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Theorem mendbas 29494
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 6111 . . . 4  |-  ( M LMHom 
M )  e.  _V
2 eqid 2438 . . . . 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 29488 . . . 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 2438 . . . . . 6  |-  ( M LMHom 
M )  =  ( M LMHom  M )
7 eqid 2438 . . . . . 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 2438 . . . . . 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 2438 . . . . . 6  |-  (Scalar `  M )  =  (Scalar `  M )
10 eqid 2438 . . . . . 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 29493 . . . . 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 2482 . . . 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 5690 . . 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 2473 . 2  |-  ( M  e.  _V  ->  ( M LMHom  M )  =  (
Base `  A )
)
15 base0 14205 . . 3  |-  (/)  =  (
Base `  (/) )
16 reldmlmhm 17083 . . . 4  |-  Rel  dom LMHom
1716ovprc1 6114 . . 3  |-  ( -.  M  e.  _V  ->  ( M LMHom  M )  =  (/) )
18 fvprc 5680 . . . . 5  |-  ( -.  M  e.  _V  ->  (MEndo `  M )  =  (/) )
195, 18syl5eq 2482 . . . 4  |-  ( -.  M  e.  _V  ->  A  =  (/) )
2019fveq2d 5690 . . 3  |-  ( -.  M  e.  _V  ->  (
Base `  A )  =  ( Base `  (/) ) )
2115, 17, 203eqtr4a 2496 . 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 1369    e. wcel 1756   _Vcvv 2967    u. cun 3321   (/)c0 3632   {csn 3872   {cpr 3874   {ctp 3876   <.cop 3878    X. cxp 4833    o. ccom 4839   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088    oFcof 6313   ndxcnx 14163   Basecbs 14166   +g cplusg 14230   .rcmulr 14231  Scalarcsca 14233   .scvsca 14234   LMHom clmhm 17077  MEndocmend 29485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-plusg 14243  df-mulr 14244  df-sca 14246  df-vsca 14247  df-lmhm 17080  df-mend 29486
This theorem is referenced by:  mendplusgfval  29495  mendmulrfval  29497  mendvscafval  29500  mendrng  29502  mendlmod  29503  mendassa  29504
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