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Theorem erngfset 34336
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 3089 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5882 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 erngset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2481 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5882 . . . . . . 7  |-  ( k  =  K  ->  ( TEndo `  k )  =  ( TEndo `  K )
)
65fveq1d 5884 . . . . . 6  |-  ( k  =  K  ->  (
( TEndo `  k ) `  w )  =  ( ( TEndo `  K ) `  w ) )
76opeq2d 4194 . . . . 5  |-  ( k  =  K  ->  <. ( Base `  ndx ) ,  ( ( TEndo `  k
) `  w ) >.  =  <. ( Base `  ndx ) ,  ( ( TEndo `  K ) `  w ) >. )
8 fveq2 5882 . . . . . . . . 9  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
98fveq1d 5884 . . . . . . . 8  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
109mpteq1d 4505 . . . . . . 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 6368 . . . . . 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 4194 . . . . 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 2423 . . . . . . 7  |-  ( k  =  K  ->  (
s  o.  t )  =  ( s  o.  t ) )
146, 6, 13mpt2eq123dv 6368 . . . . . 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 4194 . . . . 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 4094 . . . 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 4502 . . 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 34294 . . 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 5892 . . . . 5  |-  ( LHyp `  K )  e.  _V
203, 19eqeltri 2503 . . . 4  |-  H  e. 
_V
2120mptex 6152 . . 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 5965 . 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 17 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 1437    e. wcel 1872   _Vcvv 3080   {ctp 4002   <.cop 4004    |-> cmpt 4482    o. ccom 4857   ` cfv 5601    |-> cmpt2 6308   ndxcnx 15118   Basecbs 15121   +g cplusg 15190   .rcmulr 15191   LHypclh 33519   LTrncltrn 33636   TEndoctendo 34289   EDRingcedring 34290
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 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pr 4660
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 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-oprab 6310  df-mpt2 6311  df-edring 34294
This theorem is referenced by:  erngset  34337
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