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Theorem ldualsca 29247
Description: The ring of scalars of the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
Hypotheses
Ref Expression
ldualsca.f  |-  F  =  (Scalar `  W )
ldualsca.o  |-  O  =  (oppr
`  F )
ldualsca.d  |-  D  =  (LDual `  W )
ldualsca.r  |-  R  =  (Scalar `  D )
ldualsca.w  |-  ( ph  ->  W  e.  X )
Assertion
Ref Expression
ldualsca  |-  ( ph  ->  R  =  O )

Proof of Theorem ldualsca
Dummy variables  f 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2387 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2387 . . . 4  |-  ( +g  `  F )  =  ( +g  `  F )
3 eqid 2387 . . . 4  |-  (  o F ( +g  `  F
)  |`  ( (LFnl `  W )  X.  (LFnl `  W ) ) )  =  (  o F ( +g  `  F
)  |`  ( (LFnl `  W )  X.  (LFnl `  W ) ) )
4 eqid 2387 . . . 4  |-  (LFnl `  W )  =  (LFnl `  W )
5 ldualsca.d . . . 4  |-  D  =  (LDual `  W )
6 ldualsca.f . . . 4  |-  F  =  (Scalar `  W )
7 eqid 2387 . . . 4  |-  ( Base `  F )  =  (
Base `  F )
8 eqid 2387 . . . 4  |-  ( .r
`  F )  =  ( .r `  F
)
9 ldualsca.o . . . 4  |-  O  =  (oppr
`  F )
10 eqid 2387 . . . 4  |-  ( k  e.  ( Base `  F
) ,  f  e.  (LFnl `  W )  |->  ( f  o F ( .r `  F
) ( ( Base `  W )  X.  {
k } ) ) )  =  ( k  e.  ( Base `  F
) ,  f  e.  (LFnl `  W )  |->  ( f  o F ( .r `  F
) ( ( Base `  W )  X.  {
k } ) ) )
11 ldualsca.w . . . 4  |-  ( ph  ->  W  e.  X )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11ldualset 29240 . . 3  |-  ( ph  ->  D  =  ( {
<. ( Base `  ndx ) ,  (LFnl `  W
) >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  F )  |`  ( (LFnl `  W
)  X.  (LFnl `  W ) ) )
>. ,  <. (Scalar `  ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  (
Base `  F ) ,  f  e.  (LFnl `  W )  |->  ( f  o F ( .r
`  F ) ( ( Base `  W
)  X.  { k } ) ) )
>. } ) )
1312fveq2d 5672 . 2  |-  ( ph  ->  (Scalar `  D )  =  (Scalar `  ( { <. ( Base `  ndx ) ,  (LFnl `  W
) >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  F )  |`  ( (LFnl `  W
)  X.  (LFnl `  W ) ) )
>. ,  <. (Scalar `  ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  (
Base `  F ) ,  f  e.  (LFnl `  W )  |->  ( f  o F ( .r
`  F ) ( ( Base `  W
)  X.  { k } ) ) )
>. } ) ) )
14 ldualsca.r . 2  |-  R  =  (Scalar `  D )
15 fvex 5682 . . . 4  |-  (oppr `  F
)  e.  _V
169, 15eqeltri 2457 . . 3  |-  O  e. 
_V
17 eqid 2387 . . . 4  |-  ( {
<. ( Base `  ndx ) ,  (LFnl `  W
) >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  F )  |`  ( (LFnl `  W
)  X.  (LFnl `  W ) ) )
>. ,  <. (Scalar `  ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  (
Base `  F ) ,  f  e.  (LFnl `  W )  |->  ( f  o F ( .r
`  F ) ( ( Base `  W
)  X.  { k } ) ) )
>. } )  =  ( { <. ( Base `  ndx ) ,  (LFnl `  W
) >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  F )  |`  ( (LFnl `  W
)  X.  (LFnl `  W ) ) )
>. ,  <. (Scalar `  ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  (
Base `  F ) ,  f  e.  (LFnl `  W )  |->  ( f  o F ( .r
`  F ) ( ( Base `  W
)  X.  { k } ) ) )
>. } )
1817lmodsca 13523 . . 3  |-  ( O  e.  _V  ->  O  =  (Scalar `  ( { <. ( Base `  ndx ) ,  (LFnl `  W
) >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  F )  |`  ( (LFnl `  W
)  X.  (LFnl `  W ) ) )
>. ,  <. (Scalar `  ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  (
Base `  F ) ,  f  e.  (LFnl `  W )  |->  ( f  o F ( .r
`  F ) ( ( Base `  W
)  X.  { k } ) ) )
>. } ) ) )
1916, 18ax-mp 8 . 2  |-  O  =  (Scalar `  ( { <. ( Base `  ndx ) ,  (LFnl `  W
) >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  F )  |`  ( (LFnl `  W
)  X.  (LFnl `  W ) ) )
>. ,  <. (Scalar `  ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  (
Base `  F ) ,  f  e.  (LFnl `  W )  |->  ( f  o F ( .r
`  F ) ( ( Base `  W
)  X.  { k } ) ) )
>. } ) )
2013, 14, 193eqtr4g 2444 1  |-  ( ph  ->  R  =  O )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2899    u. cun 3261   {csn 3757   {ctp 3759   <.cop 3760    X. cxp 4816    |` cres 4820   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022    o Fcof 6242   ndxcnx 13393   Basecbs 13396   +g cplusg 13456   .rcmulr 13457  Scalarcsca 13459   .scvsca 13460  opprcoppr 15654  LFnlclfn 29172  LDualcld 29238
This theorem is referenced by:  ldualsbase  29248  ldualsaddN  29249  ldualsmul  29250  ldual0  29262  ldual1  29263  ldualneg  29264  lduallmodlem  29267  lduallvec  29269  ldualvsub  29270  lcdsca  31714
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-plusg 13469  df-sca 13472  df-vsca 13473  df-ldual 29239
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