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Theorem mendval 27153
Description: Value of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
Hypotheses
Ref Expression
mendval.b  |-  B  =  ( M LMHom  M )
mendval.p  |-  .+  =  ( x  e.  B ,  y  e.  B  |->  ( x  o F ( +g  `  M
) y ) )
mendval.t  |-  .X.  =  ( x  e.  B ,  y  e.  B  |->  ( x  o.  y
) )
mendval.s  |-  S  =  (Scalar `  M )
mendval.v  |-  .x.  =  ( x  e.  ( Base `  S ) ,  y  e.  B  |->  ( ( ( Base `  M
)  X.  { x } )  o F ( .s `  M
) y ) )
Assertion
Ref Expression
mendval  |-  ( M  e.  X  ->  (MEndo `  M )  =  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } ) )
Distinct variable groups:    x, y, B    x, M, y
Allowed substitution hints:    .+ ( x, y)    S( x, y)    .x. ( x, y)    .X. ( x, y)    X( x, y)

Proof of Theorem mendval
Dummy variables  m  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2900 . 2  |-  ( M  e.  X  ->  M  e.  _V )
2 oveq12 6022 . . . . . . 7  |-  ( ( m  =  M  /\  m  =  M )  ->  ( m LMHom  m )  =  ( M LMHom  M
) )
32anidms 627 . . . . . 6  |-  ( m  =  M  ->  (
m LMHom  m )  =  ( M LMHom  M ) )
4 mendval.b . . . . . 6  |-  B  =  ( M LMHom  M )
53, 4syl6eqr 2430 . . . . 5  |-  ( m  =  M  ->  (
m LMHom  m )  =  B )
65csbeq1d 3193 . . . 4  |-  ( m  =  M  ->  [_ (
m LMHom  m )  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  ( x  e.  b ,  y  e.  b 
|->  ( x  o F ( +g  `  m
) y ) )
>. ,  <. ( .r
`  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o.  y ) ) >. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  m ) >. ,  <. ( .s `  ndx ) ,  ( x  e.  ( Base `  (Scalar `  m ) ) ,  y  e.  b  |->  ( ( ( Base `  m
)  X.  { x } )  o F ( .s `  m
) y ) )
>. } )  =  [_ B  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o F ( +g  `  m
) y ) )
>. ,  <. ( .r
`  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o.  y ) ) >. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  m ) >. ,  <. ( .s `  ndx ) ,  ( x  e.  ( Base `  (Scalar `  m ) ) ,  y  e.  b  |->  ( ( ( Base `  m
)  X.  { x } )  o F ( .s `  m
) y ) )
>. } ) )
7 ovex 6038 . . . . . 6  |-  ( m LMHom 
m )  e.  _V
85, 7syl6eqelr 2469 . . . . 5  |-  ( m  =  M  ->  B  e.  _V )
9 simpr 448 . . . . . . . 8  |-  ( ( m  =  M  /\  b  =  B )  ->  b  =  B )
109opeq2d 3926 . . . . . . 7  |-  ( ( m  =  M  /\  b  =  B )  -> 
<. ( Base `  ndx ) ,  b >.  = 
<. ( Base `  ndx ) ,  B >. )
11 fveq2 5661 . . . . . . . . . . . 12  |-  ( m  =  M  ->  ( +g  `  m )  =  ( +g  `  M
) )
12 ofeq 6239 . . . . . . . . . . . 12  |-  ( ( +g  `  m )  =  ( +g  `  M
)  ->  o F
( +g  `  m )  =  o F ( +g  `  M ) )
1311, 12syl 16 . . . . . . . . . . 11  |-  ( m  =  M  ->  o F ( +g  `  m
)  =  o F ( +g  `  M
) )
1413proplem3 13836 . . . . . . . . . 10  |-  ( ( m  =  M  /\  b  =  B )  ->  ( x  o F ( +g  `  m
) y )  =  ( x  o F ( +g  `  M
) y ) )
159, 9, 14mpt2eq123dv 6068 . . . . . . . . 9  |-  ( ( m  =  M  /\  b  =  B )  ->  ( x  e.  b ,  y  e.  b 
|->  ( x  o F ( +g  `  m
) y ) )  =  ( x  e.  B ,  y  e.  B  |->  ( x  o F ( +g  `  M
) y ) ) )
16 mendval.p . . . . . . . . 9  |-  .+  =  ( x  e.  B ,  y  e.  B  |->  ( x  o F ( +g  `  M
) y ) )
1715, 16syl6eqr 2430 . . . . . . . 8  |-  ( ( m  =  M  /\  b  =  B )  ->  ( x  e.  b ,  y  e.  b 
|->  ( x  o F ( +g  `  m
) y ) )  =  .+  )
1817opeq2d 3926 . . . . . . 7  |-  ( ( m  =  M  /\  b  =  B )  -> 
<. ( +g  `  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o F ( +g  `  m
) y ) )
>.  =  <. ( +g  ` 
ndx ) ,  .+  >.
