) e. RR ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( P - 1 ) x. ( A ` k ) ) + ( C ` k ) ) e. RR )
38	37, 18, 25	redivcld 10378	. . . . . . . . 9 |- ( ( ( ( ( A ` k ) e. RR /\ ( B ` k ) e. RR ) /\ ( ( C ` k ) e. RR /\ ( T ` k ) e. RR ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( C ` k ) ) / P ) e. RR )
39	14, 38	sylan 471	. . . . . . . 8 |- ( ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ k e. ( 1 ... N ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( C ` k ) ) / P ) e. RR )
40	39	an32s 804	. . . . . . 7 |- ( ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) /\ k e. ( 1 ... N ) ) -> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( C ` k ) ) / P ) e. RR )
41	40	ralrimiva 2857	. . . . . 6 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> A. k e. ( 1 ... N ) ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( C ` k ) ) / P ) e. RR )
42		mptelee 24070	. . . . . . 7 |- ( N e. NN -> ( ( k e. ( 1 ... N ) |-> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( C ` k ) ) / P ) ) e. ( EE ` N ) <-> A. k e. ( 1 ... N ) ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( C ` k ) ) / P ) e. RR ) )
43	31, 42	syl 16	. . . . . 6 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( k e. ( 1 ... N ) |-> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( C ` k ) ) / P ) ) e. ( EE ` N ) <-> A. k e. ( 1 ... N ) ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( C ` k ) ) / P ) e. RR ) )
44	41, 43	mpbird 232	. . . . 5 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( k e. ( 1 ... N ) |-> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( C ` k ) ) / P ) ) e. ( EE ` N ) )
45	44	3adant3 1017	. . . 4 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) /\ A. i e. ( 1 ... N ) ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( T ` i ) ) ) = ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) -> ( k e. ( 1 ... N ) |-> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( C ` k ) ) / P ) ) e. ( EE ` N ) )
46		fveecn 24077	. . . . . . . . . . . 12 |- ( ( A e. ( EE ` N ) /\ i e. ( 1 ... N ) ) -> ( A ` i ) e. CC )
47	46	expcom 435	. . . . . . . . . . 11 |- ( i e. ( 1 ... N ) -> ( A e. ( EE ` N ) -> ( A ` i ) e. CC ) )
48		fveecn 24077	. . . . . . . . . . . 12 |- ( ( B e. ( EE ` N ) /\ i e. ( 1 ... N ) ) -> ( B ` i ) e. CC )
49	48	expcom 435	. . . . . . . . . . 11 |- ( i e. ( 1 ... N ) -> ( B e. ( EE ` N ) -> ( B ` i ) e. CC ) )
50	47, 49	anim12d 563	. . . . . . . . . 10 |- ( i e. ( 1 ... N ) -> ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) ) )
51		fveecn 24077	. . . . . . . . . . . 12 |- ( ( C e. ( EE ` N ) /\ i e. ( 1 ... N ) ) -> ( C ` i ) e. CC )
52	51	expcom 435	. . . . . . . . . . 11 |- ( i e. ( 1 ... N ) -> ( C e. ( EE ` N ) -> ( C ` i ) e. CC ) )
53		fveecn 24077	. . . . . . . . . . . 12 |- ( ( T e. ( EE ` N ) /\ i e. ( 1 ... N ) ) -> ( T ` i ) e. CC )
54	53	expcom 435	. . . . . . . . . . 11 |- ( i e. ( 1 ... N ) -> ( T e. ( EE ` N ) -> ( T ` i ) e. CC ) )
55	52, 54	anim12d 563	. . . . . . . . . 10 |- ( i e. ( 1 ... N ) -> ( ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) -> ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) )
56	50, 55	anim12d 563	. . . . . . . . 9 |- ( i e. ( 1 ... N ) -> ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) -> ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) ) )
57	56	impcom 430	. . . . . . . 8 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) )
58		eqcom 2452	. . . . . . . . . . . . . 14 |- ( ( T ` i ) = ( ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) / P ) <-> ( ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) / P ) = ( T ` i ) )
59		ax-1cn 9553	. . . . . . . . . . . . . . . . . . 19 |- 1 e. CC
60		simpr2 1004	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> Q e. ( 0 [,] 1 ) )
61	15	sseli 3485	. . . . . . . . . . . . . . . . . . . . 21 |- ( Q e. ( 0 [,] 1 ) -> Q e. RR )
62	61	recnd 9625	. . . . . . . . . . . . . . . . . . . 20 |- ( Q e. ( 0 [,] 1 ) -> Q e. CC )
63	60, 62	syl 16	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> Q e. CC )
64		subcl 9824	. . . . . . . . . . . . . . . . . . 19 |- ( ( 1 e. CC /\ Q e. CC ) -> ( 1 - Q ) e. CC )
65	59, 63, 64	sylancr 663	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( 1 - Q ) e. CC )
66		simpr1 1003	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> P e. ( 0 [,] 1 ) )
67	16	recnd 9625	. . . . . . . . . . . . . . . . . . . . 21 |- ( P e. ( 0 [,] 1 ) -> P e. CC )
68	66, 67	syl 16	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> P e. CC )
69		subcl 9824	. . . . . . . . . . . . . . . . . . . 20 |- ( ( P e. CC /\ 1 e. CC ) -> ( P - 1 ) e. CC )
70	68, 59, 69	sylancl 662	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( P - 1 ) e. CC )
71		simplll 759	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( A ` i ) e. CC )
