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Theorem mendsca 26820
Description: The module endomorphism algebra has the same scalars as the underlying module. (Contributed by Stefan O'Rear, 2-Sep-2015.)
Hypotheses
Ref Expression
mendsca.a  |-  A  =  (MEndo `  M )
mendsca.s  |-  S  =  (Scalar `  M )
Assertion
Ref Expression
mendsca  |-  S  =  (Scalar `  A )

Proof of Theorem mendsca
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5619 . . . . 5  |-  (Scalar `  M )  e.  _V
2 eqid 2358 . . . . . 6  |-  ( {
<. ( Base `  ndx ) ,  ( M LMHom  M ) >. ,  <. ( +g  `  ndx ) ,  ( x  e.  ( M LMHom  M ) ,  y  e.  ( M LMHom 
M )  |->  ( x  o F ( +g  `  M ) y ) ) >. ,  <. ( .r `  ndx ) ,  ( x  e.  ( M LMHom  M ) ,  y  e.  ( M LMHom 
M )  |->  ( x  o.  y ) )
>. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  M ) >. ,  <. ( .s `  ndx ) ,  ( x  e.  ( Base `  (Scalar `  M ) ) ,  y  e.  ( M LMHom 
M )  |->  ( ( ( Base `  M
)  X.  { x } )  o F ( .s `  M
) y ) )
>. } )  =  ( { <. ( Base `  ndx ) ,  ( M LMHom  M ) >. ,  <. ( +g  `  ndx ) ,  ( x  e.  ( M LMHom  M ) ,  y  e.  ( M LMHom 
M )  |->  ( x  o F ( +g  `  M ) y ) ) >. ,  <. ( .r `  ndx ) ,  ( x  e.  ( M LMHom  M ) ,  y  e.  ( M LMHom 
M )  |->  ( x  o.  y ) )
>. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  M ) >. ,  <. ( .s `  ndx ) ,  ( x  e.  ( Base `  (Scalar `  M ) ) ,  y  e.  ( M LMHom 
M )  |->  ( ( ( Base `  M
)  X.  { x } )  o F ( .s `  M
) y ) )
>. } )
32algsca 13372 . . . . 5  |-  ( (Scalar `  M )  e.  _V  ->  (Scalar `  M )  =  (Scalar `  ( { <. ( Base `  ndx ) ,  ( M LMHom  M ) >. ,  <. ( +g  `  ndx ) ,  ( x  e.  ( M LMHom  M ) ,  y  e.  ( M LMHom 
M )  |->  ( x  o F ( +g  `  M ) y ) ) >. ,  <. ( .r `  ndx ) ,  ( x  e.  ( M LMHom  M ) ,  y  e.  ( M LMHom 
M )  |->  ( x  o.  y ) )
>. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  M ) >. ,  <. ( .s `  ndx ) ,  ( x  e.  ( Base `  (Scalar `  M ) ) ,  y  e.  ( M LMHom 
M )  |->  ( ( ( Base `  M
)  X.  { x } )  o F ( .s `  M
) y ) )
>. } ) ) )
41, 3mp1i 11 . . . 4  |-  ( M  e.  _V  ->  (Scalar `  M )  =  (Scalar `  ( { <. ( Base `  ndx ) ,  ( M LMHom  M )
>. ,  <. ( +g  ` 
ndx ) ,  ( x  e.  ( M LMHom 
M ) ,  y  e.  ( M LMHom  M
)  |->  ( x  o F ( +g  `  M
) y ) )
>. ,  <. ( .r
`  ndx ) ,  ( x  e.  ( M LMHom 
M ) ,  y  e.  ( M LMHom  M
)  |->  ( x  o.  y ) ) >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  M ) >. ,  <. ( .s `  ndx ) ,  ( x  e.  ( Base `  (Scalar `  M ) ) ,  y  e.  ( M LMHom 
M )  |->  ( ( ( Base `  M
)  X.  { x } )  o F ( .s `  M
) y ) )
>. } ) ) )
5 eqid 2358 . . . . . 6  |-  ( M LMHom 
M )  =  ( M LMHom  M )
6 eqid 2358 . . . . . 6  |-  ( x  e.  ( M LMHom  M
) ,  y  e.  ( M LMHom  M ) 
|->  ( x  o F ( +g  `  M
) y ) )  =  ( x  e.  ( M LMHom  M ) ,  y  e.  ( M LMHom  M )  |->  ( x  o F ( +g  `  M ) y ) )
7 eqid 2358 . . . . . 6  |-  ( x  e.  ( M LMHom  M
) ,  y  e.  ( M LMHom  M ) 
|->  ( x  o.  y
) )  =  ( x  e.  ( M LMHom 
M ) ,  y  e.  ( M LMHom  M
)  |->  ( x  o.  y ) )
8 eqid 2358 . . . . . 6  |-  (Scalar `  M )  =  (Scalar `  M )