)
19 eqidd 2381 . . . . . . . . . 10  |-  ( ( m  =  M  /\  b  =  B )  ->  ( x  o.  y
)  =  ( x  o.  y ) )
209, 9, 19mpt2eq123dv 6068 . . . . . . . . 9  |-  ( ( m  =  M  /\  b  =  B )  ->  ( x  e.  b ,  y  e.  b 
|->  ( x  o.  y
) )  =  ( x  e.  B , 
y  e.  B  |->  ( x  o.  y ) ) )
21 mendval.t . . . . . . . . 9  |-  .X.  =  ( x  e.  B ,  y  e.  B  |->  ( x  o.  y
) )
2220, 21syl6eqr 2430 . . . . . . . 8  |-  ( ( m  =  M  /\  b  =  B )  ->  ( x  e.  b ,  y  e.  b 
|->  ( x  o.  y
) )  =  .X.  )
2322opeq2d 3926 . . . . . . 7  |-  ( ( m  =  M  /\  b  =  B )  -> 
<. ( .r `  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o.  y ) ) >.  =  <. ( .r `  ndx ) ,  .X.  >. )
2410, 18, 23tpeq123d 3834 . . . . . 6  |-  ( ( m  =  M  /\  b  =  B )  ->  { <. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o F ( +g  `  m
) y ) )
>. ,  <. ( .r
`  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o.  y ) ) >. }  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. } )
25 fveq2 5661 . . . . . . . . . 10  |-  ( m  =  M  ->  (Scalar `  m )  =  (Scalar `  M ) )
2625adantr 452 . . . . . . . . 9  |-  ( ( m  =  M  /\  b  =  B )  ->  (Scalar `  m )  =  (Scalar `  M )
)
27 mendval.s . . . . . . . . 9  |-  S  =  (Scalar `  M )
2826, 27syl6eqr 2430 . . . . . . . 8  |-  ( ( m  =  M  /\  b  =  B )  ->  (Scalar `  m )  =  S )
2928opeq2d 3926 . . . . . . 7  |-  ( ( m  =  M  /\  b  =  B )  -> 
<. (Scalar `  ndx ) ,  (Scalar `  m ) >.  =  <. (Scalar `  ndx ) ,  S >. )
3028fveq2d 5665 . . . . . . . . . 10  |-  ( ( m  =  M  /\  b  =  B )  ->  ( Base `  (Scalar `  m ) )  =  ( Base `  S
) )
31 fveq2 5661 . . . . . . . . . . . . 13  |-  ( m  =  M  ->  ( .s `  m )  =  ( .s `  M
) )
3231adantr 452 . . . . . . . . . . . 12  |-  ( ( m  =  M  /\  b  =  B )  ->  ( .s `  m
)  =  ( .s
`  M ) )
33 ofeq 6239 . . . . . . . . . . . 12  |-  ( ( .s `  m )  =  ( .s `  M )  ->  o F ( .s `  m )  =  o F ( .s `  M ) )
3432, 33syl 16 . . . . . . . . . . 11  |-  ( ( m  =  M  /\  b  =  B )  ->  o F ( .s
`  m )  =  o F ( .s
`  M ) )
35 fveq2 5661 . . . . . . . . . . . . 13  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
3635adantr 452 . . . . . . . . . . . 12  |-  ( ( m  =  M  /\  b  =  B )  ->  ( Base `  m
)  =  ( Base `  M ) )
3736xpeq1d 4834 . . . . . . . . . . 11  |-  ( ( m  =  M  /\  b  =  B )  ->  ( ( Base `  m
)  X.  { x } )  =  ( ( Base `  M
)  X.  { x } ) )
38 eqidd 2381 . . . . . . . . . . 11  |-  ( ( m  =  M  /\  b  =  B )  ->  y  =  y )
3934, 37, 38oveq123d 6034 . . . . . . . . . 10  |-  ( ( m  =  M  /\  b  =  B )  ->  ( ( ( Base `  m )  X.  {
x } )  o F ( .s `  m ) y )  =  ( ( (
Base `  M )  X.  { x } )  o F ( .s
`  M ) y ) )
4030, 9, 39mpt2eq123dv 6068 . . . . . . . . 9  |-  ( ( m  =  M  /\  b  =  B )  ->  ( x  e.  (
Base `  (Scalar `  m
) ) ,  y  e.  b  |->  ( ( ( Base `  m
)  X.  { x } )  o F ( .s `  m
) y ) )  =  ( x  e.  ( Base `  S
) ,  y  e.  B  |->  ( ( (
Base `  M )  X.  { x } )  o F ( .s
`  M ) y ) ) )
41 mendval.v . . . . . . . . 9  |-  .x.  =  ( x  e.  ( Base `  S ) ,  y  e.  B  |->  ( ( ( Base `  M
)  X.  { x } )  o F ( .s `  M
) y ) )
4240, 41syl6eqr 2430 . . . . . . . 8  |-  ( ( m  =  M  /\  b  =  B )  ->  ( x  e.  (
Base `  (Scalar `  m
) ) ,  y  e.  b  |->  ( ( ( Base `  m
)  X.  { x } )  o F ( .s `  m
) y ) )  =  .x.  )
4342opeq2d 3926 . . . . . . 7  |-  ( ( m  =  M  /\  b  =  B )  -> 
<. ( .s `  ndx ) ,  ( x  e.  ( Base `  (Scalar `  m ) ) ,  y  e.  b  |->  ( ( ( Base `  m
)  X.  { x } )  o F ( .s `  m
) y ) )
>.  =  <. ( .s
`  ndx ) ,  .x.  >.
