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Metamath Proof Explorer

Theorem dibelval2nd

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

Ref Expression
Hypotheses dibelval2nd.b 
|- B = ( Base ` K )
dibelval2nd.l 
|- .<_ = ( le ` K )
dibelval2nd.h 
|- H = ( LHyp ` K )
dibelval2nd.t 
|- T = ( ( LTrn ` K ) ` W )
dibelval2nd.o 
|- .0. = ( f e. T |-> ( _I |` B ) )
dibelval2nd.i 
|- I = ( ( DIsoB ` K ) ` W )
Assertion dibelval2nd 
|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( 2nd ` Y ) = .0. )

Proof

Step Hyp Ref Expression
1 dibelval2nd.b 
 |-  B = ( Base ` K )
2 dibelval2nd.l 
 |-  .<_ = ( le ` K )
3 dibelval2nd.h 
 |-  H = ( LHyp ` K )
4 dibelval2nd.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 dibelval2nd.o 
 |-  .0. = ( f e. T |-> ( _I |` B ) )
6 dibelval2nd.i 
 |-  I = ( ( DIsoB ` K ) ` W )
7 eqid 
 |-  ( ( DIsoA ` K ) ` W ) = ( ( DIsoA ` K ) ` W )
8 1 2 3 4 5 7 6 dibval2 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { .0. } ) )
9 8 eleq2d 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( Y e. ( I ` X ) <-> Y e. ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { .0. } ) ) )
10 9 biimp3a 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> Y e. ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { .0. } ) )
11 xp2nd 
 |-  ( Y e. ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { .0. } ) -> ( 2nd ` Y ) e. { .0. } )
12 elsni 
 |-  ( ( 2nd ` Y ) e. { .0. } -> ( 2nd ` Y ) = .0. )
13 10 11 12 3syl 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( 2nd ` Y ) = .0. )