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Theorem mendval 27594
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 2809 . 2  |-  ( M  e.  X  ->  M  e.  _V )
2 oveq12 5883 . . . . . . 7  |-  ( ( m  =  M  /\  m  =  M )  ->  ( m LMHom  m )  =  ( M LMHom  M
) )
32anidms 626 . . . . . 6  |-  ( m  =  M  ->  (
m LMHom  m )  =  ( M LMHom  M ) )
4 mendval.b . . . . . 6  |-  B  =  ( M LMHom  M )
53, 4syl6eqr 2346 . . . . 5  |-  ( m  =  M  ->  (
m LMHom  m )  =  B )
65csbeq1d 3100 . . . 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 5899 . . . . . 6  |-  ( m LMHom 
m )  e.  _V
85, 7syl6eqelr 2385 . . . . 5  |-  ( m  =  M  ->  B  e.  _V )
9 simpr 447 . . . . . . . 8  |-  ( ( m  =  M  /\  b  =  B )  ->  b  =  B )
109opeq2d 3819 . . . . . . 7  |-  ( ( m  =  M  /\  b  =  B )  -> 
<. ( Base `  ndx ) ,  b >.  = 
<. ( Base `  ndx ) ,  B >. )
11 fveq2 5541 . . . . . . . . . . . . 13  |-  ( m  =  M  ->  ( +g  `  m )  =  ( +g  `  M
) )
12 ofeq 6096 . . . . . . . . . . . . 13  |-  ( ( +g  `  m )  =  ( +g  `  M
)  ->  o F
( +g  `  m )  =  o F ( +g  `  M ) )
1311, 12syl 15 . . . . . . . . . . . 12  |-  ( m  =  M  ->  o F ( +g  `  m
)  =  o F ( +g  `  M
) )
1413oveqd 5891 . . . . . . . . . . 11  |-  ( m  =  M  ->  (
x  o F ( +g  `  m ) y )  =  ( x  o F ( +g  `  M ) y ) )
1514adantr 451 . . . . . . . . . 10  |-  ( ( m  =  M  /\  b  =  B )  ->  ( x  o F ( +g  `  m
) y )  =  ( x  o F ( +g  `  M
) y ) )
169, 9, 15mpt2eq123dv 5926 . . . . . . . . 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 ) ) )
17 mendval.p . . . . . . . . 9  |-  .+  =  ( x  e.  B ,  y  e.  B  |->  ( x  o F ( +g  `  M
) y ) )
1816, 17syl6eqr 2346 . . . . . . . 8  |-  ( ( m  =  M  /\  b  =  B )  ->  ( x  e.  b ,  y  e.  b 
|->  ( x  o F ( +g  `  m
) y ) )  =  .+  )
1918opeq2d 3819 . . . . . . 7  |-  ( ( m  =  M  /\  b  =  B )  -> 
<. ( +g  `  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o F ( +g  `  m
) y ) )
>.  =  <. ( +g  ` 
ndx ) ,  .+  >.
)
20 eqidd 2297 . . . . . . . . . 10  |-  ( ( m  =  M  /\  b  =  B )  ->  ( x  o.  y
)  =  ( x  o.  y ) )
219, 9, 20mpt2eq123dv 5926 . . . . . . . . 9  |-  ( ( m  =  M  /\  b  =  B )  ->  ( x  e.  b ,  y  e.  b 
|->  ( x  o.  y
) )  =  ( x  e.  B , 
y  e.  B  |->  ( x  o.  y ) ) )
22 mendval.t . . . . . . . . 9  |-  .X.  =  ( x  e.  B ,  y  e.  B  |->  ( x  o.  y
) )
2321, 22syl6eqr 2346 . . . . . . . 8  |-  ( ( m  =  M  /\  b  =  B )  ->  ( x  e.  b ,  y  e.  b 
|->  ( x  o.  y
) )  =  .X.  )
2423opeq2d 3819 . . . . . . 7  |-  ( ( m  =  M  /\  b  =  B )  -> 
<. ( .r `  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o.  y ) ) >.  =  <. ( .r `  ndx ) ,  .X.  >. )
2510, 19, 24tpeq123d 3734 . . . . . 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.  >. } )
26 fveq2 5541 . . . . . . . . . 10  |-  ( m  =  M  ->  (Scalar `  m )  =  (Scalar `  M ) )
2726adantr 451 . . . . . . . . 9  |-  ( ( m  =  M  /\  b  =  B )  ->  (Scalar `  m )  =  (Scalar `  M )
)
28 mendval.s . . . . . . . . 9  |-  S  =  (Scalar `  M )
2927, 28syl6eqr 2346 . . . . . . . 8  |-  ( ( m  =  M  /\  b  =  B )  ->  (Scalar `  m )  =  S )
3029opeq2d 3819 . . . . . . 7  |-  ( ( m  =  M  /\  b  =  B )  -> 
<. (Scalar `  ndx ) ,  (Scalar `  m ) >.  =  <. (Scalar `  ndx ) ,  S >. )
