| Step |
Hyp |
Ref |
Expression |
| 1 |
|
thlval.k |
|- K = ( toHL ` W ) |
| 2 |
|
thlbas.c |
|- C = ( ClSubSp ` W ) |
| 3 |
2
|
fvexi |
|- C e. _V |
| 4 |
|
eqid |
|- ( toInc ` C ) = ( toInc ` C ) |
| 5 |
4
|
ipobas |
|- ( C e. _V -> C = ( Base ` ( toInc ` C ) ) ) |
| 6 |
3 5
|
ax-mp |
|- C = ( Base ` ( toInc ` C ) ) |
| 7 |
|
baseid |
|- Base = Slot ( Base ` ndx ) |
| 8 |
|
1re |
|- 1 e. RR |
| 9 |
|
1nn |
|- 1 e. NN |
| 10 |
|
1nn0 |
|- 1 e. NN0 |
| 11 |
|
1lt10 |
|- 1 < ; 1 0 |
| 12 |
9 10 10 11
|
declti |
|- 1 < ; 1 1 |
| 13 |
8 12
|
ltneii |
|- 1 =/= ; 1 1 |
| 14 |
|
basendx |
|- ( Base ` ndx ) = 1 |
| 15 |
|
ocndx |
|- ( oc ` ndx ) = ; 1 1 |
| 16 |
14 15
|
neeq12i |
|- ( ( Base ` ndx ) =/= ( oc ` ndx ) <-> 1 =/= ; 1 1 ) |
| 17 |
13 16
|
mpbir |
|- ( Base ` ndx ) =/= ( oc ` ndx ) |
| 18 |
7 17
|
setsnid |
|- ( Base ` ( toInc ` C ) ) = ( Base ` ( ( toInc ` C ) sSet <. ( oc ` ndx ) , ( ocv ` W ) >. ) ) |
| 19 |
6 18
|
eqtri |
|- C = ( Base ` ( ( toInc ` C ) sSet <. ( oc ` ndx ) , ( ocv ` W ) >. ) ) |
| 20 |
|
eqid |
|- ( ocv ` W ) = ( ocv ` W ) |
| 21 |
1 2 4 20
|
thlval |
|- ( W e. _V -> K = ( ( toInc ` C ) sSet <. ( oc ` ndx ) , ( ocv ` W ) >. ) ) |
| 22 |
21
|
fveq2d |
|- ( W e. _V -> ( Base ` K ) = ( Base ` ( ( toInc ` C ) sSet <. ( oc ` ndx ) , ( ocv ` W ) >. ) ) ) |
| 23 |
19 22
|
eqtr4id |
|- ( W e. _V -> C = ( Base ` K ) ) |
| 24 |
|
base0 |
|- (/) = ( Base ` (/) ) |
| 25 |
|
fvprc |
|- ( -. W e. _V -> ( ClSubSp ` W ) = (/) ) |
| 26 |
2 25
|
syl5eq |
|- ( -. W e. _V -> C = (/) ) |
| 27 |
|
fvprc |
|- ( -. W e. _V -> ( toHL ` W ) = (/) ) |
| 28 |
1 27
|
syl5eq |
|- ( -. W e. _V -> K = (/) ) |
| 29 |
28
|
fveq2d |
|- ( -. W e. _V -> ( Base ` K ) = ( Base ` (/) ) ) |
| 30 |
24 26 29
|
3eqtr4a |
|- ( -. W e. _V -> C = ( Base ` K ) ) |
| 31 |
23 30
|
pm2.61i |
|- C = ( Base ` K ) |