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Theorem erngset 30989
Description: The division ring on trace-preserving endomorphisms for a fiducial co-atom  W. (Contributed by NM, 5-Jun-2013.)
Hypotheses
Ref Expression
erngset.h  |-  H  =  ( LHyp `  K
)
erngset.t  |-  T  =  ( ( LTrn `  K
) `  W )
erngset.e  |-  E  =  ( ( TEndo `  K
) `  W )
erngset.d  |-  D  =  ( ( EDRing `  K
) `  W )
Assertion
Ref Expression
erngset  |-  ( ( 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  |->  ( s  o.  t ) )
>. } )
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
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 erngset.d . . 3  |-  D  =  ( ( EDRing `  K
) `  W )
2 erngset.h . . . . 5  |-  H  =  ( LHyp `  K
)
32erngfset 30988 . . . 4  |-  ( 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 ) ) >. } ) )
43fveq1d 5527 . . 3  |-  ( K  e.  V  ->  (
( EDRing `  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
)  |->  ( s  o.  t ) ) >. } ) `  W
) )
51, 4syl5eq 2327 . 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
)  |->  ( s  o.  t ) ) >. } ) `  W
) )
6 fveq2 5525 . . . . . 6  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  ( ( TEndo `  K ) `  W ) )
76opeq2d 3803 . . . . 5  |-  ( w  =  W  ->  <. ( Base `  ndx ) ,  ( ( TEndo `  K
) `  w ) >.  =  <. ( Base `  ndx ) ,  ( ( TEndo `  K ) `  W ) >. )
8 tpeq1 3715 . . . . . 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
)  |->  ( 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 ) ) >. } )
9 erngset.e . . . . . . . 8  |-  E  =  ( ( TEndo `  K
) `  W )
109opeq2i 3800 . . . . . . 7  |-  <. ( Base `  ndx ) ,  E >.  =  <. (
Base `  ndx ) ,  ( ( TEndo `  K
) `  W ) >.
11 tpeq1 3715 . . . . . . 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 )  |->  ( 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 ) ) >. } )
1210, 11ax-mp 8 . . . . . 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 )  |->  ( 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 ) ) >. }
138, 12syl6eqr 2333 . . . . 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
)  |->  ( s  o.  t ) ) >. }  =  { <. ( 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 )  |->  ( s  o.  t
) ) >. } )
147, 13syl 15 . . . 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
)  |->  ( s  o.  t ) ) >. }  =  { <. ( 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 )  |->  ( s  o.  t
) ) >. } )
156, 9syl6eqr 2333 . . . . . . 7  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  E )
16 fveq2 5525 . . . . . . . . 9  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
17 erngset.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
1816, 17syl6eqr 2333 . . . . . . . 8  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
19 eqidd 2284 . . . . . . . 8  |-  ( w  =  W  ->  (
( s `  f
)  o.  ( t `
 f ) )  =  ( ( s `
 f )  o.  ( t `  f
) ) )
2018, 19mpteq12dv 4098 . . . . . . 7  |-  ( w  =  W  ->  (
f  e.  ( (
LTrn `  K ) `  w )  |->  ( ( s `  f )  o.  ( t `  f ) ) )  =  ( f  e.  T  |->  ( ( s `
 f )  o.  ( t `  f
) ) ) )
2115, 15, 20mpt2eq123dv 5910 . . . . . 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 3803 . . . . 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 3719 . . . 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 )  |->  ( s  o.  t
) ) >. }  =  { <. ( 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
)  |->  ( s  o.  t ) ) >. } )
24 eqidd 2284 . . . . . . 7  |-  ( w  =  W  ->  (
s  o.  t )  =  ( s  o.  t ) )
2515, 15, 24mpt2eq123dv 5910 . . . . . 6  |-  ( w  =  W  ->  (
s  e.  ( (
TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w
)  |->  ( s  o.  t ) )  =  ( s  e.  E ,  t  e.  E  |->  ( s  o.  t
) ) )
2625opeq2d 3803 . . . . 5  |-  ( w  =  W  ->  <. ( .r `  ndx ) ,  ( s  e.  ( ( TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w
)  |->  ( s  o.  t ) ) >.  =  <. ( .r `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( s  o.  t ) )
>. )
2726tpeq3d 3720 . . . 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 )  |->  ( s  o.  t
) ) >. }  =  { <. ( 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  |->  ( s  o.  t ) ) >. } )
2814, 23, 273eqtrd 2319 . . 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
)  |->  ( s  o.  t ) ) >. }  =  { <. ( 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  |->  ( s  o.  t ) )
>. } )
29 eqid 2283 . . 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 ) ) >. } )  =  ( 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 ) ) >. } )
30 tpex 4519 . . 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  |->  ( s  o.  t ) )
>. }  e.  _V
3128, 29, 30fvmpt 5602 . 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
)  |->  ( s  o.  t ) ) >. } ) `  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  |->  ( s  o.  t ) )
>. } )
325, 31sylan9eq 2335 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  |->  ( s  o.  t ) )
>. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {ctp 3642   <.cop 3643    e. cmpt 4077    o. ccom 4693   ` cfv 5255    e. cmpt2 5860   ndxcnx 13145   Basecbs 13148   +g cplusg 13208   .rcmulr 13209   LHypclh 30173   LTrncltrn 30290   TEndoctendo 30941   EDRingcedring 30942
This theorem is referenced by:  erngbase  30990  erngfplus  30991  erngfmul  30994
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-13 1686  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  ax-un 4512
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-tp 3648  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-edring 30946
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