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Theorem mendbas 29386
Description: Base set of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
Hypothesis
Ref Expression
mendbas.a  |-  A  =  (MEndo `  M )
Assertion
Ref Expression
mendbas  |-  ( M LMHom 
M )  =  (
Base `  A )

Proof of Theorem mendbas
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6105 . . . 4  |-  ( M LMHom 
M )  e.  _V
2 eqid 2433 . . . . 5  |-  ( {
<. ( Base `  ndx ) ,  ( M LMHom  M ) >. ,  <. ( +g  `  ndx ) ,  ( x  e.  ( M LMHom  M ) ,  y  e.  ( M LMHom 
M )  |->  ( x  oF ( +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 } )  oF ( .s `  M
) y ) )
>. } )  =  ( { <. ( Base `  ndx ) ,  ( M LMHom  M ) >. ,  <. ( +g  `  ndx ) ,  ( x  e.  ( M LMHom  M ) ,  y  e.  ( M LMHom 
M )  |->  ( x  oF ( +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 } )  oF ( .s `  M
) y ) )
>. } )
32algbase 29380 . . . 4  |-  ( ( M LMHom  M )  e. 
_V  ->  ( M LMHom  M
)  =  ( Base `  ( { <. ( Base `  ndx ) ,  ( M LMHom  M )
>. ,  <. ( +g  ` 
ndx ) ,  ( x  e.  ( M LMHom 
M ) ,  y  e.  ( M LMHom  M
)  |->  ( x  oF ( +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 } )  oF ( .s `  M
) y ) )
>. } ) ) )
41, 3mp1i 12 . . 3  |-  ( M  e.  _V  ->  ( M LMHom  M )  =  (
Base `  ( { <. ( Base `  ndx ) ,  ( M LMHom  M ) >. ,  <. ( +g  `  ndx ) ,  ( x  e.  ( M LMHom  M ) ,  y  e.  ( M LMHom 
M )  |->  ( x  oF ( +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 } )  oF ( .s `  M
) y ) )
>. } ) ) )
5 mendbas.a . . . . 5  |-  A  =  (MEndo `  M )
6 eqid 2433 . . . . . 6  |-  ( M LMHom 
M )  =  ( M LMHom  M )
7 eqid 2433 . . . . . 6  |-  ( x  e.  ( M LMHom  M
) ,  y  e.  ( M LMHom  M ) 
|->  ( x  oF ( +g  `  M
) y ) )  =  ( x  e.  ( M LMHom  M ) ,  y  e.  ( M LMHom  M )  |->  ( x  oF ( +g  `  M ) y ) )
8 eqid 2433 . . . . . 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 ) )
9 eqid 2433 . . . . . 6  |-  (Scalar `  M )  =  (Scalar `  M )
10 eqid 2433 . . . . . 6  |-  ( x  e.  ( Base `  (Scalar `  M ) ) ,  y  e.  ( M LMHom 
M )  |->  ( ( ( Base `  M
)  X.  { x } )  oF ( .s `  M
) y ) )  =  ( x  e.  ( Base `  (Scalar `  M ) ) ,  y  e.  ( M LMHom 
M )  |->  ( ( ( Base `  M
)  X.  { x } )  oF ( .s `  M
) y ) )
116, 7, 8, 9, 10mendval 29385 . . . . 5  |-  ( M  e.  _V  ->  (MEndo `  M )  =  ( { <. ( Base `  ndx ) ,  ( M LMHom  M ) >. ,  <. ( +g  `  ndx ) ,  ( x  e.  ( M LMHom  M ) ,  y  e.  ( M LMHom 
M )  |->  ( x  oF ( +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 } )  oF ( .s `  M
) y ) )
>. } ) )
125, 11syl5eq 2477 . . . 4  |-  ( M  e.  _V  ->  A  =  ( { <. (
Base `  ndx ) ,  ( M LMHom  M )
>. ,  <. ( +g  ` 
ndx ) ,  ( x  e.  ( M LMHom 
M ) ,  y  e.  ( M LMHom  M
)  |->  ( x  oF ( +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 } )  oF ( .s `  M
) y ) )
>. } ) )
1312fveq2d 5683 . . 3  |-  ( M  e.  _V  ->  ( Base `  A )  =  ( Base `  ( { <. ( Base `  ndx ) ,  ( M LMHom  M ) >. ,  <. ( +g  `  ndx ) ,  ( x  e.  ( M LMHom  M ) ,  y  e.  ( M LMHom 
M )  |->  ( x  oF ( +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 } )  oF ( .s `  M
) y ) )
>. } ) ) )
144, 13eqtr4d 2468 . 2  |-  ( M  e.  _V  ->  ( M LMHom  M )  =  (
Base `  A )
)
15 base0 14196 . . 3  |-  (/)  =  (
Base `  (/) )
16 reldmlmhm 17028 . . . 4  |-  Rel  dom LMHom
1716ovprc1 6108 . . 3  |-  ( -.  M  e.  _V  ->  ( M LMHom  M )  =  (/) )
18 fvprc 5673 . . . . 5  |-  ( -.  M  e.  _V  ->  (MEndo `  M )  =  (/) )
195, 18syl5eq 2477 . . . 4  |-  ( -.  M  e.  _V  ->  A  =  (/) )
2019fveq2d 5683 . . 3  |-  ( -.  M  e.  _V  ->  (
Base `  A )  =  ( Base `  (/) ) )
2115, 17, 203eqtr4a 2491 . 2  |-  ( -.  M  e.  _V  ->  ( M LMHom  M )  =  ( Base `  A
) )
2214, 21pm2.61i 164 1  |-  ( M LMHom 
M )  =  (
Base `  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1362    e. wcel 1755   _Vcvv 2962    u. cun 3314   (/)c0 3625   {csn 3865   {cpr 3867   {ctp 3869   <.cop 3871    X. cxp 4825    o. ccom 4831   ` cfv 5406  (class class class)co 6080    e. cmpt2 6082    oFcof 6307   ndxcnx 14154   Basecbs 14157   +g cplusg 14221   .rcmulr 14222  Scalarcsca 14224   .scvsca 14225   LMHom clmhm 17022  MEndocmend 29377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-n0 10568  df-z 10635  df-uz 10850  df-fz 11425  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-plusg 14234  df-mulr 14235  df-sca 14237  df-vsca 14238  df-lmhm 17025  df-mend 29378
This theorem is referenced by:  mendplusgfval  29387  mendmulrfval  29389  mendvscafval  29392  mendrng  29394  mendlmod  29395  mendassa  29396
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