| Step |
Hyp |
Ref |
Expression |
| 1 |
|
tcphval.n |
|- G = ( toCPreHil ` W ) |
| 2 |
|
tcphnmval.n |
|- N = ( norm ` G ) |
| 3 |
|
tcphnmval.v |
|- V = ( Base ` W ) |
| 4 |
|
tcphnmval.h |
|- ., = ( .i ` W ) |
| 5 |
|
eqid |
|- ( x e. V |-> ( sqrt ` ( x ., x ) ) ) = ( x e. V |-> ( sqrt ` ( x ., x ) ) ) |
| 6 |
|
fvrn0 |
|- ( sqrt ` ( x ., x ) ) e. ( ran sqrt u. { (/) } ) |
| 7 |
6
|
a1i |
|- ( x e. V -> ( sqrt ` ( x ., x ) ) e. ( ran sqrt u. { (/) } ) ) |
| 8 |
5 7
|
fmpti |
|- ( x e. V |-> ( sqrt ` ( x ., x ) ) ) : V --> ( ran sqrt u. { (/) } ) |
| 9 |
1 3 4
|
tcphval |
|- G = ( W toNrmGrp ( x e. V |-> ( sqrt ` ( x ., x ) ) ) ) |
| 10 |
|
cnex |
|- CC e. _V |
| 11 |
|
sqrtf |
|- sqrt : CC --> CC |
| 12 |
|
frn |
|- ( sqrt : CC --> CC -> ran sqrt C_ CC ) |
| 13 |
11 12
|
ax-mp |
|- ran sqrt C_ CC |
| 14 |
10 13
|
ssexi |
|- ran sqrt e. _V |
| 15 |
|
p0ex |
|- { (/) } e. _V |
| 16 |
14 15
|
unex |
|- ( ran sqrt u. { (/) } ) e. _V |
| 17 |
9 3 16
|
tngnm |
|- ( ( W e. Grp /\ ( x e. V |-> ( sqrt ` ( x ., x ) ) ) : V --> ( ran sqrt u. { (/) } ) ) -> ( x e. V |-> ( sqrt ` ( x ., x ) ) ) = ( norm ` G ) ) |
| 18 |
8 17
|
mpan2 |
|- ( W e. Grp -> ( x e. V |-> ( sqrt ` ( x ., x ) ) ) = ( norm ` G ) ) |
| 19 |
2 18
|
eqtr4id |
|- ( W e. Grp -> N = ( x e. V |-> ( sqrt ` ( x ., x ) ) ) ) |