Metamath Proof Explorer

Theorem tgrpabl

Description: The translation group is an Abelian group. Lemma G of Crawley p. 116. (Contributed by NM, 6-Jun-2013)

Ref Expression
Hypotheses tgrpgrp.h 
|- H = ( LHyp ` K )
tgrpgrp.g 
|- G = ( ( TGrp ` K ) ` W )
Assertion tgrpabl 
|- ( ( K e. HL /\ W e. H ) -> G e. Abel )

Proof

Step Hyp Ref Expression
1 tgrpgrp.h 
 |-  H = ( LHyp ` K )
2 tgrpgrp.g 
 |-  G = ( ( TGrp ` K ) ` W )
3 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
4 eqid 
 |-  ( Base ` G ) = ( Base ` G )
5 1 3 2 4 tgrpbase 
 |-  ( ( K e. HL /\ W e. H ) -> ( Base ` G ) = ( ( LTrn ` K ) ` W ) )
6 5 eqcomd 
 |-  ( ( K e. HL /\ W e. H ) -> ( ( LTrn ` K ) ` W ) = ( Base ` G ) )
7 eqidd 
 |-  ( ( K e. HL /\ W e. H ) -> ( +g ` G ) = ( +g ` G ) )
8 1 2 tgrpgrp 
 |-  ( ( K e. HL /\ W e. H ) -> G e. Grp )
9 1 3 ltrncom 
 |-  ( ( ( K e. HL /\ W e. H ) /\ f e. ( ( LTrn ` K ) ` W ) /\ g e. ( ( LTrn ` K ) ` W ) ) -> ( f o. g ) = ( g o. f ) )
10 eqid 
 |-  ( +g ` G ) = ( +g ` G )
11 1 3 2 10 tgrpov 
 |-  ( ( K e. HL /\ W e. H /\ ( f e. ( ( LTrn ` K ) ` W ) /\ g e. ( ( LTrn ` K ) ` W ) ) ) -> ( f ( +g ` G ) g ) = ( f o. g ) )
12 11 3expa 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( f e. ( ( LTrn ` K ) ` W ) /\ g e. ( ( LTrn ` K ) ` W ) ) ) -> ( f ( +g ` G ) g ) = ( f o. g ) )
13 12 3impb 
 |-  ( ( ( K e. HL /\ W e. H ) /\ f e. ( ( LTrn ` K ) ` W ) /\ g e. ( ( LTrn ` K ) ` W ) ) -> ( f ( +g ` G ) g ) = ( f o. g ) )
14 1 3 2 10 tgrpov 
 |-  ( ( K e. HL /\ W e. H /\ ( g e. ( ( LTrn ` K ) ` W ) /\ f e. ( ( LTrn ` K ) ` W ) ) ) -> ( g ( +g ` G ) f ) = ( g o. f ) )
15 14 3expa 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( g e. ( ( LTrn ` K ) ` W ) /\ f e. ( ( LTrn ` K ) ` W ) ) ) -> ( g ( +g ` G ) f ) = ( g o. f ) )
16 15 3impb 
 |-  ( ( ( K e. HL /\ W e. H ) /\ g e. ( ( LTrn ` K ) ` W ) /\ f e. ( ( LTrn ` K ) ` W ) ) -> ( g ( +g ` G ) f ) = ( g o. f ) )
17 16 3com23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ f e. ( ( LTrn ` K ) ` W ) /\ g e. ( ( LTrn ` K ) ` W ) ) -> ( g ( +g ` G ) f ) = ( g o. f ) )
18 9 13 17 3eqtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ f e. ( ( LTrn ` K ) ` W ) /\ g e. ( ( LTrn ` K ) ` W ) ) -> ( f ( +g ` G ) g ) = ( g ( +g ` G ) f ) )
19 6 7 8 18 isabld 
 |-  ( ( K e. HL /\ W e. H ) -> G e. Abel )