72	70, 71	mulcld 9619	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( P - 1 ) x. ( A ` i ) ) e. CC )
73	65, 72	mulcld 9619	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) e. CC )
74	63, 72	mulcld 9619	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) e. CC )
75	73, 74	addcld 9618	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) e. CC )
76		simpllr 760	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( B ` i ) e. CC )
77	65, 76	mulcld 9619	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( 1 - Q ) x. ( B ` i ) ) e. CC )
78		simplrl 761	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( C ` i ) e. CC )
79	63, 78	mulcld 9619	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( Q x. ( C ` i ) ) e. CC )
80	77, 79	addcld 9618	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) e. CC )
81	75, 80	addcld 9618	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) e. CC )
82		simplrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( T ` i ) e. CC )
83		simpr3 1005	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> P =/= 0 )
84	81, 68, 82, 83	divmuld 10348	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) / P ) = ( T ` i ) <-> ( P x. ( T ` i ) ) = ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) ) )
85	58, 84	syl5bb 257	. . . . . . . . . . . . 13 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( T ` i ) = ( ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) / P ) <-> ( P x. ( T ` i ) ) = ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) ) )
86		negsubdi2 9883	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 1 e. CC /\ P e. CC ) -> -u ( 1 - P ) = ( P - 1 ) )
87	59, 68, 86	sylancr 663	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> -u ( 1 - P ) = ( P - 1 ) )
88	87	oveq1d 6296	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( -u ( 1 - P ) x. ( A ` i ) ) = ( ( P - 1 ) x. ( A ` i ) ) )
89		subcl 9824	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 1 e. CC /\ P e. CC ) -> ( 1 - P ) e. CC )
90	59, 68, 89	sylancr 663	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( 1 - P ) e. CC )
91	90, 71	mulneg1d 10015	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( -u ( 1 - P ) x. ( A ` i ) ) = -u ( ( 1 - P ) x. ( A ` i ) ) )
92		npcan 9834	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( 1 e. CC /\ Q e. CC ) -> ( ( 1 - Q ) + Q ) = 1 )
93	59, 63, 92	sylancr 663	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( 1 - Q ) + Q ) = 1 )
94	93	oveq1d 6296	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( 1 - Q ) + Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) = ( 1 x. ( ( P - 1 ) x. ( A ` i ) ) ) )
95	65, 63, 72	adddird 9624	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( 1 - Q ) + Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) = ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) )
96	72	mulid2d 9617	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( 1 x. ( ( P - 1 ) x. ( A ` i ) ) ) = ( ( P - 1 ) x. ( A ` i ) ) )
97	94, 95, 96	3eqtr3rd 2493	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( P - 1 ) x. ( A ` i ) ) = ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) )
98	88, 91, 97	3eqtr3d 2492	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> -u ( ( 1 - P ) x. ( A ` i ) ) = ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) )
99	98	oveq2d 6297	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) + -u ( ( 1 - P ) x. ( A ` i ) ) ) = ( ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) + ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) ) )
100	90, 71	mulcld 9619	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( 1 - P ) x. ( A ` i ) ) e. CC )
101	80, 100	negsubd 9942	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) + -u ( ( 1 - P ) x. ( A ` i ) ) ) = ( ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) - ( ( 1 - P ) x. ( A ` i ) ) ) )
102	80, 75	addcomd 9785	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) + ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) ) = ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) )
103	99, 101, 102	3eqtr3d 2492	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) - ( ( 1 - P ) x. ( A ` i ) ) ) = ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) )
104	103	eqeq1d 2445	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) - ( ( 1 - P ) x. ( A ` i ) ) ) = ( P x. ( T ` i ) ) <-> ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) = ( P x. ( T ` i ) ) ) )
105		eqcom 2452	. . . . . . . . . . . . . 14 |- ( ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) = ( P x. ( T ` i ) ) <-> ( P x. ( T ` i ) ) = ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) )
