| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ax-icn |
|- _i e. CC |
| 2 |
|
absval |
|- ( _i e. CC -> ( abs ` _i ) = ( sqrt ` ( _i x. ( * ` _i ) ) ) ) |
| 3 |
1 2
|
ax-mp |
|- ( abs ` _i ) = ( sqrt ` ( _i x. ( * ` _i ) ) ) |
| 4 |
|
cji |
|- ( * ` _i ) = -u _i |
| 5 |
4
|
oveq2i |
|- ( _i x. ( * ` _i ) ) = ( _i x. -u _i ) |
| 6 |
1 1
|
mulneg2i |
|- ( _i x. -u _i ) = -u ( _i x. _i ) |
| 7 |
|
ixi |
|- ( _i x. _i ) = -u 1 |
| 8 |
7
|
negeqi |
|- -u ( _i x. _i ) = -u -u 1 |
| 9 |
|
negneg1e1 |
|- -u -u 1 = 1 |
| 10 |
8 9
|
eqtri |
|- -u ( _i x. _i ) = 1 |
| 11 |
5 6 10
|
3eqtri |
|- ( _i x. ( * ` _i ) ) = 1 |
| 12 |
11
|
fveq2i |
|- ( sqrt ` ( _i x. ( * ` _i ) ) ) = ( sqrt ` 1 ) |
| 13 |
|
sqrt1 |
|- ( sqrt ` 1 ) = 1 |
| 14 |
3 12 13
|
3eqtri |
|- ( abs ` _i ) = 1 |