Step |
Hyp |
Ref |
Expression |
1 |
|
ip1i.1 |
|- X = ( BaseSet ` U ) |
2 |
|
ip1i.2 |
|- G = ( +v ` U ) |
3 |
|
ip1i.4 |
|- S = ( .sOLD ` U ) |
4 |
|
ip1i.7 |
|- P = ( .iOLD ` U ) |
5 |
|
ip1i.9 |
|- U e. CPreHilOLD |
6 |
|
ipasslem1.b |
|- B e. X |
7 |
|
nn0cn |
|- ( N e. NN0 -> N e. CC ) |
8 |
7
|
negcld |
|- ( N e. NN0 -> -u N e. CC ) |
9 |
5
|
phnvi |
|- U e. NrmCVec |
10 |
1 4
|
dipcl |
|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A P B ) e. CC ) |
11 |
9 6 10
|
mp3an13 |
|- ( A e. X -> ( A P B ) e. CC ) |
12 |
|
mulcl |
|- ( ( -u N e. CC /\ ( A P B ) e. CC ) -> ( -u N x. ( A P B ) ) e. CC ) |
13 |
8 11 12
|
syl2an |
|- ( ( N e. NN0 /\ A e. X ) -> ( -u N x. ( A P B ) ) e. CC ) |
14 |
1 3
|
nvscl |
|- ( ( U e. NrmCVec /\ -u N e. CC /\ A e. X ) -> ( -u N S A ) e. X ) |
15 |
9 14
|
mp3an1 |
|- ( ( -u N e. CC /\ A e. X ) -> ( -u N S A ) e. X ) |
16 |
8 15
|
sylan |
|- ( ( N e. NN0 /\ A e. X ) -> ( -u N S A ) e. X ) |
17 |
1 4
|
dipcl |
|- ( ( U e. NrmCVec /\ ( -u N S A ) e. X /\ B e. X ) -> ( ( -u N S A ) P B ) e. CC ) |
18 |
9 6 17
|
mp3an13 |
|- ( ( -u N S A ) e. X -> ( ( -u N S A ) P B ) e. CC ) |
19 |
16 18
|
syl |
|- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N S A ) P B ) e. CC ) |
20 |
|
ax-1cn |
|- 1 e. CC |
21 |
|
mulneg2 |
|- ( ( N e. CC /\ 1 e. CC ) -> ( N x. -u 1 ) = -u ( N x. 1 ) ) |
22 |
20 21
|
mpan2 |
|- ( N e. CC -> ( N x. -u 1 ) = -u ( N x. 1 ) ) |
23 |
|
mulid1 |
|- ( N e. CC -> ( N x. 1 ) = N ) |
24 |
23
|
negeqd |
|- ( N e. CC -> -u ( N x. 1 ) = -u N ) |
25 |
22 24
|
eqtr2d |
|- ( N e. CC -> -u N = ( N x. -u 1 ) ) |
26 |
25
|
adantr |
|- ( ( N e. CC /\ A e. X ) -> -u N = ( N x. -u 1 ) ) |
27 |
26
|
oveq1d |
|- ( ( N e. CC /\ A e. X ) -> ( -u N S A ) = ( ( N x. -u 1 ) S A ) ) |
28 |
|
neg1cn |
|- -u 1 e. CC |
29 |
1 3
|
nvsass |
|- ( ( U e. NrmCVec /\ ( N e. CC /\ -u 1 e. CC /\ A e. X ) ) -> ( ( N x. -u 1 ) S A ) = ( N S ( -u 1 S A ) ) ) |
30 |
9 29
|
mpan |
|- ( ( N e. CC /\ -u 1 e. CC /\ A e. X ) -> ( ( N x. -u 1 ) S A ) = ( N S ( -u 1 S A ) ) ) |
31 |
28 30
|
mp3an2 |
|- ( ( N e. CC /\ A e. X ) -> ( ( N x. -u 1 ) S A ) = ( N S ( -u 1 S A ) ) ) |
32 |
27 31
|
eqtrd |
