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Metamath Proof Explorer

Theorem tgrpopr

Description: The group operation of the translation group is function composition. (Contributed by NM, 5-Jun-2013)

Ref Expression
Hypotheses tgrpset.h 
|- H = ( LHyp ` K )
tgrpset.t 
|- T = ( ( LTrn ` K ) ` W )
tgrpset.g 
|- G = ( ( TGrp ` K ) ` W )
tgrp.o 
|- .+ = ( +g ` G )
Assertion tgrpopr 
|- ( ( K e. V /\ W e. H ) -> .+ = ( f e. T , g e. T |-> ( f o. g ) ) )

Proof

Step Hyp Ref Expression
1 tgrpset.h 
 |-  H = ( LHyp ` K )
2 tgrpset.t 
 |-  T = ( ( LTrn ` K ) ` W )
3 tgrpset.g 
 |-  G = ( ( TGrp ` K ) ` W )
4 tgrp.o 
 |-  .+ = ( +g ` G )
5 1 2 3 tgrpset 
 |-  ( ( K e. V /\ W e. H ) -> G = { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. } )
6 5 fveq2d 
 |-  ( ( K e. V /\ W e. H ) -> ( +g ` G ) = ( +g ` { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. } ) )
7 2 fvexi 
 |-  T e. _V
8 7 7 mpoex 
 |-  ( f e. T , g e. T |-> ( f o. g ) ) e. _V
9 eqid 
 |-  { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. } = { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. }
10 9 grpplusg 
 |-  ( ( f e. T , g e. T |-> ( f o. g ) ) e. _V -> ( f e. T , g e. T |-> ( f o. g ) ) = ( +g ` { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. } ) )
11 8 10 ax-mp 
 |-  ( f e. T , g e. T |-> ( f o. g ) ) = ( +g ` { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. } )
12 6 4 11 3eqtr4g 
 |-  ( ( K e. V /\ W e. H ) -> .+ = ( f e. T , g e. T |-> ( f o. g ) ) )