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Theorem erngfset-rN 34286
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 3031 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5825 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 erngset.h-r . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2480 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5825 . . . . . . 7  |-  ( k  =  K  ->  ( TEndo `  k )  =  ( TEndo `  K )
)
65fveq1d 5827 . . . . . 6  |-  ( k  =  K  ->  (
( TEndo `  k ) `  w )  =  ( ( TEndo `  K ) `  w ) )
76opeq2d 4137 . . . . 5  |-  ( k  =  K  ->  <. ( Base `  ndx ) ,  ( ( TEndo `  k
) `  w ) >.  =  <. ( Base `  ndx ) ,  ( ( TEndo `  K ) `  w ) >. )
8 fveq2 5825 . . . . . . . . 9  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
98fveq1d 5827 . . . . . . . 8  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
109mpteq1d 4448 . . . . . . 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 6311 . . . . . 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 4137 . . . . 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 2429 . . . . . . 7  |-  ( k  =  K  ->  (
t  o.  s )  =  ( t  o.  s ) )
146, 6, 13mpt2eq123dv 6311 . . . . . 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 4137 . . . . 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 4037 . . . 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 4445 . . 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 34235 . . 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 5835 . . . . 5  |-  ( LHyp `  K )  e.  _V
203, 19eqeltri 2502 . . . 4  |-  H  e. 
_V
2120mptex 6095 . . 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 5908 . 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 17 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 1437    e. wcel 1872   _Vcvv 3022   {ctp 3945   <.cop 3947    |-> cmpt 4425    o. ccom 4800   ` cfv 5544    |-> cmpt2 6251   ndxcnx 15061   Basecbs 15064   +g cplusg 15133   .rcmulr 15134   LHypclh 33461   LTrncltrn 33578   TEndoctendo 34231   EDRingRcedring-rN 34233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-oprab 6253  df-mpt2 6254  df-edring-rN 34235
This theorem is referenced by:  erngset-rN  34287
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