106	104, 105	syl6bb 261	. . . . . . . . . . . . 13 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) - ( ( 1 - P ) x. ( A ` i ) ) ) = ( P x. ( T ` i ) ) <-> ( P x. ( T ` i ) ) = ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) ) )
107	85, 106	bitr4d 256	. . . . . . . . . . . 12 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( T ` i ) = ( ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) / P ) <-> ( ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) - ( ( 1 - P ) x. ( A ` i ) ) ) = ( P x. ( T ` i ) ) ) )
108	73, 74, 77, 79	add4d 9808	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) = ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( ( 1 - Q ) x. ( B ` i ) ) ) + ( ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( C ` i ) ) ) ) )
109	65, 72, 76	adddid 9623	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( 1 - Q ) x. ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) ) = ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( ( 1 - Q ) x. ( B ` i ) ) ) )
110	63, 72, 78	adddid 9623	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( Q x. ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) ) = ( ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( C ` i ) ) ) )
111	109, 110	oveq12d 6299	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( 1 - Q ) x. ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) ) + ( Q x. ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) ) ) = ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( ( 1 - Q ) x. ( B ` i ) ) ) + ( ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( C ` i ) ) ) ) )
112	108, 111	eqtr4d 2487	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) = ( ( ( 1 - Q ) x. ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) ) + ( Q x. ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) ) ) )
113	112	oveq1d 6296	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) / P ) = ( ( ( ( 1 - Q ) x. ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) ) + ( Q x. ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) ) ) / P ) )
114	72, 76	addcld 9618	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) e. CC )
115	65, 114	mulcld 9619	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( 1 - Q ) x. ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) ) e. CC )
116	72, 78	addcld 9618	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) e. CC )
117	63, 116	mulcld 9619	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( Q x. ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) ) e. CC )
118	115, 117, 68, 83	divdird 10364	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( 1 - Q ) x. ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) ) + ( Q x. ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) ) ) / P ) = ( ( ( ( 1 - Q ) x. ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) ) / P ) + ( ( Q x. ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) ) / P ) ) )
119	65, 114, 68, 83	divassd 10361	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( 1 - Q ) x. ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) ) / P ) = ( ( 1 - Q ) x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) )
120	63, 116, 68, 83	divassd 10361	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( Q x. ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) ) / P ) = ( Q x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) )
121	119, 120	oveq12d 6299	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( 1 - Q ) x. ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) ) / P ) + ( ( Q x. ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) ) / P ) ) = ( ( ( 1 - Q ) x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) + ( Q x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) )
122	113, 118, 121	3eqtrd 2488	. . . . . . . . . . . . 13 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) / P ) = ( ( ( 1 - Q ) x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) + ( Q x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) )
123	122	eqeq2d 2457	. . . . . . . . . . . 12 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( T ` i ) = ( ( ( ( ( 1 - Q ) x. ( ( P - 1 ) x. ( A ` i ) ) ) + ( Q x. ( ( P - 1 ) x. ( A ` i ) ) ) ) + ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) / P ) <-> ( T ` i ) = ( ( ( 1 - Q ) x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) + ( Q x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) ) )
124	68, 82	mulcld 9619	. . . . . . . . . . . . 13 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( P x. ( T ` i ) ) e. CC )
125	80, 100, 124	subaddd 9954	. . . . . . . . . . . 12 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) - ( ( 1 - P ) x. ( A ` i ) ) ) = ( P x. ( T ` i ) ) <-> ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( T ` i ) ) ) = ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) )
