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Theorem erngset-rN 34449
Description: The division ring on trace-preserving endomorphisms for a fiducial co-atom  W. (Contributed by NM, 5-Jun-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
erngset.h-r  |-  H  =  ( LHyp `  K
)
erngset.t-r  |-  T  =  ( ( LTrn `  K
) `  W )
erngset.e-r  |-  E  =  ( ( TEndo `  K
) `  W )
erngset.d-r  |-  D  =  ( ( EDRingR `  K ) `  W
)
Assertion
Ref Expression
erngset-rN  |-  ( ( K  e.  V  /\  W  e.  H )  ->  D  =  { <. (
Base `  ndx ) ,  E >. ,  <. ( +g  `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) ) >. ,  <. ( .r `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( t  o.  s ) )
>. } )
Distinct variable groups:    f, s,
t, K    f, W, s, t
Allowed substitution hints:    D( t, f, s)    T( t, f, s)    E( t, f, s)    H( t, f, s)    V( t, f, s)

Proof of Theorem erngset-rN
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 erngset.d-r . . 3  |-  D  =  ( ( EDRingR `  K ) `  W
)
2 erngset.h-r . . . . 5  |-  H  =  ( LHyp `  K
)
32erngfset-rN 34448 . . . 4  |-  ( 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 ) ) >. } ) )
43fveq1d 5691 . . 3  |-  ( K  e.  V  ->  (
( EDRingR `  K
) `  W )  =  ( ( 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 ) ) >. } ) `  W
) )
51, 4syl5eq 2485 . 2  |-  ( K  e.  V  ->  D  =  ( ( 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 ) ) >. } ) `  W
) )
6 fveq2 5689 . . . . . 6  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  ( ( TEndo `  K ) `  W ) )
76opeq2d 4064 . . . . 5  |-  ( w  =  W  ->  <. ( Base `  ndx ) ,  ( ( TEndo `  K
) `  w ) >.  =  <. ( Base `  ndx ) ,  ( ( TEndo `  K ) `  W ) >. )
8 tpeq1 3961 . . . . . 6  |-  ( <.
( Base `  ndx ) ,  ( ( TEndo `  K
) `  w ) >.  =  <. ( Base `  ndx ) ,  ( ( TEndo `  K ) `  W ) >.  ->  { <. (
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 ) ) >. } )
9 erngset.e-r . . . . . . . 8  |-  E  =  ( ( TEndo `  K
) `  W )
109opeq2i 4061 . . . . . . 7  |-  <. ( Base `  ndx ) ,  E >.  =  <. (
Base `  ndx ) ,  ( ( TEndo `  K
) `  W ) >.
11 tpeq1 3961 . . . . . . 7  |-  ( <.
( Base `  ndx ) ,  E >.  =  <. (
Base `  ndx ) ,  ( ( TEndo `  K
) `  W ) >.  ->  { <. ( Base `  ndx ) ,  E >. ,  <. ( +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 ) ) >. } )
1210, 11ax-mp 5 . . . . . 6  |-  { <. (
Base `  ndx ) ,  E >. ,  <. ( +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 ) ) >. }
138, 12syl6eqr 2491 . . . . 5  |-  ( <.
