Step |
Hyp |
Ref |
Expression |
1 |
|
ncvsprp.v |
|- V = ( Base ` W ) |
2 |
|
ncvsprp.n |
|- N = ( norm ` W ) |
3 |
|
ncvsprp.s |
|- .x. = ( .s ` W ) |
4 |
|
simpl |
|- ( ( W e. ( NrmVec i^i CVec ) /\ A e. V ) -> W e. ( NrmVec i^i CVec ) ) |
5 |
|
elin |
|- ( W e. ( NrmVec i^i CVec ) <-> ( W e. NrmVec /\ W e. CVec ) ) |
6 |
|
id |
|- ( W e. CVec -> W e. CVec ) |
7 |
6
|
cvsclm |
|- ( W e. CVec -> W e. CMod ) |
8 |
|
eqid |
|- ( Scalar ` W ) = ( Scalar ` W ) |
9 |
|
eqid |
|- ( Base ` ( Scalar ` W ) ) = ( Base ` ( Scalar ` W ) ) |
10 |
8 9
|
clmneg1 |
|- ( W e. CMod -> -u 1 e. ( Base ` ( Scalar ` W ) ) ) |
11 |
7 10
|
syl |
|- ( W e. CVec -> -u 1 e. ( Base ` ( Scalar ` W ) ) ) |
12 |
5 11
|
simplbiim |
|- ( W e. ( NrmVec i^i CVec ) -> -u 1 e. ( Base ` ( Scalar ` W ) ) ) |
13 |
12
|
adantr |
|- ( ( W e. ( NrmVec i^i CVec ) /\ A e. V ) -> -u 1 e. ( Base ` ( Scalar ` W ) ) ) |
14 |
|
simpr |
|- ( ( W e. ( NrmVec i^i CVec ) /\ A e. V ) -> A e. V ) |
15 |
1 2 3 8 9
|
ncvsprp |
|- ( ( W e. ( NrmVec i^i CVec ) /\ -u 1 e. ( Base ` ( Scalar ` W ) ) /\ A e. V ) -> ( N ` ( -u 1 .x. A ) ) = ( ( abs ` -u 1 ) x. ( N ` A ) ) ) |
16 |
4 13 14 15
|
syl3anc |
|- ( ( W e. ( NrmVec i^i CVec ) /\ A e. V ) -> ( N ` ( -u 1 .x. A ) ) = ( ( abs ` -u 1 ) x. ( N ` A ) ) ) |
17 |
|
ax-1cn |
|- 1 e. CC |
18 |
17
|
absnegi |
|- ( abs ` -u 1 ) = ( abs ` 1 ) |
19 |
|
abs1 |
|- ( abs ` 1 ) = 1 |
20 |
18 19
|
eqtri |
|- ( abs ` -u 1 ) = 1 |
21 |
20
|
oveq1i |
|- ( ( abs ` -u 1 ) x. ( N ` A ) ) = ( 1 x. ( N ` A ) ) |
22 |
|
nvcnlm |
|- ( W e. NrmVec -> W e. NrmMod ) |
23 |
|
nlmngp |
|- ( W e. NrmMod -> W e. NrmGrp ) |
24 |
22 23
|
syl |
|- ( W e. NrmVec -> W e. NrmGrp ) |
25 |
24
|
adantr |
|- ( ( W e. NrmVec /\ W e. CVec ) -> W e. NrmGrp ) |
26 |
5 25
|
sylbi |
|- ( W e. ( NrmVec i^i CVec ) -> W e. NrmGrp ) |
27 |
1 2
|
nmcl |
|- ( ( W e. NrmGrp /\ A e. V ) -> ( N ` A ) e. RR ) |
28 |
26 27
|
sylan |
|- ( ( W e. ( NrmVec i^i CVec ) /\ A e. V ) -> ( N ` A ) e. RR ) |
29 |
28
|
recnd |
|- ( ( W e. ( NrmVec i^i CVec ) /\ A e. V ) -> ( N ` A ) e. CC ) |
30 |
29
|
mulid2d |
|- ( ( W e. ( NrmVec i^i CVec ) /\ A e. V ) -> ( 1 x. ( N ` A ) ) = ( N ` A ) ) |
31 |
21 30
|
syl5eq |
|- ( ( W e. ( NrmVec i^i CVec ) /\ A e. V ) -> ( ( abs ` -u 1 ) x. ( N ` A ) ) = ( N ` A ) ) |
32 |
16 31
|
eqtrd |
|- ( ( W e. ( NrmVec i^i CVec ) /\ A e. V ) -> ( N ` ( -u 1 .x. A ) ) = ( N ` A ) ) |