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Theorem erngfset 35470
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 3115 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5857 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 erngset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2519 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5857 . . . . . . 7  |-  ( k  =  K  ->  ( TEndo `  k )  =  ( TEndo `  K )
)
65fveq1d 5859 . . . . . 6  |-  ( k  =  K  ->  (
( TEndo `  k ) `  w )  =  ( ( TEndo `  K ) `  w ) )
76opeq2d 4213 . . . . 5  |-  ( k  =  K  ->  <. ( Base `  ndx ) ,  ( ( TEndo `  k
) `  w ) >.  =  <. ( Base `  ndx ) ,  ( ( TEndo `  K ) `  w ) >. )
8 fveq2 5857 . . . . . . . . 9  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
98fveq1d 5859 . . . . . . . 8  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
109mpteq1d 4521 . . . . . . 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 6334 . . . . . 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 4213 . . . . 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 2461 . . . . . . 7  |-  ( k  =  K  ->  (
s  o.  t )  =  ( s  o.  t ) )
146, 6, 13mpt2eq123dv 6334 . . . . . 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 4213 . . . . 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 4114 . . . 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 4518 . . 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 35428 . . 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 5867 . . . . 5  |-  ( LHyp `  K )  e.  _V
203, 19eqeltri 2544 . . . 4  |-  H  e. 
_V
2120mptex 6122 . . 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 5941 . 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 1374    e. wcel 1762   _Vcvv 3106   {ctp 4024   <.cop 4026    |-> cmpt 4498    o. ccom 4996   ` cfv 5579    |-> cmpt2 6277   ndxcnx 14476   Basecbs 14479   +g cplusg 14544   .rcmulr 14545   LHypclh 34655   LTrncltrn 34772   TEndoctendo 35423   EDRingcedring 35424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-oprab 6279  df-mpt2 6280  df-edring 35428
This theorem is referenced by:  erngset  35471
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