Theorem opococ

Description: Double negative law for orthoposets. ( ococ analog.) (Contributed by NM, 13-Sep-2011)

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

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	simp2d	|- ( ( K e. OP /\ X e. B ) -> ( ._|_ ` ( ._|_ ` X ) ) = X )