Theorem dochoccl

Description: A set of vectors is closed iff it equals its double orthocomplent. (Contributed by NM, 1-Jan-2015)

	Ref	Expression
Hypotheses	dochoccl.h	|- H = ( LHyp ` K )
	dochoccl.i	|- I = ( ( DIsoH ` K ) ` W )
	dochoccl.u	|- U = ( ( DVecH ` K ) ` W )
	dochoccl.v	|- V = ( Base ` U )
	dochoccl.o	|- ._|_ = ( ( ocH ` K ) ` W )
	dochoccl.k	|- ( ph -> ( K e. HL /\ W e. H ) )
	dochoccl.g	|- ( ph -> X C_ V )
Assertion	dochoccl	|- ( ph -> ( X e. ran I <-> ( ._|_ ` ( ._|_ ` X ) ) = X ) )

Proof

Step	Hyp	Ref	Expression
1		dochoccl.h	|- H = ( LHyp ` K )
2		dochoccl.i	|- I = ( ( DIsoH ` K ) ` W )
3		dochoccl.u	|- U = ( ( DVecH ` K ) ` W )
4		dochoccl.v	|- V = ( Base ` U )
5		dochoccl.o	|- ._|_ = ( ( ocH ` K ) ` W )
6		dochoccl.k	|- ( ph -> ( K e. HL /\ W e. H ) )
7		dochoccl.g	|- ( ph -> X C_ V )
8	1 2 5	dochoc	|- ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( ._|_ ` ( ._|_ ` X ) ) = X )
9	6 8	sylan	|- ( ( ph /\ X e. ran I ) -> ( ._|_ ` ( ._|_ ` X ) ) = X )
10		simpr	|- ( ( ph /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( ._|_ ` ( ._|_ ` X ) ) = X )
11	1 3 4 5	dochssv	|- ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` X ) C_ V )
12	6 7 11	syl2anc	|- ( ph -> ( ._|_ ` X ) C_ V )
13	1 2 3 4 5	dochcl	|- ( ( ( K e. HL /\ W e. H ) /\ ( ._|_ ` X ) C_ V ) -> ( ._|_ ` ( ._|_ ` X ) ) e. ran I )
14	6 12 13	syl2anc	|- ( ph -> ( ._|_ ` ( ._|_ ` X ) ) e. ran I )
15	14	adantr	|- ( ( ph /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( ._|_ ` ( ._|_ ` X ) ) e. ran I )
16	10 15	eqeltrrd	|- ( ( ph /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> X e. ran I )
17	9 16	impbida	|- ( ph -> ( X e. ran I <-> ( ._|_ ` ( ._|_ ` X ) ) = X ) )