# Metamath Proof Explorer

## Theorem op01dm

Description: Conditions necessary for zero and unit elements to exist. (Contributed by NM, 14-Sep-2018)

Ref Expression
Hypotheses op01dm.b
`|- B = ( Base ` K )`
op01dm.u
`|- U = ( lub ` K )`
op01dm.g
`|- G = ( glb ` K )`
Assertion op01dm
`|- ( K e. OP -> ( B e. dom U /\ B e. dom G ) )`

### Proof

Step Hyp Ref Expression
1 op01dm.b
` |-  B = ( Base ` K )`
2 op01dm.u
` |-  U = ( lub ` K )`
3 op01dm.g
` |-  G = ( glb ` K )`
4 eqid
` |-  ( le ` K ) = ( le ` K )`
5 eqid
` |-  ( oc ` K ) = ( oc ` K )`
6 eqid
` |-  ( join ` K ) = ( join ` K )`
7 eqid
` |-  ( meet ` K ) = ( meet ` K )`
8 eqid
` |-  ( 0. ` K ) = ( 0. ` K )`
9 eqid
` |-  ( 1. ` K ) = ( 1. ` K )`
10 1 2 3 4 5 6 7 8 9 isopos
` |-  ( K e. OP <-> ( ( K e. Poset /\ B e. dom U /\ B e. dom G ) /\ A. x e. B A. y e. B ( ( ( ( oc ` K ) ` x ) e. B /\ ( ( oc ` K ) ` ( ( oc ` K ) ` x ) ) = x /\ ( x ( le ` K ) y -> ( ( oc ` K ) ` y ) ( le ` K ) ( ( oc ` K ) ` x ) ) ) /\ ( x ( join ` K ) ( ( oc ` K ) ` x ) ) = ( 1. ` K ) /\ ( x ( meet ` K ) ( ( oc ` K ) ` x ) ) = ( 0. ` K ) ) ) )`
11 simpl
` |-  ( ( ( B e. dom U /\ B e. dom G ) /\ A. x e. B A. y e. B ( ( ( ( oc ` K ) ` x ) e. B /\ ( ( oc ` K ) ` ( ( oc ` K ) ` x ) ) = x /\ ( x ( le ` K ) y -> ( ( oc ` K ) ` y ) ( le ` K ) ( ( oc ` K ) ` x ) ) ) /\ ( x ( join ` K ) ( ( oc ` K ) ` x ) ) = ( 1. ` K ) /\ ( x ( meet ` K ) ( ( oc ` K ) ` x ) ) = ( 0. ` K ) ) ) -> ( B e. dom U /\ B e. dom G ) )`
` |-  ( ( ( K e. Poset /\ B e. dom U /\ B e. dom G ) /\ A. x e. B A. y e. B ( ( ( ( oc ` K ) ` x ) e. B /\ ( ( oc ` K ) ` ( ( oc ` K ) ` x ) ) = x /\ ( x ( le ` K ) y -> ( ( oc ` K ) ` y ) ( le ` K ) ( ( oc ` K ) ` x ) ) ) /\ ( x ( join ` K ) ( ( oc ` K ) ` x ) ) = ( 1. ` K ) /\ ( x ( meet ` K ) ( ( oc ` K ) ` x ) ) = ( 0. ` K ) ) ) -> ( B e. dom U /\ B e. dom G ) )`
` |-  ( K e. OP -> ( B e. dom U /\ B e. dom G ) )`