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Theorem erngfset 34283
Description: The division rings on trace-preserving endomorphisms for a lattice  K. (Contributed by NM, 8-Jun-2013.)
Hypothesis
Ref Expression
erngset.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
erngfset  |-  ( K  e.  V  ->  ( EDRing `
 K )  =  ( w  e.  H  |->  { <. ( Base `  ndx ) ,  ( ( TEndo `  K ) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  t  e.  (
( TEndo `  K ) `  w )  |->  ( f  e.  ( ( LTrn `  K ) `  w
)  |->  ( ( s `
 f )  o.  ( t `  f
) ) ) )
>. ,  <. ( .r
`  ndx ) ,  ( s  e.  ( (
TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w
)  |->  ( s  o.  t ) ) >. } ) )
Distinct variable groups:    w, H    f, s, t, w, K
Allowed substitution hints:    H( t, f, s)    V( w, t, f, s)

Proof of Theorem erngfset
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2976 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5686 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 erngset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2488 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5686 . . . . . . 7  |-  ( k  =  K  ->  ( TEndo `  k )  =  ( TEndo `  K )
)
65fveq1d 5688 . . . . . 6  |-  ( k  =  K  ->  (
( TEndo `  k ) `  w )  =  ( ( TEndo `  K ) `  w ) )
76opeq2d 4061 . . . . 5  |-  ( k  =  K  ->  <. ( Base `  ndx ) ,  ( ( TEndo `  k
) `  w ) >.  =  <. ( Base `  ndx ) ,  ( ( TEndo `  K ) `  w ) >. )
8 fveq2 5686 . . . . . . . . 9  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
98fveq1d 5688 . . . . . . . 8  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
109mpteq1d 4368 . . . . . . 7  |-  ( k  =  K  ->  (
f  e.  ( (
LTrn `  k ) `  w )  |->  ( ( s `  f )  o.  ( t `  f ) ) )  =  ( f  e.  ( ( LTrn `  K
) `  w )  |->  ( ( s `  f )  o.  (
t `  f )
) ) )
116, 6, 10mpt2eq123dv 6143 . . . . . 6  |-  ( k  =  K  ->  (
s  e.  ( (
TEndo `  k ) `  w ) ,  t  e.  ( ( TEndo `  k ) `  w
)  |->  ( f  e.  ( ( LTrn `  k
) `  w )  |->  ( ( s `  f )  o.  (
t `  f )
) ) )  =  ( s  e.  ( ( TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w
)  |->  ( f  e.  ( ( LTrn `  K
) `  w )  |->  ( ( s `  f )  o.  (
t `  f )
) ) ) )
1211opeq2d 4061 . . . . 5  |-  ( k  =  K  ->  <. ( +g  `  ndx ) ,  ( s  e.  ( ( TEndo `  k ) `  w ) ,  t  e.  ( ( TEndo `  k ) `  w
)  |->  ( f  e.  ( ( LTrn `  k
) `  w )  |->  ( ( s `  f )  o.  (
t `  f )
) ) ) >.  =  <. ( +g  `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  t  e.  (
( TEndo `  K ) `  w )  |->  ( f  e.  ( ( LTrn `  K ) `  w
)  |->  ( ( s `
 f )  o.  ( t `  f
) ) ) )
>. )
13 eqidd 2439 . . . . . . 7  |-  ( k  =  K  ->  (
s  o.  t )  =  ( s  o.  t ) )
146, 6, 13mpt2eq123dv 6143 . . . . . 6  |-  ( k  =  K  ->  (
s  e.  ( (
TEndo `  k ) `  w ) ,  t  e.  ( ( TEndo `  k ) `  w
)  |->  ( s  o.  t ) )  =  ( s  e.  ( ( TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w
)  |->  ( s  o.  t ) ) )
1514opeq2d 4061 . . . . 5  |-  ( k  =  K  ->  <. ( .r `  ndx ) ,  ( s  e.  ( ( TEndo `  k ) `  w ) ,  t  e.  ( ( TEndo `  k ) `  w
)  |->  ( s  o.  t ) ) >.  =  <. ( .r `  ndx ) ,  ( s  e.  ( ( TEndo `  K ) `  w
) ,  t  e.  ( ( TEndo `  K
) `  w )  |->  ( s  o.  t
) ) >. )
167, 12, 15tpeq123d 3964 . . . 4  |-  ( k  =  K  ->  { <. (
Base `  ndx ) ,  ( ( TEndo `  k
) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( s  e.  ( ( TEndo `  k
) `  w ) ,  t  e.  (
( TEndo `  k ) `  w )  |->  ( f  e.  ( ( LTrn `  k ) `  w
)  |->  ( ( s `
 f )  o.  ( t `  f
) ) ) )
>. ,  <. ( .r