126	107, 123, 125	3bitr3rd 284	. . . . . . . . . . 11 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( T ` i ) ) ) = ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) <-> ( T ` i ) = ( ( ( 1 - Q ) x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) + ( Q x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) ) )
127	126	biimpd 207	. . . . . . . . . 10 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( T ` i ) ) ) = ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) -> ( T ` i ) = ( ( ( 1 - Q ) x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) + ( Q x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) ) )
128		npncan2 9851	. . . . . . . . . . . . . . . . 17 |- ( ( 1 e. CC /\ P e. CC ) -> ( ( 1 - P ) + ( P - 1 ) ) = 0 )
129	59, 68, 128	sylancr 663	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( 1 - P ) + ( P - 1 ) ) = 0 )
130	129	oveq1d 6296	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( 1 - P ) + ( P - 1 ) ) x. ( A ` i ) ) = ( 0 x. ( A ` i ) ) )
131	90, 70, 71	adddird 9624	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( 1 - P ) + ( P - 1 ) ) x. ( A ` i ) ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( ( P - 1 ) x. ( A ` i ) ) ) )
132		mul02 9761	. . . . . . . . . . . . . . . 16 |- ( ( A ` i ) e. CC -> ( 0 x. ( A ` i ) ) = 0 )
133	132	ad3antrrr 729	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( 0 x. ( A ` i ) ) = 0 )
134	130, 131, 133	3eqtr3d 2492	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( 1 - P ) x. ( A ` i ) ) + ( ( P - 1 ) x. ( A ` i ) ) ) = 0 )
135	134	oveq1d 6296	. . . . . . . . . . . . 13 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( 1 - P ) x. ( A ` i ) ) + ( ( P - 1 ) x. ( A ` i ) ) ) + ( B ` i ) ) = ( 0 + ( B ` i ) ) )
136	100, 72, 76	addassd 9621	. . . . . . . . . . . . 13 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( 1 - P ) x. ( A ` i ) ) + ( ( P - 1 ) x. ( A ` i ) ) ) + ( B ` i ) ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) ) )
137	76	addid2d 9784	. . . . . . . . . . . . 13 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( 0 + ( B ` i ) ) = ( B ` i ) )
138	135, 136, 137	3eqtr3rd 2493	. . . . . . . . . . . 12 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( B ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) ) )
139	114, 68, 83	divcan2d 10328	. . . . . . . . . . . . 13 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) = ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) )
140	139	oveq2d 6297	. . . . . . . . . . . 12 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) ) )
141	138, 140	eqtr4d 2487	. . . . . . . . . . 11 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( B ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) ) )
142	134	oveq1d 6296	. . . . . . . . . . . . 13 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( 1 - P ) x. ( A ` i ) ) + ( ( P - 1 ) x. ( A ` i ) ) ) + ( C ` i ) ) = ( 0 + ( C ` i ) ) )
143	100, 72, 78	addassd 9621	. . . . . . . . . . . . 13 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( 1 - P ) x. ( A ` i ) ) + ( ( P - 1 ) x. ( A ` i ) ) ) + ( C ` i ) ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) ) )
144	78	addid2d 9784	. . . . . . . . . . . . 13 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( 0 + ( C ` i ) ) = ( C ` i ) )
145	142, 143, 144	3eqtr3rd 2493	. . . . . . . . . . . 12 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( C ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) ) )
146	116, 68, 83	divcan2d 10328	. . . . . . . . . . . . 13 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) = ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) )
147	146	oveq2d 6297	. . . . . . . . . . . 12 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) ) )
148	145, 147	eqtr4d 2487	. . . . . . . . . . 11 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( C ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) )
149	141, 148	jca 532	. . . . . . . . . 10 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( B ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) ) /\ ( C ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) ) )
150	127, 149	jctild 543	. . . . . . . . 9 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( T ` i ) ) ) = ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) -> ( ( ( B ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) ) /\ ( C ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) ) /\ ( T ` i ) = ( ( ( 1 - Q ) x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) + ( Q x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) ) ) )