9 eqid 2358 . . . . . 6  |-  ( x  e.  ( Base `  (Scalar `  M ) ) ,  y  e.  ( M LMHom 
M )  |->  ( ( ( Base `  M
)  X.  { x } )  o F ( .s `  M
) y ) )  =  ( x  e.  ( Base `  (Scalar `  M ) ) ,  y  e.  ( M LMHom 
M )  |->  ( ( ( Base `  M
)  X.  { x } )  o F ( .s `  M
) y ) )
105, 6, 7, 8, 9mendval 26814 . . . . 5  |-  ( M  e.  _V  ->  (MEndo `  M )  =  ( { <. ( Base `  ndx ) ,  ( M LMHom  M ) >. ,  <. ( +g  `  ndx ) ,  ( x  e.  ( M LMHom  M ) ,  y  e.  ( M LMHom 
M )  |->  ( x  o F ( +g  `  M ) y ) ) >. ,  <. ( .r `  ndx ) ,  ( x  e.  ( M LMHom  M ) ,  y  e.  ( M LMHom 
M )  |->  ( x  o.  y ) )
>. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  M ) >. ,  <. ( .s `  ndx ) ,  ( x  e.  ( Base `  (Scalar `  M ) ) ,  y  e.  ( M LMHom 
M )  |->  ( ( ( Base `  M
)  X.  { x } )  o F ( .s `  M
) y ) )
>. } ) )
1110fveq2d 5609 . . . 4  |-  ( M  e.  _V  ->  (Scalar `  (MEndo `  M )
)  =  (Scalar `  ( { <. ( Base `  ndx ) ,  ( M LMHom  M ) >. ,  <. ( +g  `  ndx ) ,  ( x  e.  ( M LMHom  M ) ,  y  e.  ( M LMHom 
M )  |->  ( x  o F ( +g  `  M ) y ) ) >. ,  <. ( .r `  ndx ) ,  ( x  e.  ( M LMHom  M ) ,  y  e.  ( M LMHom 
M )  |->  ( x  o.  y ) )
>. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  M ) >. ,  <. ( .s `  ndx ) ,  ( x  e.  ( Base `  (Scalar `  M ) ) ,  y  e.  ( M LMHom 
M )  |->  ( ( ( Base `  M
)  X.  { x } )  o F ( .s `  M
) y ) )
>. } ) ) )
124, 11eqtr4d 2393 . . 3  |-  ( M  e.  _V  ->  (Scalar `  M )  =  (Scalar `  (MEndo `  M )
) )
13 df-sca 13315 . . . . 5  |- Scalar  = Slot  5
1413str0 13275 . . . 4  |-  (/)  =  (Scalar `  (/) )
15 fvprc 5599 . . . 4  |-  ( -.  M  e.  _V  ->  (Scalar `  M )  =  (/) )
16 fvprc 5599 . . . . 5  |-  ( -.  M  e.  _V  ->  (MEndo `  M )  =  (/) )
1716fveq2d 5609 . . . 4  |-  ( -.  M  e.  _V  ->  (Scalar `  (MEndo `  M )
)  =  (Scalar `  (/) ) )
1814, 15, 173eqtr4a 2416 . . 3  |-  ( -.  M  e.  _V  ->  (Scalar `  M )  =  (Scalar `  (MEndo `  M )
) )
1912, 18pm2.61i 156 . 2  |-  (Scalar `  M )  =  (Scalar `  (MEndo `  M )
)
20 mendsca.s . 2  |-  S  =  (Scalar `  M )
21 mendsca.a . . 3  |-  A  =  (MEndo `  M )
2221fveq2i 5608 . 2  |-  (Scalar `  A )  =  (Scalar `  (MEndo `  M )
)
2319, 20, 223eqtr4i 2388 1  |-  S  =  (Scalar `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1642    e. wcel 1710   _Vcvv 2864    u. cun 3226   (/)c0 3531   {csn 3716   {cpr 3717   {ctp 3718   <.cop 3719    X. cxp 4766    o. ccom 4772   ` cfv 5334  (class class class)co 5942    e. cmpt2 5944    o Fcof 6160   5c5 9885   ndxcnx 13236   Basecbs 13239   +g cplusg 13299   .rcmulr 13300  Scalarcsca 13302   .scvsca 13303   LMHom clmhm 15869  MEndocmend 26812
This theorem is referenced by:  mendlmod  26824  mendassa  26825
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-of 6162  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-n0 10055  df-z 10114  df-uz 10320  df-fz 10872  df-struct 13241  df-ndx 13242  df-slot 13243  df-base 13244  df-plusg 13312  df-mulr 13313  df-sca 13315  df-vsca 13316  df-mend 26813
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