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Metamath Proof Explorer

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 ) )