Step |
Hyp |
Ref |
Expression |
1 |
|
rrxdim.1 |
|- H = ( RR^ ` I ) |
2 |
1
|
rrxval |
|- ( I e. V -> H = ( toCPreHil ` ( RRfld freeLMod I ) ) ) |
3 |
|
eqid |
|- ( toCPreHil ` ( RRfld freeLMod I ) ) = ( toCPreHil ` ( RRfld freeLMod I ) ) |
4 |
|
eqid |
|- ( Base ` ( RRfld freeLMod I ) ) = ( Base ` ( RRfld freeLMod I ) ) |
5 |
|
eqid |
|- ( .i ` ( RRfld freeLMod I ) ) = ( .i ` ( RRfld freeLMod I ) ) |
6 |
3 4 5
|
tcphval |
|- ( toCPreHil ` ( RRfld freeLMod I ) ) = ( ( RRfld freeLMod I ) toNrmGrp ( x e. ( Base ` ( RRfld freeLMod I ) ) |-> ( sqrt ` ( x ( .i ` ( RRfld freeLMod I ) ) x ) ) ) ) |
7 |
2 6
|
eqtrdi |
|- ( I e. V -> H = ( ( RRfld freeLMod I ) toNrmGrp ( x e. ( Base ` ( RRfld freeLMod I ) ) |-> ( sqrt ` ( x ( .i ` ( RRfld freeLMod I ) ) x ) ) ) ) ) |
8 |
7
|
fveq2d |
|- ( I e. V -> ( dim ` H ) = ( dim ` ( ( RRfld freeLMod I ) toNrmGrp ( x e. ( Base ` ( RRfld freeLMod I ) ) |-> ( sqrt ` ( x ( .i ` ( RRfld freeLMod I ) ) x ) ) ) ) ) ) |
9 |
|
resubdrg |
|- ( RR e. ( SubRing ` CCfld ) /\ RRfld e. DivRing ) |
10 |
9
|
simpri |
|- RRfld e. DivRing |
11 |
|
eqid |
|- ( RRfld freeLMod I ) = ( RRfld freeLMod I ) |
12 |
11
|
frlmlvec |
|- ( ( RRfld e. DivRing /\ I e. V ) -> ( RRfld freeLMod I ) e. LVec ) |
13 |
10 12
|
mpan |
|- ( I e. V -> ( RRfld freeLMod I ) e. LVec ) |
14 |
4
|
tcphex |
|- ( x e. ( Base ` ( RRfld freeLMod I ) ) |-> ( sqrt ` ( x ( .i ` ( RRfld freeLMod I ) ) x ) ) ) e. _V |
15 |
|
eqid |
|- ( ( RRfld freeLMod I ) toNrmGrp ( x e. ( Base ` ( RRfld freeLMod I ) ) |-> ( sqrt ` ( x ( .i ` ( RRfld freeLMod I ) ) x ) ) ) ) = ( ( RRfld freeLMod I ) toNrmGrp ( x e. ( Base ` ( RRfld freeLMod I ) ) |-> ( sqrt ` ( x ( .i ` ( RRfld freeLMod I ) ) x ) ) ) ) |
16 |
15
|
tngdim |
|- ( ( ( RRfld freeLMod I ) e. LVec /\ ( x e. ( Base ` ( RRfld freeLMod I ) ) |-> ( sqrt ` ( x ( .i ` ( RRfld freeLMod I ) ) x ) ) ) e. _V ) -> ( dim ` ( RRfld freeLMod I ) ) = ( dim ` ( ( RRfld freeLMod I ) toNrmGrp ( x e. ( Base ` ( RRfld freeLMod I ) ) |-> ( sqrt ` ( x ( .i ` ( RRfld freeLMod I ) ) x ) ) ) ) ) ) |
17 |
13 14 16
|
sylancl |
|- ( I e. V -> ( dim ` ( RRfld freeLMod I ) ) = ( dim ` ( ( RRfld freeLMod I ) toNrmGrp ( x e. ( Base ` ( RRfld freeLMod I ) ) |-> ( sqrt ` ( x ( .i ` ( RRfld freeLMod I ) ) x ) ) ) ) ) ) |
18 |
11
|
frlmdim |
|- ( ( RRfld e. DivRing /\ I e. V ) -> ( dim ` ( RRfld freeLMod I ) ) = ( # ` I ) ) |
19 |
10 18
|
mpan |
|- ( I e. V -> ( dim ` ( RRfld freeLMod I ) ) = ( # ` I ) ) |
20 |
8 17 19
|
3eqtr2d |
|- ( I e. V -> ( dim ` H ) = ( # ` I ) ) |