( Base `  ndx ) ,  ( ( TEndo `  K
) `  w ) >.  =  <. ( Base `  ndx ) ,  ( ( TEndo `  K ) `  W ) >.  ->  { <. (
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 ) ,  E >. ,  <. ( +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
) ) >. } )
147, 13syl 16 . . . 4  |-  ( w  =  W  ->  { <. (
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 ) ,  E >. ,  <. ( +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
) ) >. } )
156, 9syl6eqr 2491 . . . . . . 7  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  E )
16 fveq2 5689 . . . . . . . . 9  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
17 erngset.t-r . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
1816, 17syl6eqr 2491 . . . . . . . 8  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
19 eqidd 2442 . . . . . . . 8  |-  ( w  =  W  ->  (
( s `  f
)  o.  ( t `
 f ) )  =  ( ( s `
 f )  o.  ( t `  f
) ) )
2018, 19mpteq12dv 4368 . . . . . . 7  |-  ( w  =  W  ->  (
f  e.  ( (
LTrn `  K ) `  w )  |->  ( ( s `  f )  o.  ( t `  f ) ) )  =  ( f  e.  T  |->  ( ( s `
 f )  o.  ( t `  f
) ) ) )
2115, 15, 20mpt2eq123dv 6146 . . . . . 6  |-  ( w  =  W  ->  (
s  e.  ( (
TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w
)  |->  ( f  e.  ( ( LTrn `  K
) `  w )  |->  ( ( s `  f )  o.  (
t `  f )
) ) )  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) ) )
2221opeq2d 4064 . . . . 5  |-  ( w  =  W  ->  <. ( +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.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `
 f )  o.  ( t `  f
) ) ) )
>. )
2322tpeq2d 3965 . . . 4  |-  ( w  =  W  ->  { <. (
Base `  ndx ) ,  E >. ,  <. ( +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 ) ,  E >. , 
<. ( +g  `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `
 f )  o.  ( t `  f
) ) ) )
>. ,  <. ( .r
`  ndx ) ,  ( s  e.  ( (
TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w
)  |->  ( t  o.  s ) ) >. } )
24 eqidd 2442 . . . . . . 7  |-  ( w  =  W  ->  (
t  o.  s )  =  ( t  o.  s ) )
2515, 15, 24mpt2eq123dv 6146 . . . . . 6  |-  ( w  =  W  ->  (
s  e.  ( (
TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w
)  |->  ( t  o.  s ) )  =  ( s  e.  E ,  t  e.  E  |->  ( t  o.  s
) ) )
2625opeq2d 4064 . . . . 5  |-  ( w  =  W  ->  <. ( .r `  ndx ) ,  ( s  e.  ( ( TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w
)  |->  ( t  o.  s ) ) >.  =  <. ( .r `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( t  o.  s ) )
>. )
2726tpeq3d 3966 . . . 4  |-  ( w  =  W  ->  { <. (
Base `  ndx ) ,  E >. ,  <. ( +g  `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) ) >. ,  <. ( .r `  ndx ) ,  ( s  e.  ( ( TEndo `  K ) `  w
) ,  t  e.  ( ( TEndo `  K
) `  w )  |->  ( t  o.  s
) ) >. }  =  { <. ( Base `  ndx ) ,  E >. , 
<. ( +g  `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `
 f )  o.  ( t `  f
) ) ) )
>. ,  <. ( .r
`  ndx ) ,  ( s  e.  E , 
t  e.  E  |->  ( t  o.  s ) ) >. } )
2814, 23, 273eqtrd 2477 . . 3  |-  ( w  =  W  ->  { <. (
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 ) ,  E >. ,  <. ( +g  `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) ) >. ,  <. ( .r `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( t  o.  s ) )
>. } )
29 eqid 2441 . . 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 ) ) >. } )  =  ( 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 ) ) >. } )
30 tpex 6377 . . 3  |-  { <. (
Base `  ndx ) ,  E >. ,  <. ( +g  `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) ) >. ,  <. ( .r `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( t  o.  s ) )
>. }  e.  _V
3128, 29, 30fvmpt 5772 . 2  |-  ( W  e.  H  ->  (
( 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 ) ) >. } ) `  W
)  =  { <. (
Base `  ndx ) ,  E >. ,  <. ( +g  `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) ) >. ,  <. ( .r `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( t  o.  s ) )
>. } )
325, 31sylan9eq 2493 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  D  =  { <. (
Base `  ndx ) ,  E >. ,  <. ( +g  `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) ) >. ,  <. ( .r `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( t  o.  s ) )
>. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {ctp 3879   <.cop 3881    e. cmpt 4348    o. ccom 4842   ` cfv 5416    e. cmpt2 6091   ndxcnx 14169   Basecbs 14172   +g cplusg 14236   .rcmulr 14237   LHypclh 33625   LTrncltrn 33742   TEndoctendo 34393   EDRingRcedring-rN 34395
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pr 4529  ax-un 6370
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-oprab 6093  df-mpt2 6094  df-edring-rN 34397
This theorem is referenced by:  erngbase-rN  34450  erngfplus-rN  34451  erngfmul-rN  34454
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