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Theorem tgrpset 30861
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 30860 . . . 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 5672 . . 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 5670 . . . . . . 7  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
65opeq2d 3935 . . . . . 6  |-  ( w  =  W  ->  <. ( Base `  ndx ) ,  ( ( LTrn `  K
) `  w ) >.  =  <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  W ) >. )
7 eqidd 2390 . . . . . . . 8  |-  ( w  =  W  ->  (
f  o.  g )  =  ( f  o.  g ) )
85, 5, 7mpt2eq123dv 6077 . . . . . . 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 3935 . . . . . 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 3836 . . . . 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 2389 . . . . 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 4349 . . . . 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 5747 . . . 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 3932 . . . . 5  |-  <. ( Base `  ndx ) ,  T >.  =  <. (
Base `  ndx ) ,  ( ( LTrn `  K
) `  W ) >.
16 eqid 2389 . . . . . . 7  |-  ( f  o.  g )  =  ( f  o.  g
)
1714, 14, 16mpt2eq123i 6078 . . . . . 6  |-  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g ) )  =  ( f  e.  ( ( LTrn `  K
) `  W ) ,  g  e.  (
( LTrn `  K ) `  W )  |->  ( f  o.  g ) )
1817opeq2i 3932 . . . . 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 3833 . . . 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 2439 . . 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 2441 . 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 2433 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 1649    e. wcel 1717   {cpr 3760   <.cop 3762    e. cmpt 4209    o. ccom 4824   ` cfv 5396    e. cmpt2 6024   ndxcnx 13395   Basecbs 13398   +g cplusg 13458   LHypclh 30100   LTrncltrn 30217   TGrpctgrp 30858
This theorem is referenced by:  tgrpbase  30862  tgrpopr  30863  dvaabl  31141
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-oprab 6026  df-mpt2 6027  df-tgrp 30859
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