Metamath Proof Explorer

Theorem ax5seglem2

Description: Lemma for ax5seg . Rexpress another congruence sum given betweenness. (Contributed by Scott Fenton, 11-Jun-2013)

Ref Expression
Assertion ax5seglem2 
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( B ` j ) - ( C ` j ) ) ^ 2 ) = ( ( ( 1 - T ) ^ 2 ) x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) )

Proof

Step Hyp Ref Expression
1 simpl2l 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) /\ j e. ( 1 ... N ) ) -> A e. ( EE ` N ) )
2 fveecn 
 |-  ( ( A e. ( EE ` N ) /\ j e. ( 1 ... N ) ) -> ( A ` j ) e. CC )
3 1 2 sylancom 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) /\ j e. ( 1 ... N ) ) -> ( A ` j ) e. CC )
4 simpl2r 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) /\ j e. ( 1 ... N ) ) -> C e. ( EE ` N ) )
5 fveecn 
 |-  ( ( C e. ( EE ` N ) /\ j e. ( 1 ... N ) ) -> ( C ` j ) e. CC )
6 4 5 sylancom 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) /\ j e. ( 1 ... N ) ) -> ( C ` j ) e. CC )
7 elicc01 
 |-  ( T e. ( 0 [,] 1 ) <-> ( T e. RR /\ 0 <_ T /\ T <_ 1 ) )
8 7 simp1bi 
 |-  ( T e. ( 0 [,] 1 ) -> T e. RR )
9 8 recnd 
 |-  ( T e. ( 0 [,] 1 ) -> T e. CC )
10 9 adantr 
 |-  ( ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) -> T e. CC )
11 10 3ad2ant3 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> T e. CC )
12 11 adantr 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) /\ j e. ( 1 ... N ) ) -> T e. CC )
13 fveq2 
 |-  ( i = j -> ( B ` i ) = ( B ` j ) )
14 fveq2 
 |-  ( i = j -> ( A ` i ) = ( A ` j ) )
15 14 oveq2d 
 |-  ( i = j -> ( ( 1 - T ) x. ( A ` i ) ) = ( ( 1 - T ) x. ( A ` j ) ) )
16 fveq2 
 |-  ( i = j -> ( C ` i ) = ( C ` j ) )
17 16 oveq2d 
 |-  ( i = j -> ( T x. ( C ` i ) ) = ( T x. ( C ` j ) ) )
18 15 17 oveq12d 
 |-  ( i = j -> ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) = ( ( ( 1 - T ) x. ( A ` j ) ) + ( T x. ( C ` j ) ) ) )
19 13 18 eqeq12d 
 |-  ( i = j -> ( ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) <-> ( B ` j ) = ( ( ( 1 - T ) x. ( A ` j ) ) + ( T x. ( C ` j ) ) ) ) )
20 19 rspccva 
 |-  ( ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) /\ j e. ( 1 ... N ) ) -> ( B ` j ) = ( ( ( 1 - T ) x. ( A ` j ) ) + ( T x. ( C ` j ) ) ) )
21 20 adantll 
 |-  ( ( ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) /\ j e. ( 1 ... N ) ) -> ( B ` j ) = ( ( ( 1 - T ) x. ( A ` j ) ) + ( T x. ( C ` j ) ) ) )
22 21 3ad2antl3 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) /\ j e. ( 1 ... N ) ) -> ( B ` j ) = ( ( ( 1 - T ) x. ( A ` j ) ) + ( T x. ( C ` j ) ) ) )
23 oveq1 
 |-  ( ( B ` j ) = ( ( ( 1 - T ) x. ( A ` j ) ) + ( T x. ( C ` j ) ) ) -> ( ( B ` j ) - ( C ` j ) ) = ( ( ( ( 1 - T ) x. ( A ` j ) ) + ( T x. ( C ` j ) ) ) - ( C ` j ) ) )
24 23 oveq1d 
 |-  ( ( B ` j ) = ( ( ( 1 - T ) x. ( A ` j ) ) + ( T x. ( C ` j ) ) ) -> ( ( ( B ` j ) - ( C ` j ) ) ^ 2 ) = ( ( ( ( ( 1 - T ) x. ( A ` j ) ) + ( T x. ( C ` j ) ) ) - ( C ` j ) ) ^ 2 ) )
25 ax-1cn 
 |-  1 e. CC
26 subcl 
 |-  ( ( 1 e. CC /\ T e. CC ) -> ( 1 - T ) e. CC )
27 25 26 mpan 
 |-  ( T e. CC -> ( 1 - T ) e. CC )
28 27 3ad2ant3 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> ( 1 - T ) e. CC )
29 simp1 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> ( A ` j ) e. CC )
