Step |
Hyp |
Ref |
Expression |
1 |
|
dip0r.1 |
|- X = ( BaseSet ` U ) |
2 |
|
dip0r.5 |
|- Z = ( 0vec ` U ) |
3 |
|
dip0r.7 |
|- P = ( .iOLD ` U ) |
4 |
|
eqid |
|- ( normCV ` U ) = ( normCV ` U ) |
5 |
1 4 3
|
ipidsq |
|- ( ( U e. NrmCVec /\ A e. X ) -> ( A P A ) = ( ( ( normCV ` U ) ` A ) ^ 2 ) ) |
6 |
5
|
eqeq1d |
|- ( ( U e. NrmCVec /\ A e. X ) -> ( ( A P A ) = 0 <-> ( ( ( normCV ` U ) ` A ) ^ 2 ) = 0 ) ) |
7 |
1 4
|
nvcl |
|- ( ( U e. NrmCVec /\ A e. X ) -> ( ( normCV ` U ) ` A ) e. RR ) |
8 |
7
|
recnd |
|- ( ( U e. NrmCVec /\ A e. X ) -> ( ( normCV ` U ) ` A ) e. CC ) |
9 |
|
sqeq0 |
|- ( ( ( normCV ` U ) ` A ) e. CC -> ( ( ( ( normCV ` U ) ` A ) ^ 2 ) = 0 <-> ( ( normCV ` U ) ` A ) = 0 ) ) |
10 |
8 9
|
syl |
|- ( ( U e. NrmCVec /\ A e. X ) -> ( ( ( ( normCV ` U ) ` A ) ^ 2 ) = 0 <-> ( ( normCV ` U ) ` A ) = 0 ) ) |
11 |
1 2 4
|
nvz |
|- ( ( U e. NrmCVec /\ A e. X ) -> ( ( ( normCV ` U ) ` A ) = 0 <-> A = Z ) ) |
12 |
6 10 11
|
3bitrd |
|- ( ( U e. NrmCVec /\ A e. X ) -> ( ( A P A ) = 0 <-> A = Z ) ) |