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Metamath Proof Explorer

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 )