| Step |
Hyp |
Ref |
Expression |
| 1 |
|
p0val.b |
|- B = ( Base ` K ) |
| 2 |
|
p0val.g |
|- G = ( glb ` K ) |
| 3 |
|
p0val.z |
|- .0. = ( 0. ` K ) |
| 4 |
|
elex |
|- ( K e. V -> K e. _V ) |
| 5 |
|
fveq2 |
|- ( p = K -> ( glb ` p ) = ( glb ` K ) ) |
| 6 |
5 2
|
eqtr4di |
|- ( p = K -> ( glb ` p ) = G ) |
| 7 |
|
fveq2 |
|- ( p = K -> ( Base ` p ) = ( Base ` K ) ) |
| 8 |
7 1
|
eqtr4di |
|- ( p = K -> ( Base ` p ) = B ) |
| 9 |
6 8
|
fveq12d |
|- ( p = K -> ( ( glb ` p ) ` ( Base ` p ) ) = ( G ` B ) ) |
| 10 |
|
df-p0 |
|- 0. = ( p e. _V |-> ( ( glb ` p ) ` ( Base ` p ) ) ) |
| 11 |
|
fvex |
|- ( G ` B ) e. _V |
| 12 |
9 10 11
|
fvmpt |
|- ( K e. _V -> ( 0. ` K ) = ( G ` B ) ) |
| 13 |
3 12
|
eqtrid |
|- ( K e. _V -> .0. = ( G ` B ) ) |
| 14 |
4 13
|
syl |
|- ( K e. V -> .0. = ( G ` B ) ) |