30 28 29 mulcld 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> ( ( 1 - T ) x. ( A ` j ) ) e. CC )
31 simp3 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> T e. CC )
32 simp2 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> ( C ` j ) e. CC )
33 31 32 mulcld 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> ( T x. ( C ` j ) ) e. CC )
34 30 33 32 addsubassd 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> ( ( ( ( 1 - T ) x. ( A ` j ) ) + ( T x. ( C ` j ) ) ) - ( C ` j ) ) = ( ( ( 1 - T ) x. ( A ` j ) ) + ( ( T x. ( C ` j ) ) - ( C ` j ) ) ) )
35 subdi 
 |-  ( ( ( 1 - T ) e. CC /\ ( A ` j ) e. CC /\ ( C ` j ) e. CC ) -> ( ( 1 - T ) x. ( ( A ` j ) - ( C ` j ) ) ) = ( ( ( 1 - T ) x. ( A ` j ) ) - ( ( 1 - T ) x. ( C ` j ) ) ) )
36 27 35 syl3an1 
 |-  ( ( T e. CC /\ ( A ` j ) e. CC /\ ( C ` j ) e. CC ) -> ( ( 1 - T ) x. ( ( A ` j ) - ( C ` j ) ) ) = ( ( ( 1 - T ) x. ( A ` j ) ) - ( ( 1 - T ) x. ( C ` j ) ) ) )
37 36 3coml 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> ( ( 1 - T ) x. ( ( A ` j ) - ( C ` j ) ) ) = ( ( ( 1 - T ) x. ( A ` j ) ) - ( ( 1 - T ) x. ( C ` j ) ) ) )
38 subdir 
 |-  ( ( 1 e. CC /\ T e. CC /\ ( C ` j ) e. CC ) -> ( ( 1 - T ) x. ( C ` j ) ) = ( ( 1 x. ( C ` j ) ) - ( T x. ( C ` j ) ) ) )
39 25 38 mp3an1 
 |-  ( ( T e. CC /\ ( C ` j ) e. CC ) -> ( ( 1 - T ) x. ( C ` j ) ) = ( ( 1 x. ( C ` j ) ) - ( T x. ( C ` j ) ) ) )
40 39 ancoms 
 |-  ( ( ( C ` j ) e. CC /\ T e. CC ) -> ( ( 1 - T ) x. ( C ` j ) ) = ( ( 1 x. ( C ` j ) ) - ( T x. ( C ` j ) ) ) )
41 40 3adant1 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> ( ( 1 - T ) x. ( C ` j ) ) = ( ( 1 x. ( C ` j ) ) - ( T x. ( C ` j ) ) ) )
42 mullid 
 |-  ( ( C ` j ) e. CC -> ( 1 x. ( C ` j ) ) = ( C ` j ) )
43 42 oveq1d 
 |-  ( ( C ` j ) e. CC -> ( ( 1 x. ( C ` j ) ) - ( T x. ( C ` j ) ) ) = ( ( C ` j ) - ( T x. ( C ` j ) ) ) )
44 43 3ad2ant2 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> ( ( 1 x. ( C ` j ) ) - ( T x. ( C ` j ) ) ) = ( ( C ` j ) - ( T x. ( C ` j ) ) ) )
45 41 44 eqtrd 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> ( ( 1 - T ) x. ( C ` j ) ) = ( ( C ` j ) - ( T x. ( C ` j ) ) ) )
46 45 oveq2d 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> ( ( ( 1 - T ) x. ( A ` j ) ) - ( ( 1 - T ) x. ( C ` j ) ) ) = ( ( ( 1 - T ) x. ( A ` j ) ) - ( ( C ` j ) - ( T x. ( C ` j ) ) ) ) )
47 30 32 33 subsub2d 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> ( ( ( 1 - T ) x. ( A ` j ) ) - ( ( C ` j ) - ( T x. ( C ` j ) ) ) ) = ( ( ( 1 - T ) x. ( A ` j ) ) + ( ( T x. ( C ` j ) ) - ( C ` j ) ) ) )
48 37 46 47 3eqtrd 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> ( ( 1 - T ) x. ( ( A ` j ) - ( C ` j ) ) ) = ( ( ( 1 - T ) x. ( A ` j ) ) + ( ( T x. ( C ` j ) ) - ( C ` j ) ) ) )
49 34 48 eqtr4d 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> ( ( ( ( 1 - T ) x. ( A ` j ) ) + ( T x. ( C ` j ) ) ) - ( C ` j ) ) = ( ( 1 - T ) x. ( ( A ` j ) - ( C ` j ) ) ) )
50 49 oveq1d 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> ( ( ( ( ( 1 - T ) x. ( A ` j ) ) + ( T x. ( C ` j ) ) ) - ( C ` j ) ) ^ 2 ) = ( ( ( 1 - T ) x. ( ( A ` j ) - ( C ` j ) ) ) ^ 2 ) )