3129fveq2d 5545 . . . . . . . . . 10  |-  ( ( m  =  M  /\  b  =  B )  ->  ( Base `  (Scalar `  m ) )  =  ( Base `  S
) )
32 fveq2 5541 . . . . . . . . . . . . 13  |-  ( m  =  M  ->  ( .s `  m )  =  ( .s `  M
) )
3332adantr 451 . . . . . . . . . . . 12  |-  ( ( m  =  M  /\  b  =  B )  ->  ( .s `  m
)  =  ( .s
`  M ) )
34 ofeq 6096 . . . . . . . . . . . 12  |-  ( ( .s `  m )  =  ( .s `  M )  ->  o F ( .s `  m )  =  o F ( .s `  M ) )
3533, 34syl 15 . . . . . . . . . . 11  |-  ( ( m  =  M  /\  b  =  B )  ->  o F ( .s
`  m )  =  o F ( .s
`  M ) )
36 fveq2 5541 . . . . . . . . . . . . 13  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
3736adantr 451 . . . . . . . . . . . 12  |-  ( ( m  =  M  /\  b  =  B )  ->  ( Base `  m
)  =  ( Base `  M ) )
3837xpeq1d 4728 . . . . . . . . . . 11  |-  ( ( m  =  M  /\  b  =  B )  ->  ( ( Base `  m
)  X.  { x } )  =  ( ( Base `  M
)  X.  { x } ) )
39 eqidd 2297 . . . . . . . . . . 11  |-  ( ( m  =  M  /\  b  =  B )  ->  y  =  y )
4035, 38, 39oveq123d 5895 . . . . . . . . . 10  |-  ( ( m  =  M  /\  b  =  B )  ->  ( ( ( Base `  m )  X.  {
x } )  o F ( .s `  m ) y )  =  ( ( (
Base `  M )  X.  { x } )  o F ( .s
`  M ) y ) )
4131, 9, 40mpt2eq123dv 5926 . . . . . . . . 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 ) ) )
42 mendval.v . . . . . . . . 9  |-  .x.  =  ( x  e.  ( Base `  S ) ,  y  e.  B  |->  ( ( ( Base `  M
)  X.  { x } )  o F ( .s `  M
) y ) )
4341, 42syl6eqr 2346 . . . . . . . 8  |-  ( ( m  =  M  /\  b  =  B )  ->  ( x  e.  (
Base `  (Scalar `  m
) ) ,  y  e.  b  |->  ( ( ( Base `  m
)  X.  { x } )  o F ( .s `  m
) y ) )  =  .x.  )
4443opeq2d 3819 . . . . . . 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.  >.
)
4530, 44preq12d 3727 . . . . . 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.  >. } )
4625, 45uneq12d 3343 . . . . 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.  >. } ) )
478, 46csbied 3136 . . . 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.  >. } ) )
486, 47eqtrd 2328 . . 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.  >. } ) )
49 df-mend 27593 . . 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 ) )
>. } ) )
50 tpex 4535 . . . 4  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. ( .r `  ndx ) , 
.X.  >. }  e.  _V
51 prex 4233 . . . 4  |-  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  .x.  >. }  e.  _V
5250, 51unex 4534 . . 3  |-  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } )  e. 
_V
5348, 49, 52fvmpt 5618 . 2  |-  ( M  e.  _V  ->  (MEndo `  M )  =  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } ) )
541, 53syl 15 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 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   [_csb 3094    u. cun 3163   {csn 3653   {cpr 3654   {ctp 3655   <.cop 3656    X. cxp 4703    o. ccom 4709   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876    o Fcof 6092   ndxcnx 13161   Basecbs 13164   +g cplusg 13224   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228   LMHom clmhm 15792  MEndocmend 27592
This theorem is referenced by:  mendbas  27595  mendplusgfval  27596  mendmulrfval  27598  mendsca  27600  mendvscafval  27601
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-mend 27593
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