)
4429, 43preq12d 3827 . . . . . 6  |-  ( ( m  =  M  /\  b  =  B )  ->  { <. (Scalar `  ndx ) ,  (Scalar `  m
) >. ,  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  (Scalar `  m
) ) ,  y  e.  b  |->  ( ( ( Base `  m
)  X.  { x } )  o F ( .s `  m
) y ) )
>. }  =  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  .x.  >. } )
4524, 44uneq12d 3438 . . . . 5  |-  ( ( m  =  M  /\  b  =  B )  ->  ( { <. ( Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  ( x  e.  b ,  y  e.  b 
|->  ( x  o F ( +g  `  m
) y ) )
>. ,  <. ( .r
`  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o.  y ) ) >. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  m ) >. ,  <. ( .s `  ndx ) ,  ( x  e.  ( Base `  (Scalar `  m ) ) ,  y  e.  b  |->  ( ( ( Base `  m
)  X.  { x } )  o F ( .s `  m
) y ) )
>. } )  =  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } ) )
468, 45csbied 3229 . . . 4  |-  ( m  =  M  ->  [_ B  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o F ( +g  `  m
) y ) )
>. ,  <. ( .r
`  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o.  y ) ) >. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  m ) >. ,  <. ( .s `  ndx ) ,  ( x  e.  ( Base `  (Scalar `  m ) ) ,  y  e.  b  |->  ( ( ( Base `  m
)  X.  { x } )  o F ( .s `  m
) y ) )
>. } )  =  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } ) )
476, 46eqtrd 2412 . . 3  |-  ( m  =  M  ->  [_ (
m LMHom  m )  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  ( x  e.  b ,  y  e.  b 
|->  ( x  o F ( +g  `  m
) y ) )
>. ,  <. ( .r
`  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o.  y ) ) >. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  m ) >. ,  <. ( .s `  ndx ) ,  ( x  e.  ( Base `  (Scalar `  m ) ) ,  y  e.  b  |->  ( ( ( Base `  m
)  X.  { x } )  o F ( .s `  m
) y ) )
>. } )  =  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } ) )
48 df-mend 27152 . . 3  |- MEndo  =  ( m  e.  _V  |->  [_ ( m LMHom  m )  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o F ( +g  `  m
) y ) )
>. ,  <. ( .r
`  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o.  y ) ) >. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  m ) >. ,  <. ( .s `  ndx ) ,  ( x  e.  ( Base `  (Scalar `  m ) ) ,  y  e.  b  |->  ( ( ( Base `  m
)  X.  { x } )  o F ( .s `  m
) y ) )
>. } ) )
49 tpex 4641 . . . 4  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. ( .r `  ndx ) , 
.X.  >. }  e.  _V
50 prex 4340 . . . 4  |-  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  .x.  >. }  e.  _V
5149, 50unex 4640 . . 3  |-  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } )  e. 
_V
5247, 48, 51fvmpt 5738 . 2  |-  ( M  e.  _V  ->  (MEndo `  M )  =  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } ) )
531, 52syl 16 1  |-  ( M  e.  X  ->  (MEndo `  M )  =  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2892   [_csb 3187    u. cun 3254   {csn 3750   {cpr 3751   {ctp 3752   <.cop 3753    X. cxp 4809    o. ccom 4815   ` cfv 5387  (class class class)co 6013    e. cmpt2 6015    o Fcof 6235   ndxcnx 13386   Basecbs 13389   +g cplusg 13449   .rcmulr 13450  Scalarcsca 13452   .scvsca 13453   LMHom clmhm 16015  MEndocmend 27151
This theorem is referenced by:  mendbas  27154  mendplusgfval  27155  mendmulrfval  27157  mendsca  27159  mendvscafval  27160
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-mend 27152
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