Description: An orthoposet has a unit element. ( helch analog.) (Contributed by NM, 22-Oct-2011)
Ref | Expression | ||
---|---|---|---|
Hypotheses | op1cl.b | |- B = ( Base ` K ) |
|
op1cl.u | |- .1. = ( 1. ` K ) |
||
Assertion | op1cl | |- ( K e. OP -> .1. e. B ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | op1cl.b | |- B = ( Base ` K ) |
|
2 | op1cl.u | |- .1. = ( 1. ` K ) |
|
3 | eqid | |- ( lub ` K ) = ( lub ` K ) |
|
4 | 1 3 2 | p1val | |- ( K e. OP -> .1. = ( ( lub ` K ) ` B ) ) |
5 | id | |- ( K e. OP -> K e. OP ) |
|
6 | eqid | |- ( glb ` K ) = ( glb ` K ) |
|
7 | 1 3 6 | op01dm | |- ( K e. OP -> ( B e. dom ( lub ` K ) /\ B e. dom ( glb ` K ) ) ) |
8 | 7 | simpld | |- ( K e. OP -> B e. dom ( lub ` K ) ) |
9 | 1 3 5 8 | lubcl | |- ( K e. OP -> ( ( lub ` K ) ` B ) e. B ) |
10 | 4 9 | eqeltrd | |- ( K e. OP -> .1. e. B ) |