51 subcl 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC ) -> ( ( A ` j ) - ( C ` j ) ) e. CC )
52 51 3adant3 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> ( ( A ` j ) - ( C ` j ) ) e. CC )
53 28 52 sqmuld 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> ( ( ( 1 - T ) x. ( ( A ` j ) - ( C ` j ) ) ) ^ 2 ) = ( ( ( 1 - T ) ^ 2 ) x. ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) )
54 50 53 eqtrd 
 |-  ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) -> ( ( ( ( ( 1 - T ) x. ( A ` j ) ) + ( T x. ( C ` j ) ) ) - ( C ` j ) ) ^ 2 ) = ( ( ( 1 - T ) ^ 2 ) x. ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) )
55 24 54 sylan9eqr 
 |-  ( ( ( ( A ` j ) e. CC /\ ( C ` j ) e. CC /\ T e. CC ) /\ ( B ` j ) = ( ( ( 1 - T ) x. ( A ` j ) ) + ( T x. ( C ` j ) ) ) ) -> ( ( ( B ` j ) - ( C ` j ) ) ^ 2 ) = ( ( ( 1 - T ) ^ 2 ) x. ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) )
56 3 6 12 22 55 syl31anc 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) /\ j e. ( 1 ... N ) ) -> ( ( ( B ` j ) - ( C ` j ) ) ^ 2 ) = ( ( ( 1 - T ) ^ 2 ) x. ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) )
57 56 sumeq2dv 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( B ` j ) - ( C ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( 1 - T ) ^ 2 ) x. ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) )
58 fzfid 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> ( 1 ... N ) e. Fin )
59 1re 
 |-  1 e. RR
60 resubcl 
 |-  ( ( 1 e. RR /\ T e. RR ) -> ( 1 - T ) e. RR )
61 59 8 60 sylancr 
 |-  ( T e. ( 0 [,] 1 ) -> ( 1 - T ) e. RR )
62 61 resqcld 
 |-  ( T e. ( 0 [,] 1 ) -> ( ( 1 - T ) ^ 2 ) e. RR )
63 62 recnd 
 |-  ( T e. ( 0 [,] 1 ) -> ( ( 1 - T ) ^ 2 ) e. CC )
64 63 adantr 
 |-  ( ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) -> ( ( 1 - T ) ^ 2 ) e. CC )
65 64 3ad2ant3 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> ( ( 1 - T ) ^ 2 ) e. CC )
66 2 3adant1 
 |-  ( ( N e. NN /\ A e. ( EE ` N ) /\ j e. ( 1 ... N ) ) -> ( A ` j ) e. CC )
67 66 3adant2r 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ j e. ( 1 ... N ) ) -> ( A ` j ) e. CC )
68 5 3adant1 
 |-  ( ( N e. NN /\ C e. ( EE ` N ) /\ j e. ( 1 ... N ) ) -> ( C ` j ) e. CC )
69 68 3adant2l 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ j e. ( 1 ... N ) ) -> ( C ` j ) e. CC )
70 67 69 subcld 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ j e. ( 1 ... N ) ) -> ( ( A ` j ) - ( C ` j ) ) e. CC )
71 70 sqcld 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ j e. ( 1 ... N ) ) -> ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) e. CC )
72 71 3expa 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) e. CC )
73 72 3adantl3 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) /\ j e. ( 1 ... N ) ) -> ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) e. CC )
74 58 65 73 fsummulc2 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> ( ( ( 1 - T ) ^ 2 ) x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) = sum_ j e. ( 1 ... N ) ( ( ( 1 - T ) ^ 2 ) x. ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) )
75 57 74 eqtr4d 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( B ` j ) - ( C ` j ) ) ^ 2 ) = ( ( ( 1 - T ) ^ 2 ) x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) )