| Step |
Hyp |
Ref |
Expression |
| 1 |
|
tcphval.n |
|- G = ( toCPreHil ` W ) |
| 2 |
|
tcphval.v |
|- V = ( Base ` W ) |
| 3 |
|
tcphval.h |
|- ., = ( .i ` W ) |
| 4 |
|
id |
|- ( w = W -> w = W ) |
| 5 |
|
fveq2 |
|- ( w = W -> ( Base ` w ) = ( Base ` W ) ) |
| 6 |
5 2
|
eqtr4di |
|- ( w = W -> ( Base ` w ) = V ) |
| 7 |
|
fveq2 |
|- ( w = W -> ( .i ` w ) = ( .i ` W ) ) |
| 8 |
7 3
|
eqtr4di |
|- ( w = W -> ( .i ` w ) = ., ) |
| 9 |
8
|
oveqd |
|- ( w = W -> ( x ( .i ` w ) x ) = ( x ., x ) ) |
| 10 |
9
|
fveq2d |
|- ( w = W -> ( sqrt ` ( x ( .i ` w ) x ) ) = ( sqrt ` ( x ., x ) ) ) |
| 11 |
6 10
|
mpteq12dv |
|- ( w = W -> ( x e. ( Base ` w ) |-> ( sqrt ` ( x ( .i ` w ) x ) ) ) = ( x e. V |-> ( sqrt ` ( x ., x ) ) ) ) |
| 12 |
4 11
|
oveq12d |
|- ( w = W -> ( w toNrmGrp ( x e. ( Base ` w ) |-> ( sqrt ` ( x ( .i ` w ) x ) ) ) ) = ( W toNrmGrp ( x e. V |-> ( sqrt ` ( x ., x ) ) ) ) ) |
| 13 |
|
df-tcph |
|- toCPreHil = ( w e. _V |-> ( w toNrmGrp ( x e. ( Base ` w ) |-> ( sqrt ` ( x ( .i ` w ) x ) ) ) ) ) |
| 14 |
|
ovex |
|- ( W toNrmGrp ( x e. V |-> ( sqrt ` ( x ., x ) ) ) ) e. _V |
| 15 |
12 13 14
|
fvmpt |
|- ( W e. _V -> ( toCPreHil ` W ) = ( W toNrmGrp ( x e. V |-> ( sqrt ` ( x ., x ) ) ) ) ) |
| 16 |
|
fvprc |
|- ( -. W e. _V -> ( toCPreHil ` W ) = (/) ) |
| 17 |
|
reldmtng |
|- Rel dom toNrmGrp |
| 18 |
17
|
ovprc1 |
|- ( -. W e. _V -> ( W toNrmGrp ( x e. V |-> ( sqrt ` ( x ., x ) ) ) ) = (/) ) |
| 19 |
16 18
|
eqtr4d |
|- ( -. W e. _V -> ( toCPreHil ` W ) = ( W toNrmGrp ( x e. V |-> ( sqrt ` ( x ., x ) ) ) ) ) |
| 20 |
15 19
|
pm2.61i |
|- ( toCPreHil ` W ) = ( W toNrmGrp ( x e. V |-> ( sqrt ` ( x ., x ) ) ) ) |
| 21 |
1 20
|
eqtri |
|- G = ( W toNrmGrp ( x e. V |-> ( sqrt ` ( x ., x ) ) ) ) |