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Metamath Proof Explorer

Theorem dibfval

Description: The partial isomorphism B for a lattice K . (Contributed by NM, 8-Dec-2013)

Ref Expression
Hypotheses dibval.b 
|- B = ( Base ` K )
dibval.h 
|- H = ( LHyp ` K )
dibval.t 
|- T = ( ( LTrn ` K ) ` W )
dibval.o 
|- .0. = ( f e. T |-> ( _I |` B ) )
dibval.j 
|- J = ( ( DIsoA ` K ) ` W )
dibval.i 
|- I = ( ( DIsoB ` K ) ` W )
Assertion dibfval 
|- ( ( K e. V /\ W e. H ) -> I = ( x e. dom J |-> ( ( J ` x ) X. { .0. } ) ) )

Proof

Step Hyp Ref Expression
1 dibval.b 
 |-  B = ( Base ` K )
2 dibval.h 
 |-  H = ( LHyp ` K )
3 dibval.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 dibval.o 
 |-  .0. = ( f e. T |-> ( _I |` B ) )
5 dibval.j 
 |-  J = ( ( DIsoA ` K ) ` W )
6 dibval.i 
 |-  I = ( ( DIsoB ` K ) ` W )
7 1 2 dibffval 
 |-  ( K e. V -> ( DIsoB ` K ) = ( w e. H |-> ( x e. dom ( ( DIsoA ` K ) ` w ) |-> ( ( ( ( DIsoA ` K ) ` w ) ` x ) X. { ( f e. ( ( LTrn ` K ) ` w ) |-> ( _I |` B ) ) } ) ) ) )
8 7 fveq1d 
 |-  ( K e. V -> ( ( DIsoB ` K ) ` W ) = ( ( w e. H |-> ( x e. dom ( ( DIsoA ` K ) ` w ) |-> ( ( ( ( DIsoA ` K ) ` w ) ` x ) X. { ( f e. ( ( LTrn ` K ) ` w ) |-> ( _I |` B ) ) } ) ) ) ` W ) )
9 6 8 eqtrid 
 |-  ( K e. V -> I = ( ( w e. H |-> ( x e. dom ( ( DIsoA ` K ) ` w ) |-> ( ( ( ( DIsoA ` K ) ` w ) ` x ) X. { ( f e. ( ( LTrn ` K ) ` w ) |-> ( _I |` B ) ) } ) ) ) ` W ) )
10 fveq2 
 |-  ( w = W -> ( ( DIsoA ` K ) ` w ) = ( ( DIsoA ` K ) ` W ) )
11 10 5 eqtr4di 
 |-  ( w = W -> ( ( DIsoA ` K ) ` w ) = J )
12 11 dmeqd 
 |-  ( w = W -> dom ( ( DIsoA ` K ) ` w ) = dom J )
13 11 fveq1d 
 |-  ( w = W -> ( ( ( DIsoA ` K ) ` w ) ` x ) = ( J ` x ) )
14 fveq2 
 |-  ( w = W -> ( ( LTrn ` K ) ` w ) = ( ( LTrn ` K ) ` W ) )
15 14 3 eqtr4di 
 |-  ( w = W -> ( ( LTrn ` K ) ` w ) = T )
16 eqidd 
 |-  ( w = W -> ( _I |` B ) = ( _I |` B ) )
17 15 16 mpteq12dv 
 |-  ( w = W -> ( f e. ( ( LTrn ` K ) ` w ) |-> ( _I |` B ) ) = ( f e. T |-> ( _I |` B ) ) )
18 17 4 eqtr4di 
 |-  ( w = W -> ( f e. ( ( LTrn ` K ) ` w ) |-> ( _I |` B ) ) = .0. )
19 18 sneqd 
 |-  ( w = W -> { ( f e. ( ( LTrn ` K ) ` w ) |-> ( _I |` B ) ) } = { .0. } )
20 13 19 xpeq12d 
 |-  ( w = W -> ( ( ( ( DIsoA ` K ) ` w ) ` x ) X. { ( f e. ( ( LTrn ` K ) ` w ) |-> ( _I |` B ) ) } ) = ( ( J ` x ) X. { .0. } ) )
21 12 20 mpteq12dv 
 |-  ( w = W -> ( x e. dom ( ( DIsoA ` K ) ` w ) |-> ( ( ( ( DIsoA ` K ) ` w ) ` x ) X. { ( f e. ( ( LTrn ` K ) ` w ) |-> ( _I |` B ) ) } ) ) = ( x e. dom J |-> ( ( J ` x ) X. { .0. } ) ) )
22 eqid 
 |-  ( w e. H |-> ( x e. dom ( ( DIsoA ` K ) ` w ) |-> ( ( ( ( DIsoA ` K ) ` w ) ` x ) X. { ( f e. ( ( LTrn ` K ) ` w ) |-> ( _I |` B ) ) } ) ) ) = ( w e. H |-> ( x e. dom ( ( DIsoA ` K ) ` w ) |-> ( ( ( ( DIsoA ` K ) ` w ) ` x ) X. { ( f e. ( ( LTrn ` K ) ` w ) |-> ( _I |` B ) ) } ) ) )
23 5 fvexi 
 |-  J e. _V
24 23 dmex 
 |-  dom J e. _V
25 24 mptex 
 |-  ( x e. dom J |-> ( ( J ` x ) X. { .0. } ) ) e. _V
26 21 22 25 fvmpt 
 |-  ( W e. H -> ( ( w e. H |-> ( x e. dom ( ( DIsoA ` K ) ` w ) |-> ( ( ( ( DIsoA ` K ) ` w ) ` x ) X. { ( f e. ( ( LTrn ` K ) ` w ) |-> ( _I |` B ) ) } ) ) ) ` W ) = ( x e. dom J |-> ( ( J ` x ) X. { .0. } ) ) )
27 9 26 sylan9eq 
 |-  ( ( K e. V /\ W e. H ) -> I = ( x e. dom J |-> ( ( J ` x ) X. { .0. } ) ) )