Theorem dihord6a

Description: Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014)

	Ref	Expression
Hypotheses	dihord3.b	|- B = ( Base ` K )
	dihord3.l	|- .<_ = ( le ` K )
	dihord3.h	|- H = ( LHyp ` K )
	dihord3.i	|- I = ( ( DIsoH ` K ) ` W )
Assertion	dihord6a	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) /\ ( I ` X ) C_ ( I ` Y ) ) -> X .<_ Y )

Proof

Step	Hyp	Ref	Expression
1		dihord3.b	|- B = ( Base ` K )
2		dihord3.l	|- .<_ = ( le ` K )
3		dihord3.h	|- H = ( LHyp ` K )
4		dihord3.i	|- I = ( ( DIsoH ` K ) ` W )
5		eqid	|- ( Atoms ` K ) = ( Atoms ` K )
6		eqid	|- ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
7		eqid	|- ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) = ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) )
8		eqid	|- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
9		eqid	|- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
10		eqid	|- ( ( DVecH ` K ) ` W ) = ( ( DVecH ` K ) ` W )
11		eqid	|- ( LSSum ` ( ( DVecH ` K ) ` W ) ) = ( LSSum ` ( ( DVecH ` K ) ` W ) )
12		eqid	|- ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` ( ( oc ` K ) ` W ) ) = q ) = ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` ( ( oc ` K ) ` W ) ) = q )
13	1 2 5 3 6 7 8 9 4 10 11 12	dihord6apre	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) /\ ( I ` X ) C_ ( I ` Y ) ) -> X .<_ Y )