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Definition df-mend 30719
Description: Define the endomorphism algebra of a module. (Contributed by Stefan O'Rear, 2-Sep-2015.)
Assertion
Ref Expression
df-mend  |- MEndo  =  ( m  e.  _V  |->  [_ ( m LMHom  m )  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  oF ( +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 } )  oF ( .s `  m
) y ) )
>. } ) )
Distinct variable group:    m, b, x, y

Detailed syntax breakdown of Definition df-mend
StepHypRef Expression
1 cmend 30718 . 2  class MEndo
2 vm . . 3  setvar  m
3 cvv 3106 . . 3  class  _V
4 vb . . . 4  setvar  b
52cv 1373 . . . . 5  class  m
6 clmhm 17441 . . . . 5  class LMHom
75, 5, 6co 6275 . . . 4  class  ( m LMHom 
m )
8 cnx 14476 . . . . . . . 8  class  ndx
9 cbs 14479 . . . . . . . 8  class  Base
108, 9cfv 5579 . . . . . . 7  class  ( Base `  ndx )
114cv 1373 . . . . . . 7  class  b
1210, 11cop 4026 . . . . . 6  class  <. ( Base `  ndx ) ,  b >.
13 cplusg 14544 . . . . . . . 8  class  +g
148, 13cfv 5579 . . . . . . 7  class  ( +g  ` 
ndx )
15 vx . . . . . . . 8  setvar  x
16 vy . . . . . . . 8  setvar  y
1715cv 1373 . . . . . . . . 9  class  x
1816cv 1373 . . . . . . . . 9  class  y
195, 13cfv 5579 . . . . . . . . . 10  class  ( +g  `  m )
2019cof 6513 . . . . . . . . 9  class  oF ( +g  `  m
)
2117, 18, 20co 6275 . . . . . . . 8  class  ( x  oF ( +g  `  m ) y )
2215, 16, 11, 11, 21cmpt2 6277 . . . . . . 7  class  ( x  e.  b ,  y  e.  b  |->  ( x  oF ( +g  `  m ) y ) )
2314, 22cop 4026 . . . . . 6  class  <. ( +g  `  ndx ) ,  ( x  e.  b ,  y  e.  b 
|->  ( x  oF ( +g  `  m
) y ) )
>.
24 cmulr 14545 . . . . . . . 8  class  .r
258, 24cfv 5579 . . . . . . 7  class  ( .r
`  ndx )
2617, 18ccom 4996 . . . . . . . 8  class  ( x  o.  y )
2715, 16, 11, 11, 26cmpt2 6277 . . . . . . 7  class  ( x  e.  b ,  y  e.  b  |->  ( x  o.  y ) )
2825, 27cop 4026 . . . . . 6  class  <. ( .r `  ndx ) ,  ( x  e.  b ,  y  e.  b 
|->  ( x  o.  y
) ) >.
2912, 23, 28ctp 4024 . . . . 5  class  { <. (
Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  ( x  e.  b ,  y  e.  b 
|->  ( x  oF ( +g  `  m
) y ) )
>. ,  <. ( .r
`  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o.  y ) ) >. }
30 csca 14547 . . . . . . . 8  class Scalar
318, 30cfv 5579 . . . . . . 7  class  (Scalar `  ndx )
325, 30cfv 5579 . . . . . . 7  class  (Scalar `  m )
3331, 32cop 4026 . . . . . 6  class  <. (Scalar ` 
ndx ) ,  (Scalar `  m ) >.
34 cvsca 14548 . . . . . . . 8  class  .s
358, 34cfv 5579 . . . . . . 7  class  ( .s
`  ndx )
3632, 9cfv 5579 . . . . . . . 8  class  ( Base `  (Scalar `  m )
)
375, 9cfv 5579 . . . . . . . . . 10  class  ( Base `  m )
3817csn 4020 . . . . . . . . . 10  class  { x }
3937, 38cxp 4990 . . . . . . . . 9  class  ( (
Base `  m )  X.  { x } )
405, 34cfv 5579 . . . . . . . . . 10  class  ( .s
`  m )
4140cof 6513 . . . . . . . . 9  class  oF ( .s `  m
)
4239, 18, 41co 6275 . . . . . . . 8  class  ( ( ( Base `  m
)  X.  { x } )  oF ( .s `  m
) y )
4315, 16, 36, 11, 42cmpt2 6277 . . . . . . 7  class  ( x  e.  ( Base `  (Scalar `  m ) ) ,  y  e.  b  |->  ( ( ( Base `  m
)  X.  { x } )  oF ( .s `  m
) y ) )
4435, 43cop 4026 . . . . . 6  class  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  (Scalar `  m
) ) ,  y  e.  b  |->  ( ( ( Base `  m
)  X.  { x } )  oF ( .s `  m
) y ) )
>.
4533, 44cpr 4022 . . . . 5  class  { <. (Scalar `  ndx ) ,  (Scalar `  m ) >. ,  <. ( .s `  ndx ) ,  ( x  e.  ( Base `  (Scalar `  m ) ) ,  y  e.  b  |->  ( ( ( Base `  m
)  X.  { x } )  oF ( .s `  m
) y ) )
>. }
4629, 45cun 3467 . . . 4  class  ( {
<. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  oF ( +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 } )  oF ( .s `  m
) y ) )
>. } )
474, 7, 46csb 3428 . . 3  class  [_ (
m LMHom  m )  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  ( x  e.  b ,  y  e.  b 
|->  ( x  oF ( +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 } )  oF ( .s `  m
) y ) )
>. } )
482, 3, 47cmpt 4498 . 2  class  ( m  e.  _V  |->  [_ (
m LMHom  m )  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  ( x  e.  b ,  y  e.  b 
|->  ( x  oF ( +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 } )  oF ( .s `  m
) y ) )
>. } ) )
491, 48wceq 1374 1  wff MEndo  =  ( m  e.  _V  |->  [_ ( m LMHom  m )  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  oF ( +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 } )  oF ( .s `  m
) y ) )
>. } ) )
Colors of variables: wff setvar class
This definition is referenced by:  mendval  30726
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