Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mendbas Structured version   Unicode version

Theorem mendbas 31374
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 6298 . . . 4  |-  ( M LMHom 
M )  e.  _V
2 eqid 2454 . . . . 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 31368 . . . 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 2454 . . . . . 6  |-  ( M LMHom 
M )  =  ( M LMHom  M )
7 eqid 2454 . . . . . 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 2454 . . . . . 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 2454 . . . . . 6  |-  (Scalar `  M )  =  (Scalar `  M )
10 eqid 2454 . . . . . 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 31373 . . . . 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 2507 . . . 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 5852 . . 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 2498 . 2  |-  ( M  e.  _V  ->  ( M LMHom  M )  =  (
Base `  A )
)
15 base0 14757 . . 3  |-  (/)  =  (
Base `  (/) )
16 reldmlmhm 17866 . . . 4  |-  Rel  dom LMHom
1716ovprc1 6301 . . 3  |-  ( -.  M  e.  _V  ->  ( M LMHom  M )  =  (/) )
18 fvprc 5842 . . . . 5  |-  ( -.  M  e.  _V  ->  (MEndo `  M )  =  (/) )
195, 18syl5eq 2507 . . . 4  |-  ( -.  M  e.  _V  ->  A  =  (/) )
2019fveq2d 5852 . . 3  |-  ( -.  M  e.  _V  ->  (
Base `  A )  =  ( Base `  (/) ) )
2115, 17, 203eqtr4a 2521 . 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 1398    e. wcel 1823   _Vcvv 3106    u. cun 3459   (/)c0 3783   {csn 4016   {cpr 4018   {ctp 4020   <.cop 4022    X. cxp 4986    o. ccom 4992   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272    oFcof 6511   ndxcnx 14713   Basecbs 14716   +g cplusg 14784   .rcmulr 14785  Scalarcsca 14787   .scvsca 14788   LMHom clmhm 17860  MEndocmend 31365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-plusg 14797  df-mulr 14798  df-sca 14800  df-vsca 14801  df-lmhm 17863  df-mend 31366
This theorem is referenced by:  mendplusgfval  31375  mendmulrfval  31377  mendvscafval  31380  mendring  31382  mendlmod  31383  mendassa  31384
  Copyright terms: Public domain W3C validator