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Theorem tgrpset 31479
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 31478 . . . 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 5722 . . 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 5720 . . . . . . 7  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
65opeq2d 3983 . . . . . 6  |-  ( w  =  W  ->  <. ( Base `  ndx ) ,  ( ( LTrn `  K
) `  w ) >.  =  <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. )
7 eqidd 2436 . . . . . . . 8  |-  ( w  =  W  ->  (
f  o.  g )  =  ( f  o.  g ) )
85, 5, 7mpt2eq123dv 6128 . . . . . . 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 3983 . . . . . 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 3883 . . . . 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 2435 . . . . 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 4398 . . . . 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 5798 . . . 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 3980 . . . . 5  |-  <. ( Base `  ndx ) ,  T >.  =  <. (
Base `  ndx ) ,  ( ( LTrn `  K
) `  W ) >.
16 eqid 2435 . . . . . . 7  |-  ( f  o.  g )  =  ( f  o.  g
)
1714, 14, 16mpt2eq123i 6129 . . . . . 6  |-  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g ) )  =  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
1817opeq2i 3980 . . . . 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 3880 . . . 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 2485 . . 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 2487 . 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 2479 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 359    = wceq 1652    e. wcel 1725   {cpr 3807   <.cop 3809    e. cmpt 4258    o. ccom 4874   ` cfv 5446    e. cmpt2 6075   ndxcnx 13458   Basecbs 13461   +g cplusg 13521   LHypclh 30718   LTrncltrn 30835   TGrpctgrp 31476
This theorem is referenced by:  tgrpbase  31480  tgrpopr  31481  dvaabl  31759
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-oprab 6077  df-mpt2 6078  df-tgrp 31477
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