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Theorem ldualfvadd 29926
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  |-  .+^  =  (  o F  .+  |`  ( 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 2436 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 ldualvadd.a . . . 4  |-  .+  =  ( +g  `  R )
3 ldualfvadd.q . . . 4  |-  .+^  =  (  o F  .+  |`  ( 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 2436 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
8 eqid 2436 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
9 eqid 2436 . . . 4  |-  (oppr `  R
)  =  (oppr `  R
)
10 eqid 2436 . . . 4  |-  ( k  e.  ( Base `  R
) ,  f  e.  F  |->  ( f  o F ( .r `  R ) ( (
Base `  W )  X.  { k } ) ) )  =  ( k  e.  ( Base `  R ) ,  f  e.  F  |->  ( f  o F ( .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 29923 . . 3  |-  ( ph  ->  D  =  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  .+^  >. ,  <. (Scalar `  ndx ) ,  (oppr `  R ) >. }  u.  {
<. ( .s `  ndx ) ,  ( k  e.  ( Base `  R
) ,  f  e.  F  |->  ( f  o F ( .r `  R ) ( (
Base `  W )  X.  { k } ) ) ) >. } ) )
1312fveq2d 5732 . 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  o F ( .r `  R ) ( (
Base `  W )  X.  { k } ) ) ) >. } ) ) )
14 ldualvadd.p . 2  |-  .+b  =  ( +g  `  D )
15 fvex 5742 . . . . . 6  |-  (LFnl `  W )  e.  _V
164, 15eqeltri 2506 . . . . 5  |-  F  e. 
_V
17 id 20 . . . . . 6  |-  ( F  e.  _V  ->  F  e.  _V )
1817, 17ofmresex 6345 . . . . 5  |-  ( F  e.  _V  ->  (  o F  .+  |`  ( F  X.  F ) )  e.  _V )
1916, 18ax-mp 8 . . . 4  |-  (  o F  .+  |`  ( F  X.  F ) )  e.  _V
203, 19eqeltri 2506 . . 3  |-  .+^  e.  _V
21 eqid 2436 . . . 4  |-  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  .+^  >. ,  <. (Scalar `  ndx ) ,  (oppr `  R ) >. }  u.  {
<. ( .s `  ndx ) ,  ( k  e.  ( Base `  R
) ,  f  e.  F  |->  ( f  o F ( .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  o F ( .r `  R ) ( (
Base `  W )  X.  { k } ) ) ) >. } )
2221lmodplusg 13595 . . 3  |-  (  .+^  e.  _V  ->  .+^  =  ( +g  `  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  .+^  >. ,  <. (Scalar `  ndx ) ,  (oppr `  R ) >. }  u.  {
<. ( .s `  ndx ) ,  ( k  e.  ( Base `  R
) ,  f  e.  F  |->  ( f  o F ( .r `  R ) ( (
Base `  W )  X.  { k } ) ) ) >. } ) ) )
2320, 22ax-mp 8 . 2  |-  .+^  =  ( +g  `  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  .+^  >. ,  <. (Scalar `  ndx ) ,  (oppr `  R ) >. }  u.  {
<. ( .s `  ndx ) ,  ( k  e.  ( Base `  R
) ,  f  e.  F  |->  ( f  o F ( .r `  R ) ( (
Base `  W )  X.  { k } ) ) ) >. } ) )
2413, 14, 233eqtr4g 2493 1  |-  ( ph  -> 
.+b  =  .+^  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2956    u. cun 3318   {csn 3814   {ctp 3816   <.cop 3817    X. cxp 4876    |` cres 4880   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083    o Fcof 6303   ndxcnx 13466   Basecbs 13469   +g cplusg 13529   .rcmulr 13530  Scalarcsca 13532   .scvsca 13533  opprcoppr 15727  LFnlclfn 29855  LDualcld 29921
This theorem is referenced by:  ldualvadd  29927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-sca 13545  df-vsca 13546  df-ldual 29922
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