`  ndx ) ,  ( s  e.  ( (
TEndo `  k ) `  w ) ,  t  e.  ( ( TEndo `  k ) `  w
)  |->  ( s  o.  t ) ) >. }  =  { <. ( Base `  ndx ) ,  ( ( TEndo `  K
) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  t  e.  (
( TEndo `  K ) `  w )  |->  ( f  e.  ( ( LTrn `  K ) `  w
)  |->  ( ( s `
 f )  o.  ( t `  f
) ) ) )
>. ,  <. ( .r
`  ndx ) ,  ( s  e.  ( (
TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w
)  |->  ( s  o.  t ) ) >. } )
174, 16mpteq12dv 4365 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  { <. (
Base `  ndx ) ,  ( ( TEndo `  k
) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( s  e.  ( ( TEndo `  k
) `  w ) ,  t  e.  (
( TEndo `  k ) `  w )  |->  ( f  e.  ( ( LTrn `  k ) `  w
)  |->  ( ( s `
 f )  o.  ( t `  f
) ) ) )
>. ,  <. ( .r
`  ndx ) ,  ( s  e.  ( (
TEndo `  k ) `  w ) ,  t  e.  ( ( TEndo `  k ) `  w
)  |->  ( s  o.  t ) ) >. } )  =  ( w  e.  H  |->  {
<. ( Base `  ndx ) ,  ( ( TEndo `  K ) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  t  e.  (
( TEndo `  K ) `  w )  |->  ( f  e.  ( ( LTrn `  K ) `  w
)  |->  ( ( s `
 f )  o.  ( t `  f
) ) ) )
>. ,  <. ( .r
`  ndx ) ,  ( s  e.  ( (
TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w
)  |->  ( s  o.  t ) ) >. } ) )
18 df-edring 34241 . . 3  |-  EDRing  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  { <. (
Base `  ndx ) ,  ( ( TEndo `  k
) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( s  e.  ( ( TEndo `  k
) `  w ) ,  t  e.  (
( TEndo `  k ) `  w )  |->  ( f  e.  ( ( LTrn `  k ) `  w
)  |->  ( ( s `
 f )  o.  ( t `  f
) ) ) )
>. ,  <. ( .r
`  ndx ) ,  ( s  e.  ( (
TEndo `  k ) `  w ) ,  t  e.  ( ( TEndo `  k ) `  w
)  |->  ( s  o.  t ) ) >. } ) )
19 fvex 5696 . . . . 5  |-  ( LHyp `  K )  e.  _V
203, 19eqeltri 2508 . . . 4  |-  H  e. 
_V
2120mptex 5943 . . 3  |-  ( w  e.  H  |->  { <. (
Base `  ndx ) ,  ( ( TEndo `  K
) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  t  e.  (
( TEndo `  K ) `  w )  |->  ( f  e.  ( ( LTrn `  K ) `  w
)  |->  ( ( s `
 f )  o.  ( t `  f
) ) ) )
>. ,  <. ( .r
`  ndx ) ,  ( s  e.  ( (
TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w
)  |->  ( s  o.  t ) ) >. } )  e.  _V
2217, 18, 21fvmpt 5769 . 2  |-  ( K  e.  _V  ->  ( EDRing `
 K )  =  ( w  e.  H  |->  { <. ( Base `  ndx ) ,  ( ( TEndo `  K ) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  t  e.  (
( TEndo `  K ) `  w )  |->  ( f  e.  ( ( LTrn `  K ) `  w
)  |->  ( ( s `
 f )  o.  ( t `  f
) ) ) )
>. ,  <. ( .r
`  ndx ) ,  ( s  e.  ( (
TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w
)  |->  ( s  o.  t ) ) >. } ) )
231, 22syl 16 1  |-  ( K  e.  V  ->  ( EDRing `
 K )  =  ( w  e.  H  |->  { <. ( Base `  ndx ) ,  ( ( TEndo `  K ) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  t  e.  (
( TEndo `  K ) `  w )  |->  ( f  e.  ( ( LTrn `  K ) `  w
)  |->  ( ( s `
 f )  o.  ( t `  f
) ) ) )
>. ,  <. ( .r
`  ndx ) ,  ( s  e.  ( (
TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w
)  |->  ( s  o.  t ) ) >. } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2967   {ctp 3876   <.cop 3878    e. cmpt 4345    o. ccom 4839   ` cfv 5413    e. cmpt2 6088   ndxcnx 14163   Basecbs 14166   +g cplusg 14230   .rcmulr 14231   LHypclh 33468   LTrncltrn 33585   TEndoctendo 34236   EDRingcedring 34237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-oprab 6090  df-mpt2 6091  df-edring 34241
This theorem is referenced by:  erngset  34284
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