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Theorem ldualvbase 35264
Description: The vectors of a dual space are functionals of the original space. (Contributed by NM, 18-Oct-2014.)
Hypotheses
Ref Expression
ldualvbase.f  |-  F  =  (LFnl `  W )
ldualvbase.d  |-  D  =  (LDual `  W )
ldualvbase.v  |-  V  =  ( Base `  D
)
ldualvbase.w  |-  ( ph  ->  W  e.  X )
Assertion
Ref Expression
ldualvbase  |-  ( ph  ->  V  =  F )

Proof of Theorem ldualvbase
Dummy variables  f 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2382 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2382 . . . 4  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
3 eqid 2382 . . . 4  |-  (  oF ( +g  `  (Scalar `  W ) )  |`  ( F  X.  F
) )  =  (  oF ( +g  `  (Scalar `  W )
)  |`  ( F  X.  F ) )
4 ldualvbase.f . . . 4  |-  F  =  (LFnl `  W )
5 ldualvbase.d . . . 4  |-  D  =  (LDual `  W )
6 eqid 2382 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
7 eqid 2382 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
8 eqid 2382 . . . 4  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
9 eqid 2382 . . . 4  |-  (oppr `  (Scalar `  W ) )  =  (oppr
`  (Scalar `  W )
)
10 eqid 2382 . . . 4  |-  ( k  e.  ( Base `  (Scalar `  W ) ) ,  f  e.  F  |->  ( f  oF ( .r `  (Scalar `  W ) ) ( ( Base `  W
)  X.  { k } ) ) )  =  ( k  e.  ( Base `  (Scalar `  W ) ) ,  f  e.  F  |->  ( f  oF ( .r `  (Scalar `  W ) ) ( ( Base `  W
)  X.  { k } ) ) )
11 ldualvbase.w . . . 4  |-  ( ph  ->  W  e.  X )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11ldualset 35263 . . 3  |-  ( ph  ->  D  =  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  (  oF ( +g  `  (Scalar `  W ) )  |`  ( F  X.  F
) ) >. ,  <. (Scalar `  ndx ) ,  (oppr `  (Scalar `  W ) )
>. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  ( Base `  (Scalar `  W ) ) ,  f  e.  F  |->  ( f  oF ( .r `  (Scalar `  W ) ) ( ( Base `  W
)  X.  { k } ) ) )
>. } ) )
1312fveq2d 5778 . 2  |-  ( ph  ->  ( Base `  D
)  =  ( Base `  ( { <. ( Base `  ndx ) ,  F >. ,  <. ( +g  `  ndx ) ,  (  oF ( +g  `  (Scalar `  W ) )  |`  ( F  X.  F
) ) >. ,  <. (Scalar `  ndx ) ,  (oppr `  (Scalar `  W ) )
>. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  ( Base `  (Scalar `  W ) ) ,  f  e.  F  |->  ( f  oF ( .r `  (Scalar `  W ) ) ( ( Base `  W
)  X.  { k } ) ) )
>. } ) ) )
14 ldualvbase.v . 2  |-  V  =  ( Base `  D
)
15 fvex 5784 . . . 4  |-  (LFnl `  W )  e.  _V
164, 15eqeltri 2466 . . 3  |-  F  e. 
_V
17 eqid 2382 . . . 4  |-  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  (  oF ( +g  `  (Scalar `  W ) )  |`  ( F  X.  F
) ) >. ,  <. (Scalar `  ndx ) ,  (oppr `  (Scalar `  W ) )
>. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  ( Base `  (Scalar `  W ) ) ,  f  e.  F  |->  ( f  oF ( .r `  (Scalar `  W ) ) ( ( Base `  W
)  X.  { k } ) ) )
>. } )  =  ( { <. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  (  oF ( +g  `  (Scalar `  W ) )  |`  ( F  X.  F
) ) >. ,  <. (Scalar `  ndx ) ,  (oppr `  (Scalar `  W ) )
>. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  ( Base `  (Scalar `  W ) ) ,  f  e.  F  |->  ( f  oF ( .r `  (Scalar `  W ) ) ( ( Base `  W
)  X.  { k } ) ) )
>. } )
1817lmodbase 14771 . . 3  |-  ( F  e.  _V  ->  F  =  ( Base `  ( { <. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  (  oF ( +g  `  (Scalar `  W ) )  |`  ( F  X.  F
) ) >. ,  <. (Scalar `  ndx ) ,  (oppr `  (Scalar `  W ) )
>. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  ( Base `  (Scalar `  W ) ) ,  f  e.  F  |->  ( f  oF ( .r `  (Scalar `  W ) ) ( ( Base `  W
)  X.  { k } ) ) )
>. } ) ) )
1916, 18ax-mp 5 . 2  |-  F  =  ( Base `  ( { <. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  (  oF ( +g  `  (Scalar `  W ) )  |`  ( F  X.  F
) ) >. ,  <. (Scalar `  ndx ) ,  (oppr `  (Scalar `  W ) )
>. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  ( Base `  (Scalar `  W ) ) ,  f  e.  F  |->  ( f  oF ( .r `  (Scalar `  W ) ) ( ( Base `  W
)  X.  { k } ) ) )
>. } ) )
2013, 14, 193eqtr4g 2448 1  |-  ( ph  ->  V  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1826   _Vcvv 3034    u. cun 3387   {csn 3944   {ctp 3948   <.cop 3950    X. cxp 4911    |` cres 4915   ` cfv 5496  (class class class)co 6196    |-> cmpt2 6198    oFcof 6437   ndxcnx 14631   Basecbs 14634   +g cplusg 14702   .rcmulr 14703  Scalarcsca 14705   .scvsca 14706  opprcoppr 17384  LFnlclfn 35195  LDualcld 35261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-plusg 14715  df-sca 14718  df-vsca 14719  df-ldual 35262
This theorem is referenced by:  ldualelvbase  35265  ldualgrplem  35283  lduallmodlem  35290  lclkr  37673  lclkrs  37679  lcfrvalsnN  37681  lcfrlem4  37685  lcfrlem5  37686  lcfrlem6  37687  lcfrlem16  37698  lcfr  37725  lcdvbase  37733  mapdunirnN  37790
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