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Metamath Proof Explorer

Theorem dibelval1st1

Description: Membership in value of the partial isomorphism B for a lattice K . (Contributed by NM, 13-Feb-2014)

Ref Expression
Hypotheses dibelval1st1.b 
|- B = ( Base ` K )
dibelval1st1.l 
|- .<_ = ( le ` K )
dibelval1st1.h 
|- H = ( LHyp ` K )
dibelval1st1.t 
|- T = ( ( LTrn ` K ) ` W )
dibelval1st1.i 
|- I = ( ( DIsoB ` K ) ` W )
Assertion dibelval1st1 
|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( 1st ` Y ) e. T )

Proof

Step Hyp Ref Expression
1 dibelval1st1.b 
 |-  B = ( Base ` K )
2 dibelval1st1.l 
 |-  .<_ = ( le ` K )
3 dibelval1st1.h 
 |-  H = ( LHyp ` K )
4 dibelval1st1.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 dibelval1st1.i 
 |-  I = ( ( DIsoB ` K ) ` W )
6 eqid 
 |-  ( ( DIsoA ` K ) ` W ) = ( ( DIsoA ` K ) ` W )
7 1 2 3 6 5 dibelval1st 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( 1st ` Y ) e. ( ( ( DIsoA ` K ) ` W ) ` X ) )
8 1 2 3 4 6 diael 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( 1st ` Y ) e. ( ( ( DIsoA ` K ) ` W ) ` X ) ) -> ( 1st ` Y ) e. T )
9 7 8 syld3an3 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( 1st ` Y ) e. T )