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Theorem erngfset-rN 34451
Description: The division rings on trace-preserving endomorphisms for a lattice  K. (Contributed by NM, 8-Jun-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
erngset.h-r  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
erngfset-rN  |-  ( K  e.  V  ->  ( EDRingR `  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
)  |->  ( t  o.  s ) ) >. } ) )
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-rN
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2981 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5691 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 erngset.h-r . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2493 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5691 . . . . . . 7  |-  ( k  =  K  ->  ( TEndo `  k )  =  ( TEndo `  K )
)
65fveq1d 5693 . . . . . 6  |-  ( k  =  K  ->  (
( TEndo `  k ) `  w )  =  ( ( TEndo `  K ) `  w ) )
76opeq2d 4066 . . . . 5  |-  ( k  =  K  ->  <. ( Base `  ndx ) ,  ( ( TEndo `  k
) `  w ) >.  =  <. ( Base `  ndx ) ,  ( ( TEndo `  K ) `  w ) >. )
8 fveq2 5691 . . . . . . . . 9  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
98fveq1d 5693 . . . . . . . 8  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
109mpteq1d 4373 . . . . . . 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 6148 . . . . . 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 4066 . . . . 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 2444 . . . . . . 7  |-  ( k  =  K  ->  (
t  o.  s )  =  ( t  o.  s ) )
146, 6, 13mpt2eq123dv 6148 . . . . . 6  |-  ( k  =  K  ->  (
s  e.  ( (
TEndo `  k ) `  w ) ,  t  e.  ( ( TEndo `  k ) `  w
)  |->  ( t  o.  s ) )  =  ( s  e.  ( ( TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w
)  |->  ( t  o.  s ) ) )
1514opeq2d 4066 . . . . 5  |-  ( k  =  K  ->  <. ( .r `  ndx ) ,  ( s  e.  ( ( TEndo `  k ) `  w ) ,  t  e.  ( ( TEndo `  k ) `  w
)  |->  ( t  o.  s ) ) >.  =  <. ( .r `  ndx ) ,  ( s  e.  ( ( TEndo `  K ) `  w
) ,  t  e.  ( ( TEndo `  K
) `  w )  |->  ( t  o.  s
) ) >. )
167, 12, 15tpeq123d 3969 . . . 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
)  |->  ( t  o.  s ) ) >. }  =  { <. ( 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
)  |->  ( t  o.  s ) ) >. } )
174, 16mpteq12dv 4370 . . 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
)  |->  ( t  o.  s ) ) >. } )  =  ( 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
)  |->  ( t  o.  s ) ) >. } ) )
18 df-edring-rN 34400 . . 3  |-  EDRingR  =  ( 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
)  |->  ( t  o.  s ) ) >. } ) )
19 fvex 5701 . . . . 5  |-  ( LHyp `  K )  e.  _V
203, 19eqeltri 2513 . . . 4  |-  H  e. 
_V
2120mptex 5948 . . 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
)  |->  ( t  o.  s ) ) >. } )  e.  _V
2217, 18, 21fvmpt 5774 . 2  |-  ( K  e.  _V  ->  ( EDRingR `  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
)  |->  ( t  o.  s ) ) >. } ) )
231, 22syl 16 1  |-  ( K  e.  V  ->  ( EDRingR `  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
)  |->  ( t  o.  s ) ) >. } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2972   {ctp 3881   <.cop 3883    e. cmpt 4350    o. ccom 4844   ` cfv 5418    e. cmpt2 6093   ndxcnx 14171   Basecbs 14174   +g cplusg 14238   .rcmulr 14239   LHypclh 33628   LTrncltrn 33745   TEndoctendo 34396   EDRingRcedring-rN 34398
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pr 4531
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-oprab 6095  df-mpt2 6096  df-edring-rN 34400
This theorem is referenced by:  erngset-rN  34452
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