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Metamath Proof Explorer

Theorem diafval

Description: The partial isomorphism A for a lattice K . (Contributed by NM, 15-Oct-2013)

Ref Expression
Hypotheses diaval.b 
|- B = ( Base ` K )
diaval.l 
|- .<_ = ( le ` K )
diaval.h 
|- H = ( LHyp ` K )
diaval.t 
|- T = ( ( LTrn ` K ) ` W )
diaval.r 
|- R = ( ( trL ` K ) ` W )
diaval.i 
|- I = ( ( DIsoA ` K ) ` W )
Assertion diafval 
|- ( ( K e. V /\ W e. H ) -> I = ( x e. { y e. B | y .<_ W } |-> { f e. T | ( R ` f ) .<_ x } ) )

Proof

Step Hyp Ref Expression
1 diaval.b 
 |-  B = ( Base ` K )
2 diaval.l 
 |-  .<_ = ( le ` K )
3 diaval.h 
 |-  H = ( LHyp ` K )
4 diaval.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 diaval.r 
 |-  R = ( ( trL ` K ) ` W )
6 diaval.i 
 |-  I = ( ( DIsoA ` K ) ` W )
7 1 2 3 diaffval 
 |-  ( K e. V -> ( DIsoA ` K ) = ( w e. H |-> ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) ) )
8 7 fveq1d 
 |-  ( K e. V -> ( ( DIsoA ` K ) ` W ) = ( ( w e. H |-> ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) ) ` W ) )
9 6 8 eqtrid 
 |-  ( K e. V -> I = ( ( w e. H |-> ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) ) ` W ) )
10 breq2 
 |-  ( w = W -> ( y .<_ w <-> y .<_ W ) )
11 10 rabbidv 
 |-  ( w = W -> { y e. B | y .<_ w } = { y e. B | y .<_ W } )
12 fveq2 
 |-  ( w = W -> ( ( LTrn ` K ) ` w ) = ( ( LTrn ` K ) ` W ) )
13 12 4 eqtr4di 
 |-  ( w = W -> ( ( LTrn ` K ) ` w ) = T )
14 fveq2 
 |-  ( w = W -> ( ( trL ` K ) ` w ) = ( ( trL ` K ) ` W ) )
15 14 5 eqtr4di 
 |-  ( w = W -> ( ( trL ` K ) ` w ) = R )
16 15 fveq1d 
 |-  ( w = W -> ( ( ( trL ` K ) ` w ) ` f ) = ( R ` f ) )
17 16 breq1d 
 |-  ( w = W -> ( ( ( ( trL ` K ) ` w ) ` f ) .<_ x <-> ( R ` f ) .<_ x ) )
18 13 17 rabeqbidv 
 |-  ( w = W -> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } = { f e. T | ( R ` f ) .<_ x } )
19 11 18 mpteq12dv 
 |-  ( w = W -> ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) = ( x e. { y e. B | y .<_ W } |-> { f e. T | ( R ` f ) .<_ x } ) )
20 eqid 
 |-  ( w e. H |-> ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) ) = ( w e. H |-> ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) )
21 1 fvexi 
 |-  B e. _V
22 21 mptrabex 
 |-  ( x e. { y e. B | y .<_ W } |-> { f e. T | ( R ` f ) .<_ x } ) e. _V
23 19 20 22 fvmpt 
 |-  ( W e. H -> ( ( w e. H |-> ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) ) ` W ) = ( x e. { y e. B | y .<_ W } |-> { f e. T | ( R ` f ) .<_ x } ) )
24 9 23 sylan9eq 
 |-  ( ( K e. V /\ W e. H ) -> I = ( x e. { y e. B | y .<_ W } |-> { f e. T | ( R ` f ) .<_ x } ) )