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Theorem tgrpfset 30933
Description: The translation group maps for a lattice  K. (Contributed by NM, 5-Jun-2013.)
Hypothesis
Ref Expression
tgrpset.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
tgrpfset  |-  ( 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 ) )
>. } ) )
Distinct variable groups:    w, H    f, g, w, K
Allowed substitution hints:    H( f, g)    V( w, f, g)

Proof of Theorem tgrpfset
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5525 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 tgrpset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2333 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5525 . . . . . . 7  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
65fveq1d 5527 . . . . . 6  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
76opeq2d 3803 . . . . 5  |-  ( k  =  K  ->  <. ( Base `  ndx ) ,  ( ( LTrn `  k
) `  w ) >.  =  <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  w ) >. )
8 eqidd 2284 . . . . . . 7  |-  ( k  =  K  ->  (
f  o.  g )  =  ( f  o.  g ) )
96, 6, 8mpt2eq123dv 5910 . . . . . 6  |-  ( k  =  K  ->  (
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 ) ) )
109opeq2d 3803 . . . . 5  |-  ( k  =  K  ->  <. ( +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 ) ) >.
)
117, 10preq12d 3714 . . . 4  |-  ( k  =  K  ->  { <. (
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 ) )
>. } )
124, 11mpteq12dv 4098 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  { <. (
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 ) )
>. } ) )
13 df-tgrp 30932 . . 3  |-  TGrp  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  {
<. ( Base `  ndx ) ,  ( ( LTrn `  k ) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  k
) `  w ) ,  g  e.  (
( LTrn `  k ) `  w )  |->  ( f  o.  g ) )
>. } ) )
14 fvex 5539 . . . . 5  |-  ( LHyp `  K )  e.  _V
153, 14eqeltri 2353 . . . 4  |-  H  e. 
_V
1615mptex 5746 . . 3  |-  ( w  e.  H  |->  { <. (
Base `  ndx ) ,  ( ( LTrn `  K
) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  w ) ,  g  e.  (
( LTrn `  K ) `  w )  |->  ( f  o.  g ) )
>. } )  e.  _V
1712, 13, 16fvmpt 5602 . 2  |-  ( 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 ) )
>. } ) )
181, 17syl 15 1  |-  ( 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 ) )
>. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   {cpr 3641   <.cop 3643    e. cmpt 4077    o. ccom 4693   ` cfv 5255    e. cmpt2 5860   ndxcnx 13145   Basecbs 13148   +g cplusg 13208   LHypclh 30173   LTrncltrn 30290   TGrpctgrp 30931
This theorem is referenced by:  tgrpset  30934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-tgrp 30932
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