|- ( ( N e. CC /\ A e. X ) -> ( -u N S A ) = ( N S ( -u 1 S A ) ) ) |
33 |
7 32
|
sylan |
|- ( ( N e. NN0 /\ A e. X ) -> ( -u N S A ) = ( N S ( -u 1 S A ) ) ) |
34 |
33
|
oveq1d |
|- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N S A ) P B ) = ( ( N S ( -u 1 S A ) ) P B ) ) |
35 |
1 3
|
nvscl |
|- ( ( U e. NrmCVec /\ -u 1 e. CC /\ A e. X ) -> ( -u 1 S A ) e. X ) |
36 |
9 28 35
|
mp3an12 |
|- ( A e. X -> ( -u 1 S A ) e. X ) |
37 |
1 2 3 4 5 6
|
ipasslem1 |
|- ( ( N e. NN0 /\ ( -u 1 S A ) e. X ) -> ( ( N S ( -u 1 S A ) ) P B ) = ( N x. ( ( -u 1 S A ) P B ) ) ) |
38 |
36 37
|
sylan2 |
|- ( ( N e. NN0 /\ A e. X ) -> ( ( N S ( -u 1 S A ) ) P B ) = ( N x. ( ( -u 1 S A ) P B ) ) ) |
39 |
34 38
|
eqtrd |
|- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N S A ) P B ) = ( N x. ( ( -u 1 S A ) P B ) ) ) |
40 |
39
|
oveq2d |
|- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N x. ( A P B ) ) - ( ( -u N S A ) P B ) ) = ( ( -u N x. ( A P B ) ) - ( N x. ( ( -u 1 S A ) P B ) ) ) ) |
41 |
1 4
|
dipcl |
|- ( ( U e. NrmCVec /\ ( -u 1 S A ) e. X /\ B e. X ) -> ( ( -u 1 S A ) P B ) e. CC ) |
42 |
9 6 41
|
mp3an13 |
|- ( ( -u 1 S A ) e. X -> ( ( -u 1 S A ) P B ) e. CC ) |
43 |
36 42
|
syl |
|- ( A e. X -> ( ( -u 1 S A ) P B ) e. CC ) |
44 |
|
mulcl |
|- ( ( N e. CC /\ ( ( -u 1 S A ) P B ) e. CC ) -> ( N x. ( ( -u 1 S A ) P B ) ) e. CC ) |
45 |
7 43 44
|
syl2an |
|- ( ( N e. NN0 /\ A e. X ) -> ( N x. ( ( -u 1 S A ) P B ) ) e. CC ) |
46 |
13 45
|
negsubd |
|- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N x. ( A P B ) ) + -u ( N x. ( ( -u 1 S A ) P B ) ) ) = ( ( -u N x. ( A P B ) ) - ( N x. ( ( -u 1 S A ) P B ) ) ) ) |
47 |
|
mulneg1 |
|- ( ( N e. CC /\ ( ( -u 1 S A ) P B ) e. CC ) -> ( -u N x. ( ( -u 1 S A ) P B ) ) = -u ( N x. ( ( -u 1 S A ) P B ) ) ) |
48 |
7 43 47
|
syl2an |
|- ( ( N e. NN0 /\ A e. X ) -> ( -u N x. ( ( -u 1 S A ) P B ) ) = -u ( N x. ( ( -u 1 S A ) P B ) ) ) |
49 |
48
|
oveq2d |
|- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N x. ( A P B ) ) + ( -u N x. ( ( -u 1 S A ) P B ) ) ) = ( ( -u N x. ( A P B ) ) + -u ( N x. ( ( -u 1 S A ) P B ) ) ) ) |
50 |
8
|
adantr |
|- ( ( N e. NN0 /\ A e. X ) -> -u N e. CC ) |
51 |
11
|
adantl |
