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Theorem mendsca 27365
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 5701 . . . . 5  |-  (Scalar `  M )  e.  _V
2 eqid 2404 . . . . . 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 13557 . . . . 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 12 . . . 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 2404 . . . . . 6  |-  ( M LMHom 
M )  =  ( M LMHom  M )
6 eqid 2404 . . . . . 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 2404 . . . . . 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 2404 . . . . . 6  |-  (Scalar `  M )  =  (Scalar `  M )
9 eqid 2404 . . . . . 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 27359 . . . . 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 5691 . . . 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 2439 . . 3  |-  ( M  e.  _V  ->  (Scalar `  M )  =  (Scalar `  (MEndo `  M )
) )
13 df-sca 13500 . . . . 5  |- Scalar  = Slot  5
1413str0 13460 . . . 4  |-  (/)  =  (Scalar `  (/) )
15 fvprc 5681 . . . 4  |-  ( -.  M  e.  _V  ->  (Scalar `  M )  =  (/) )
16 fvprc 5681 . . . . 5  |-  ( -.  M  e.  _V  ->  (MEndo `  M )  =  (/) )
1716fveq2d 5691 . . . 4  |-  ( -.  M  e.  _V  ->  (Scalar `  (MEndo `  M )
)  =  (Scalar `  (/) ) )
1814, 15, 173eqtr4a 2462 . . 3  |-  ( -.  M  e.  _V  ->  (Scalar `  M )  =  (Scalar `  (MEndo `  M )
) )
1912, 18pm2.61i 158 . 2  |-  (Scalar `  M )  =  (Scalar `  (MEndo `  M )
)
20 mendsca.s . 2  |-  S  =  (Scalar `  M )
21 mendsca.a . . 3  |-  A  =  (MEndo `  M )
2221fveq2i 5690 . 2  |-  (Scalar `  A )  =  (Scalar `  (MEndo `  M )
)
2319, 20, 223eqtr4i 2434 1  |-  S  =  (Scalar `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278   (/)c0 3588   {csn 3774   {cpr 3775   {ctp 3776   <.cop 3777    X. cxp 4835    o. ccom 4841   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042    o Fcof 6262   5c5 10008   ndxcnx 13421   Basecbs 13424   +g cplusg 13484   .rcmulr 13485  Scalarcsca 13487   .scvsca 13488   LMHom clmhm 16050  MEndocmend 27357
This theorem is referenced by:  mendlmod  27369  mendassa  27370
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-mend 27358
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