Theorem dihord3

Description: The isomorphism H for a lattice K is order-preserving in the region under co-atom W . (Contributed by NM, 6-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	dihord3	|- ( ( ( 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	|- ( ( DIsoB ` K ) ` W ) = ( ( DIsoB ` K ) ` W )
6	1 2 3 4 5	dihvalb	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( ( ( DIsoB ` K ) ` W ) ` X ) )
7	6	3adant3	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( I ` X ) = ( ( ( DIsoB ` K ) ` W ) ` X ) )
8	1 2 3 4 5	dihvalb	|- ( ( ( K e. HL /\ W e. H ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( I ` Y ) = ( ( ( DIsoB ` K ) ` W ) ` Y ) )
9	8	3adant2	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( I ` Y ) = ( ( ( DIsoB ` K ) ` W ) ` Y ) )
10	7 9	sseq12d	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( I ` X ) C_ ( I ` Y ) <-> ( ( ( DIsoB ` K ) ` W ) ` X ) C_ ( ( ( DIsoB ` K ) ` W ) ` Y ) ) )
11	1 2 3 5	dibord	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( ( ( DIsoB ` K ) ` W ) ` X ) C_ ( ( ( DIsoB ` K ) ` W ) ` Y ) <-> X .<_ Y ) )
12	10 11	bitrd	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( I ` X ) C_ ( I ` Y ) <-> X .<_ Y ) )