Metamath Proof Explorer

Theorem opoccl

Description: Closure of orthocomplement operation. ( choccl analog.) (Contributed by NM, 20-Oct-2011)

Ref Expression
Hypotheses opoccl.b 
|- B = ( Base ` K )
opoccl.o 
|- ._|_ = ( oc ` K )
Assertion opoccl 
|- ( ( K e. OP /\ X e. B ) -> ( ._|_ ` X ) e. B )

Proof

Step Hyp Ref Expression
1 opoccl.b 
 |-  B = ( Base ` K )
2 opoccl.o 
 |-  ._|_ = ( oc ` K )
3 eqid 
 |-  ( le ` K ) = ( le ` K )
4 eqid 
 |-  ( join ` K ) = ( join ` K )
5 eqid 
 |-  ( meet ` K ) = ( meet ` K )
6 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
7 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
8 1 3 2 4 5 6 7 oposlem 
 |-  ( ( K e. OP /\ X e. B /\ X e. B ) -> ( ( ( ._|_ ` X ) e. B /\ ( ._|_ ` ( ._|_ ` X ) ) = X /\ ( X ( le ` K ) X -> ( ._|_ ` X ) ( le ` K ) ( ._|_ ` X ) ) ) /\ ( X ( join ` K ) ( ._|_ ` X ) ) = ( 1. ` K ) /\ ( X ( meet ` K ) ( ._|_ ` X ) ) = ( 0. ` K ) ) )
9 8 3anidm23 
 |-  ( ( K e. OP /\ X e. B ) -> ( ( ( ._|_ ` X ) e. B /\ ( ._|_ ` ( ._|_ ` X ) ) = X /\ ( X ( le ` K ) X -> ( ._|_ ` X ) ( le ` K ) ( ._|_ ` X ) ) ) /\ ( X ( join ` K ) ( ._|_ ` X ) ) = ( 1. ` K ) /\ ( X ( meet ` K ) ( ._|_ ` X ) ) = ( 0. ` K ) ) )
10 9 simp1d 
 |-  ( ( K e. OP /\ X e. B ) -> ( ( ._|_ ` X ) e. B /\ ( ._|_ ` ( ._|_ ` X ) ) = X /\ ( X ( le ` K ) X -> ( ._|_ ` X ) ( le ` K ) ( ._|_ ` X ) ) ) )
11 10 simp1d 
 |-  ( ( K e. OP /\ X e. B ) -> ( ._|_ ` X ) e. B )