|- ( ( N e. NN0 /\ A e. X ) -> ( A P B ) e. CC ) |
52 |
43
|
adantl |
|- ( ( N e. NN0 /\ A e. X ) -> ( ( -u 1 S A ) P B ) e. CC ) |
53 |
50 51 52
|
adddid |
|- ( ( N e. NN0 /\ A e. X ) -> ( -u N x. ( ( A P B ) + ( ( -u 1 S A ) P B ) ) ) = ( ( -u N x. ( A P B ) ) + ( -u N x. ( ( -u 1 S A ) P B ) ) ) ) |
54 |
1 2 3 4 5
|
ipdiri |
|- ( ( A e. X /\ ( -u 1 S A ) e. X /\ B e. X ) -> ( ( A G ( -u 1 S A ) ) P B ) = ( ( A P B ) + ( ( -u 1 S A ) P B ) ) ) |
55 |
6 54
|
mp3an3 |
|- ( ( A e. X /\ ( -u 1 S A ) e. X ) -> ( ( A G ( -u 1 S A ) ) P B ) = ( ( A P B ) + ( ( -u 1 S A ) P B ) ) ) |
56 |
36 55
|
mpdan |
|- ( A e. X -> ( ( A G ( -u 1 S A ) ) P B ) = ( ( A P B ) + ( ( -u 1 S A ) P B ) ) ) |
57 |
|
eqid |
|- ( 0vec ` U ) = ( 0vec ` U ) |
58 |
1 2 3 57
|
nvrinv |
|- ( ( U e. NrmCVec /\ A e. X ) -> ( A G ( -u 1 S A ) ) = ( 0vec ` U ) ) |
59 |
9 58
|
mpan |
|- ( A e. X -> ( A G ( -u 1 S A ) ) = ( 0vec ` U ) ) |
60 |
59
|
oveq1d |
|- ( A e. X -> ( ( A G ( -u 1 S A ) ) P B ) = ( ( 0vec ` U ) P B ) ) |
61 |
1 57 4
|
dip0l |
|- ( ( U e. NrmCVec /\ B e. X ) -> ( ( 0vec ` U ) P B ) = 0 ) |
62 |
9 6 61
|
mp2an |
|- ( ( 0vec ` U ) P B ) = 0 |
63 |
60 62
|
eqtrdi |
|- ( A e. X -> ( ( A G ( -u 1 S A ) ) P B ) = 0 ) |
64 |
56 63
|
eqtr3d |
|- ( A e. X -> ( ( A P B ) + ( ( -u 1 S A ) P B ) ) = 0 ) |
65 |
64
|
oveq2d |
|- ( A e. X -> ( -u N x. ( ( A P B ) + ( ( -u 1 S A ) P B ) ) ) = ( -u N x. 0 ) ) |
66 |
8
|
mul01d |
|- ( N e. NN0 -> ( -u N x. 0 ) = 0 ) |
67 |
65 66
|
sylan9eqr |
|- ( ( N e. NN0 /\ A e. X ) -> ( -u N x. ( ( A P B ) + ( ( -u 1 S A ) P B ) ) ) = 0 ) |
68 |
53 67
|
eqtr3d |
|- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N x. ( A P B ) ) + ( -u N x. ( ( -u 1 S A ) P B ) ) ) = 0 ) |
69 |
49 68
|
eqtr3d |
|- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N x. ( A P B ) ) + -u ( N x. ( ( -u 1 S A ) P B ) ) ) = 0 ) |
70 |
40 46 69
|
3eqtr2d |
|- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N x. ( A P B ) ) - ( ( -u N S A ) P B ) ) = 0 ) |
71 |
13 19 70
|
subeq0d |
|- ( ( N e. NN0 /\ A e. X ) -> ( -u N x. ( A P B ) ) = ( ( -u N S A ) P B ) ) |
72 |
71
|
eqcomd |
|- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N S A ) P B ) = ( -u N x. ( A P B ) ) ) |