Step |
Hyp |
Ref |
Expression |
1 |
|
op0le.b |
|- B = ( Base ` K ) |
2 |
|
op0le.l |
|- .<_ = ( le ` K ) |
3 |
|
op0le.z |
|- .0. = ( 0. ` K ) |
4 |
|
eqid |
|- ( glb ` K ) = ( glb ` K ) |
5 |
|
simpl |
|- ( ( K e. OP /\ X e. B ) -> K e. OP ) |
6 |
|
simpr |
|- ( ( K e. OP /\ X e. B ) -> X e. B ) |
7 |
|
eqid |
|- ( lub ` K ) = ( lub ` K ) |
8 |
1 7 4
|
op01dm |
|- ( K e. OP -> ( B e. dom ( lub ` K ) /\ B e. dom ( glb ` K ) ) ) |
9 |
8
|
simprd |
|- ( K e. OP -> B e. dom ( glb ` K ) ) |
10 |
9
|
adantr |
|- ( ( K e. OP /\ X e. B ) -> B e. dom ( glb ` K ) ) |
11 |
1 4 2 3 5 6 10
|
p0le |
|- ( ( K e. OP /\ X e. B ) -> .0. .<_ X ) |