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Theorem tgrpset 31616
Description: The translation group for a fiducial co-atom  W. (Contributed by NM, 5-Jun-2013.)
Hypotheses
Ref Expression
tgrpset.h  |-  H  =  ( LHyp `  K
)
tgrpset.t  |-  T  =  ( ( LTrn `  K
) `  W )
tgrpset.g  |-  G  =  ( ( TGrp `  K
) `  W )
Assertion
Ref Expression
tgrpset  |-  ( ( K  e.  V  /\  W  e.  H )  ->  G  =  { <. (
Base `  ndx ) ,  T >. ,  <. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g
) ) >. } )
Distinct variable groups:    f, g, K    T, f, g    f, W, g
Allowed substitution hints:    G( f, g)    H( f, g)    V( f, g)

Proof of Theorem tgrpset
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 tgrpset.g . 2  |-  G  =  ( ( TGrp `  K
) `  W )
2 tgrpset.h . . . . 5  |-  H  =  ( LHyp `  K
)
32tgrpfset 31615 . . . 4  |-  ( 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 ) )
>. } ) )
43fveq1d 5733 . . 3  |-  ( K  e.  V  ->  (
( TGrp `  K ) `  W )  =  ( ( w  e.  H  |->  { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  w ) ,  g  e.  (
( LTrn `  K ) `  w )  |->  ( f  o.  g ) )
>. } ) `  W
) )
5 fveq2 5731 . . . . . . 7  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
65opeq2d 3993 . . . . . 6  |-  ( w  =  W  ->  <. ( Base `  ndx ) ,  ( ( LTrn `  K
) `  w ) >.  =  <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. )
7 eqidd 2439 . . . . . . . 8  |-  ( w  =  W  ->  (
f  o.  g )  =  ( f  o.  g ) )
85, 5, 7mpt2eq123dv 6139 . . . . . . 7  |-  ( w  =  W  ->  (
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 ) ) )
98opeq2d 3993 . . . . . 6  |-  ( w  =  W  ->  <. ( +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 ) ) >.
)
106, 9preq12d 3893 . . . . 5  |-  ( w  =  W  ->  { <. (
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 ) )
>. } )
11 eqid 2438 . . . . 5  |-  ( w  e.  H  |->  { <. (
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 ) )
>. } )
12 prex 4409 . . . . 5  |-  { <. (
Base `  ndx ) ,  ( ( LTrn `  K
) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. }  e.  _V
1310, 11, 12fvmpt 5809 . . . 4  |-  ( W  e.  H  ->  (
( w  e.  H  |->  { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  w ) ,  g  e.  (
( LTrn `  K ) `  w )  |->  ( f  o.  g ) )
>. } ) `  W
)  =  { <. (
Base `  ndx ) ,  ( ( LTrn `  K
) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. } )
14 tgrpset.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
1514opeq2i 3990 . . . . 5  |-  <. ( Base `  ndx ) ,  T >.  =  <. (
Base `  ndx ) ,  ( ( LTrn `  K
) `  W ) >.
16 eqid 2438 . . . . . . 7  |-  ( f  o.  g )  =  ( f  o.  g
)
1714, 14, 16mpt2eq123i 6140 . . . . . 6  |-  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g ) )  =  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
1817opeq2i 3990 . . . . 5  |-  <. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g
) ) >.  =  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>.
1915, 18preq12i 3890 . . . 4  |-  { <. (
Base `  ndx ) ,  T >. ,  <. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g
) ) >. }  =  { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
>. }
2013, 19syl6eqr 2488 . . 3  |-  ( W  e.  H  ->  (
( w  e.  H  |->  { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  w ) ,  g  e.  (
( LTrn `  K ) `  w )  |->  ( f  o.  g ) )
>. } ) `  W
)  =  { <. (
Base `  ndx ) ,  T >. ,  <. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g
) ) >. } )
214, 20sylan9eq 2490 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( ( TGrp `  K
) `  W )  =  { <. ( Base `  ndx ) ,  T >. , 
<. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g ) ) >. } )
221, 21syl5eq 2482 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  G  =  { <. (
Base `  ndx ) ,  T >. ,  <. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g
) ) >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {cpr 3817   <.cop 3819    e. cmpt 4269    o. ccom 4885   ` cfv 5457    e. cmpt2 6086   ndxcnx 13471   Basecbs 13474   +g cplusg 13534   LHypclh 30855   LTrncltrn 30972   TGrpctgrp 31613
This theorem is referenced by:  tgrpbase  31617  tgrpopr  31618  dvaabl  31896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-oprab 6088  df-mpt2 6089  df-tgrp 31614
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