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Theorem tgrpfset 31380
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 2956 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5719 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 tgrpset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2485 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5719 . . . . . . 7  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
65fveq1d 5721 . . . . . 6  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
76opeq2d 3983 . . . . 5  |-  ( k  =  K  ->  <. ( Base `  ndx ) ,  ( ( LTrn `  k
) `  w ) >.  =  <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  w ) >. )
8 eqidd 2436 . . . . . . 7  |-  ( k  =  K  ->  (
f  o.  g )  =  ( f  o.  g ) )
96, 6, 8mpt2eq123dv 6127 . . . . . 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 3983 . . . . 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 3883 . . . 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 4279 . . 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 31379 . . 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 5733 . . . . 5  |-  ( LHyp `  K )  e.  _V
153, 14eqeltri 2505 . . . 4  |-  H  e. 
_V
1615mptex 5957 . . 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 5797 . 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 16 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 1652    e. wcel 1725   _Vcvv 2948   {cpr 3807   <.cop 3809    e. cmpt 4258    o. ccom 4873   ` cfv 5445    e. cmpt2 6074   ndxcnx 13454   Basecbs 13457   +g cplusg 13517   LHypclh 30620   LTrncltrn 30737   TGrpctgrp 31378
This theorem is referenced by:  tgrpset  31381
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 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-oprab 6076  df-mpt2 6077  df-tgrp 31379
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