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Definition df-mend 27593
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  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 ) )
>. } ) )
Distinct variable group:    m, b, x, y

Detailed syntax breakdown of Definition df-mend
StepHypRef Expression
1 cmend 27592 . 2  class MEndo
2 vm . . 3  set  m
3 cvv 2801 . . 3  class  _V
4 vb . . . 4  set  b
52cv 1631 . . . . 5  class  m
6 clmhm 15792 . . . . 5  class LMHom
75, 5, 6co 5874 . . . 4  class  ( m LMHom 
m )
8 cnx 13161 . . . . . . . 8  class  ndx
9 cbs 13164 . . . . . . . 8  class  Base
108, 9cfv 5271 . . . . . . 7  class  ( Base `  ndx )
114cv 1631 . . . . . . 7  class  b
1210, 11cop 3656 . . . . . 6  class  <. ( Base `  ndx ) ,  b >.
13 cplusg 13224 . . . . . . . 8  class  +g
148, 13cfv 5271 . . . . . . 7  class  ( +g  ` 
ndx )
15 vx . . . . . . . 8  set  x
16 vy . . . . . . . 8  set  y
1715cv 1631 . . . . . . . . 9  class  x
1816cv 1631 . . . . . . . . 9  class  y
195, 13cfv 5271 . . . . . . . . . 10  class  ( +g  `  m )
2019cof 6092 . . . . . . . . 9  class  o F ( +g  `  m
)
2117, 18, 20co 5874 . . . . . . . 8  class  ( x  o F ( +g  `  m ) y )
2215, 16, 11, 11, 21cmpt2 5876 . . . . . . 7  class  ( x  e.  b ,  y  e.  b  |->  ( x  o F ( +g  `  m ) y ) )
2314, 22cop 3656 . . . . . 6  class  <. ( +g  `  ndx ) ,  ( x  e.  b ,  y  e.  b 
|->  ( x  o F ( +g  `  m
) y ) )
>.
24 cmulr 13225 . . . . . . . 8  class  .r
258, 24cfv 5271 . . . . . . 7  class  ( .r
`  ndx )
2617, 18ccom 4709 . . . . . . . 8  class  ( x  o.  y )
2715, 16, 11, 11, 26cmpt2 5876 . . . . . . 7  class  ( x  e.  b ,  y  e.  b  |->  ( x  o.  y ) )
2825, 27cop 3656 . . . . . 6  class  <. ( .r `  ndx ) ,  ( x  e.  b ,  y  e.  b 
|->  ( x  o.  y
) ) >.
2912, 23, 28ctp 3655 . . . . 5  class  { <. (
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 ) ) >. }
30 csca 13227 . . . . . . . 8  class Scalar
318, 30cfv 5271 . . . . . . 7  class  (Scalar `  ndx )
325, 30cfv 5271 . . . . . . 7  class  (Scalar `  m )
3331, 32cop 3656 . . . . . 6  class  <. (Scalar ` 
ndx ) ,  (Scalar `  m ) >.
34 cvsca 13228 . . . . . . . 8  class  .s
358, 34cfv 5271 . . . . . . 7  class  ( .s
`  ndx )
3632, 9cfv 5271 . . . . . . . 8  class  ( Base `  (Scalar `  m )
)
375, 9cfv 5271 . . . . . . . . . 10  class  ( Base `  m )
3817csn 3653 . . . . . . . . . 10  class  { x }
3937, 38cxp 4703 . . . . . . . . 9  class  ( (
Base `  m )  X.  { x } )
405, 34cfv 5271 . . . . . . . . . 10  class  ( .s
`  m )
4140cof 6092 . . . . . . . . 9  class  o F ( .s `  m
)
4239, 18, 41co 5874 . . . . . . . 8  class  ( ( ( Base `  m
)  X.  { x } )  o F ( .s `  m
) y )
4315, 16, 36, 11, 42cmpt2 5876 . . . . . . 7  class  ( x  e.  ( Base `  (Scalar `  m ) ) ,  y  e.  b  |->  ( ( ( Base `  m
)  X.  { x } )  o F ( .s `  m
) y ) )
4435, 43cop 3656 . . . . . 6  class  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  (Scalar `  m
) ) ,  y  e.  b  |->  ( ( ( Base `  m
)  X.  { x } )  o F ( .s `  m
) y ) )
>.
4533, 44cpr 3654 . . . . 5  class  { <. (Scalar `  ndx ) ,  (Scalar `  m ) >. ,  <. ( .s `  ndx ) ,  ( x  e.  ( Base `  (Scalar `  m ) ) ,  y  e.  b  |->  ( ( ( Base `  m
)  X.  { x } )  o F ( .s `  m
) y ) )
>. }
4629, 45cun 3163 . . . 4  class  ( {
<. ( 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 ) )
>. } )
474, 7, 46csb 3094 . . 3  class  [_ (
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 ) )
>. } )
482, 3, 47cmpt 4093 . 2  class  ( 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 ) )
>. } ) )
491, 48wceq 1632 1  wff 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 ) )
>. } ) )
Colors of variables: wff set class
This definition is referenced by:  mendval  27594
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