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Theorem ldualvbase 33800
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 2462 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2462 . . . 4  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
3 eqid 2462 . . . 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 2462 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
7 eqid 2462 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
8 eqid 2462 . . . 4  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
9 eqid 2462 . . . 4  |-  (oppr `  (Scalar `  W ) )  =  (oppr
`  (Scalar `  W )
)
10 eqid 2462 . . . 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 33799 . . 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 5863 . 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 5869 . . . 4  |-  (LFnl `  W )  e.  _V
164, 15eqeltri 2546 . . 3  |-  F  e. 
_V
17 eqid 2462 . . . 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 14611 . . 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 2528 1  |-  ( ph  ->  V  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   _Vcvv 3108    u. cun 3469   {csn 4022   {ctp 4026   <.cop 4028    X. cxp 4992    |` cres 4996   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279    oFcof 6515   ndxcnx 14478   Basecbs 14481   +g cplusg 14546   .rcmulr 14547  Scalarcsca 14549   .scvsca 14550  opprcoppr 17050  LFnlclfn 33731  LDualcld 33797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-plusg 14559  df-sca 14562  df-vsca 14563  df-ldual 33798
This theorem is referenced by:  ldualelvbase  33801  ldualgrplem  33819  lduallmodlem  33826  lclkr  36207  lclkrs  36213  lcfrvalsnN  36215  lcfrlem4  36219  lcfrlem5  36220  lcfrlem6  36221  lcfrlem16  36232  lcfr  36259  lcdvbase  36267  mapdunirnN  36324
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