151		df-3an 976	. . . . . . . . 9 |- ( ( ( B ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) ) /\ ( C ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) /\ ( T ` i ) = ( ( ( 1 - Q ) x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) + ( Q x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) ) <-> ( ( ( B ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) ) /\ ( C ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) ) /\ ( T ` i ) = ( ( ( 1 - Q ) x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) + ( Q x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) ) )
152	150, 151	syl6ibr 227	. . . . . . . 8 |- ( ( ( ( ( A ` i ) e. CC /\ ( B ` i ) e. CC ) /\ ( ( C ` i ) e. CC /\ ( T ` i ) e. CC ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( T ` i ) ) ) = ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) -> ( ( B ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) ) /\ ( C ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) /\ ( T ` i ) = ( ( ( 1 - Q ) x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) + ( Q x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) ) ) )
153	57, 152	sylan 471	. . . . . . 7 |- ( ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ i e. ( 1 ... N ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( T ` i ) ) ) = ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) -> ( ( B ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) ) /\ ( C ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) /\ ( T ` i ) = ( ( ( 1 - Q ) x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) + ( Q x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) ) ) )
154	153	an32s 804	. . . . . 6 |- ( ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( T ` i ) ) ) = ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) -> ( ( B ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) ) /\ ( C ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) /\ ( T ` i ) = ( ( ( 1 - Q ) x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) + ( Q x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) ) ) )
155	154	ralimdva 2851	. . . . 5 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) ) -> ( A. i e. ( 1 ... N ) ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( T ` i ) ) ) = ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) -> A. i e. ( 1 ... N ) ( ( B ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) ) /\ ( C ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) /\ ( T ` i ) = ( ( ( 1 - Q ) x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) + ( Q x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) ) ) )
156	155	3impia 1194	. . . 4 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) /\ A. i e. ( 1 ... N ) ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( T ` i ) ) ) = ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) -> A. i e. ( 1 ... N ) ( ( B ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) ) /\ ( C ` i ) = ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) /\ ( T ` i ) = ( ( ( 1 - Q ) x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) + ( Q x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( C ` i ) ) / P ) ) ) ) )
157		fveq1 5855	. . . . . . . 8 |- ( x = ( k e. ( 1 ... N ) |-> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) / P ) ) -> ( x ` i ) = ( ( k e. ( 1 ... N ) |-> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) / P ) ) ` i ) )
158		fveq2 5856	. . . . . . . . . . . 12 |- ( k = i -> ( A ` k ) = ( A ` i ) )
159	158	oveq2d 6297	. . . . . . . . . . 11 |- ( k = i -> ( ( P - 1 ) x. ( A ` k ) ) = ( ( P - 1 ) x. ( A ` i ) ) )
160		fveq2 5856	. . . . . . . . . . 11 |- ( k = i -> ( B ` k ) = ( B ` i ) )
161	159, 160	oveq12d 6299	. . . . . . . . . 10 |- ( k = i -> ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) = ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) )
162	161	oveq1d 6296	. . . . . . . . 9 |- ( k = i -> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) / P ) = ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) )
163		eqid 2443	. . . . . . . . 9 |- ( k e. ( 1 ... N ) |-> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) / P ) ) = ( k e. ( 1 ... N ) |-> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) / P ) )
164		ovex 6309	. . . . . . . . 9 |- ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) e. _V
165	162, 163, 164	fvmpt 5941	. . . . . . . 8 |- ( i e. ( 1 ... N ) -> ( ( k e. ( 1 ... N ) |-> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) / P ) ) ` i ) = ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) )
166	157, 165	sylan9eq 2504	. . . . . . 7 |- ( ( x = ( k e. ( 1 ... N ) |-> ( ( ( ( P - 1 ) x. ( A ` k ) ) + ( B ` k ) ) / P ) ) /\ i e. ( 1 ... N ) ) -> ( x ` i ) = ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) )
167		oveq2 6289	. . . . . . . . . 10 |- ( ( x ` i ) = ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) -> ( P x. ( x ` i ) ) = ( P x. ( ( ( ( P - 1 ) x. ( A ` i ) ) + ( B ` i ) ) / P ) ) )
168	167	oveq2d 6297	. . . . . . . . 9 |- ( ( x ` i ) = ( ( ( ( P - 1 ) x.