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Metamath Proof Explorer

Theorem tgrpbase

Description: The base set of the translation group is the set of all translations (for a fiducial co-atom W ). (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.c 
|- C = ( Base ` G )
Assertion tgrpbase 
|- ( ( K e. V /\ W e. H ) -> C = T )

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.c 
 |-  C = ( Base ` 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 ) -> ( Base ` G ) = ( Base ` { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. } ) )
7 2 fvexi 
 |-  T e. _V
8 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 ) ) >. }
9 8 grpbase 
 |-  ( T e. _V -> T = ( Base ` { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. } ) )
10 7 9 ax-mp 
 |-  T = ( Base ` { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. } )
11 6 4 10 3eqtr4g 
 |-  ( ( K e. V /\ W e. H ) -> C = T )