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Metamath Proof Explorer

Theorem tgrpfset

Description: The translation group maps for a lattice K . (Contributed by NM, 5-Jun-2013)

Ref Expression
Hypothesis tgrpset.h 
|- H = ( LHyp ` K )
Assertion tgrpfset 
|- ( K e. V -> ( TGrp ` K ) = ( w e. H |-> { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. } ) )

Proof

Step Hyp Ref Expression
1 tgrpset.h 
 |-  H = ( LHyp ` K )
2 elex 
 |-  ( K e. V -> K e. _V )
3 fveq2 
 |-  ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
4 3 1 eqtr4di 
 |-  ( k = K -> ( LHyp ` k ) = H )
5 fveq2 
 |-  ( k = K -> ( LTrn ` k ) = ( LTrn ` K ) )
6 5 fveq1d 
 |-  ( k = K -> ( ( LTrn ` k ) ` w ) = ( ( LTrn ` K ) ` w ) )
7 6 opeq2d 
 |-  ( k = K -> <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >. = <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. )
8 eqidd 
 |-  ( k = K -> ( f o. g ) = ( f o. g ) )
9 6 6 8 mpoeq123dv 
 |-  ( k = K -> ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) = ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) )
10 9 opeq2d 
 |-  ( k = K -> <. ( +g ` ndx ) , ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) >. = <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. )
11 7 10 preq12d 
 |-  ( k = K -> { <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) >. } = { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. } )
12 4 11 mpteq12dv 
 |-  ( k = K -> ( w e. ( LHyp ` k ) |-> { <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) >. } ) = ( w e. H |-> { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. } ) )
13 df-tgrp 
 |-  TGrp = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) >. } ) )
14 12 13 1 mptfvmpt 
 |-  ( K e. _V -> ( TGrp ` K ) = ( w e. H |-> { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. } ) )
15 2 14 syl 
 |-  ( K e. V -> ( TGrp ` K ) = ( w e. H |-> { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. } ) )