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Theorem ldualfvadd 33081
Description: Vector addition in the dual of a vector space. (Contributed by NM, 21-Oct-2014.)
Hypotheses
Ref Expression
ldualvadd.f  |-  F  =  (LFnl `  W )
ldualvadd.r  |-  R  =  (Scalar `  W )
ldualvadd.a  |-  .+  =  ( +g  `  R )
ldualvadd.d  |-  D  =  (LDual `  W )
ldualvadd.p  |-  .+b  =  ( +g  `  D )
ldualvadd.w  |-  ( ph  ->  W  e.  X )
ldualfvadd.q  |-  .+^  =  (  oF  .+  |`  ( F  X.  F ) )
Assertion
Ref Expression
ldualfvadd  |-  ( ph  -> 
.+b  =  .+^  )

Proof of Theorem ldualfvadd
Dummy variables  f 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2451 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 ldualvadd.a . . . 4  |-  .+  =  ( +g  `  R )
3 ldualfvadd.q . . . 4  |-  .+^  =  (  oF  .+  |`  ( F  X.  F ) )
4 ldualvadd.f . . . 4  |-  F  =  (LFnl `  W )
5 ldualvadd.d . . . 4  |-  D  =  (LDual `  W )
6 ldualvadd.r . . . 4  |-  R  =  (Scalar `  W )
7 eqid 2451 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
8 eqid 2451 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
9 eqid 2451 . . . 4  |-  (oppr `  R
)  =  (oppr `  R
)
10 eqid 2451 . . . 4  |-  ( k  e.  ( Base `  R
) ,  f  e.  F  |->  ( f  oF ( .r `  R ) ( (
Base `  W )  X.  { k } ) ) )  =  ( k  e.  ( Base `  R ) ,  f  e.  F  |->  ( f  oF ( .r
`  R ) ( ( Base `  W
)  X.  { k } ) ) )
11 ldualvadd.w . . . 4  |-  ( ph  ->  W  e.  X )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11ldualset 33078 . . 3  |-  ( ph  ->  D  =  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  .+^  >. ,  <. (Scalar `  ndx ) ,  (oppr `  R ) >. }  u.  {
<. ( .s `  ndx ) ,  ( k  e.  ( Base `  R
) ,  f  e.  F  |->  ( f  oF ( .r `  R ) ( (
Base `  W )  X.  { k } ) ) ) >. } ) )
1312fveq2d 5795 . 2  |-  ( ph  ->  ( +g  `  D
)  =  ( +g  `  ( { <. ( Base `  ndx ) ,  F >. ,  <. ( +g  `  ndx ) , 
.+^  >. ,  <. (Scalar ` 
ndx ) ,  (oppr `  R ) >. }  u.  {
<. ( .s `  ndx ) ,  ( k  e.  ( Base `  R
) ,  f  e.  F  |->  ( f  oF ( .r `  R ) ( (
Base `  W )  X.  { k } ) ) ) >. } ) ) )
14 ldualvadd.p . 2  |-  .+b  =  ( +g  `  D )
15 fvex 5801 . . . . . 6  |-  (LFnl `  W )  e.  _V
164, 15eqeltri 2535 . . . . 5  |-  F  e. 
_V
17 id 22 . . . . . 6  |-  ( F  e.  _V  ->  F  e.  _V )
1817, 17ofmresex 6676 . . . . 5  |-  ( F  e.  _V  ->  (  oF  .+  |`  ( F  X.  F ) )  e.  _V )
1916, 18ax-mp 5 . . . 4  |-  (  oF  .+  |`  ( F  X.  F ) )  e.  _V
203, 19eqeltri 2535 . . 3  |-  .+^  e.  _V
21 eqid 2451 . . . 4  |-  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  .+^  >. ,  <. (Scalar `  ndx ) ,  (oppr `  R ) >. }  u.  {
<. ( .s `  ndx ) ,  ( k  e.  ( Base `  R
) ,  f  e.  F  |->  ( f  oF ( .r `  R ) ( (
Base `  W )  X.  { k } ) ) ) >. } )  =  ( { <. (
Base `  ndx ) ,  F >. ,  <. ( +g  `  ndx ) , 
.+^  >. ,  <. (Scalar ` 
ndx ) ,  (oppr `  R ) >. }  u.  {
<. ( .s `  ndx ) ,  ( k  e.  ( Base `  R
) ,  f  e.  F  |->  ( f  oF ( .r `  R ) ( (
Base `  W )  X.  { k } ) ) ) >. } )
2221lmodplusg 14408 . . 3  |-  (  .+^  e.  _V  ->  .+^  =  ( +g  `  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  .+^  >. ,  <. (Scalar `  ndx ) ,  (oppr `  R ) >. }  u.  {
<. ( .s `  ndx ) ,  ( k  e.  ( Base `  R
) ,  f  e.  F  |->  ( f  oF ( .r `  R ) ( (
Base `  W )  X.  { k } ) ) ) >. } ) ) )
2320, 22ax-mp 5 . 2  |-  .+^  =  ( +g  `  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  .+^  >. ,  <. (Scalar `  ndx ) ,  (oppr `  R ) >. }  u.  {
<. ( .s `  ndx ) ,  ( k  e.  ( Base `  R
) ,  f  e.  F  |->  ( f  oF ( .r `  R ) ( (
Base `  W )  X.  { k } ) ) ) >. } ) )
2413, 14, 233eqtr4g 2517 1  |-  ( ph  -> 
.+b  =  .+^  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3070    u. cun 3426   {csn 3977   {ctp 3981   <.cop 3983    X. cxp 4938    |` cres 4942   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194    oFcof 6420   ndxcnx 14275   Basecbs 14278   +g cplusg 14342   .rcmulr 14343  Scalarcsca 14345   .scvsca 14346  opprcoppr 16822  LFnlclfn 33010  LDualcld 33076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-of 6422  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-plusg 14355  df-sca 14358  df-vsca 14359  df-ldual 33077
This theorem is referenced by:  ldualvadd  33082
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