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Theorem ldualvsdi1 29258
Description: Distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.)
Hypotheses
Ref Expression
ldualvsdi1.f  |-  F  =  (LFnl `  W )
ldualvsdi1.r  |-  R  =  (Scalar `  W )
ldualvsdi1.k  |-  K  =  ( Base `  R
)
ldualvsdi1.d  |-  D  =  (LDual `  W )
ldualvsdi1.p  |-  .+  =  ( +g  `  D )
ldualvsdi1.s  |-  .x.  =  ( .s `  D )
ldualvsdi1.w  |-  ( ph  ->  W  e.  LMod )
ldualvsdi1.x  |-  ( ph  ->  X  e.  K )
ldualvsdi1.g  |-  ( ph  ->  G  e.  F )
ldualvsdi1.h  |-  ( ph  ->  H  e.  F )
Assertion
Ref Expression
ldualvsdi1  |-  ( ph  ->  ( X  .x.  ( G  .+  H ) )  =  ( ( X 
.x.  G )  .+  ( X  .x.  H ) ) )

Proof of Theorem ldualvsdi1
StepHypRef Expression
1 ldualvsdi1.f . . . 4  |-  F  =  (LFnl `  W )
2 eqid 2387 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
3 ldualvsdi1.r . . . 4  |-  R  =  (Scalar `  W )
4 ldualvsdi1.k . . . 4  |-  K  =  ( Base `  R
)
5 eqid 2387 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
6 ldualvsdi1.d . . . 4  |-  D  =  (LDual `  W )
7 ldualvsdi1.s . . . 4  |-  .x.  =  ( .s `  D )
8 ldualvsdi1.w . . . 4  |-  ( ph  ->  W  e.  LMod )
9 ldualvsdi1.x . . . 4  |-  ( ph  ->  X  e.  K )
10 ldualvsdi1.g . . . 4  |-  ( ph  ->  G  e.  F )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10ldualvs 29252 . . 3  |-  ( ph  ->  ( X  .x.  G
)  =  ( G  o F ( .r
`  R ) ( ( Base `  W
)  X.  { X } ) ) )
12 ldualvsdi1.h . . . 4  |-  ( ph  ->  H  e.  F )
131, 2, 3, 4, 5, 6, 7, 8, 9, 12ldualvs 29252 . . 3  |-  ( ph  ->  ( X  .x.  H
)  =  ( H  o F ( .r
`  R ) ( ( Base `  W
)  X.  { X } ) ) )
1411, 13oveq12d 6038 . 2  |-  ( ph  ->  ( ( X  .x.  G )  o F ( +g  `  R
) ( X  .x.  H ) )  =  ( ( G  o F ( .r `  R ) ( (
Base `  W )  X.  { X } ) )  o F ( +g  `  R ) ( H  o F ( .r `  R
) ( ( Base `  W )  X.  { X } ) ) ) )
15 eqid 2387 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
16 ldualvsdi1.p . . 3  |-  .+  =  ( +g  `  D )
171, 3, 4, 6, 7, 8, 9, 10ldualvscl 29254 . . 3  |-  ( ph  ->  ( X  .x.  G
)  e.  F )
181, 3, 4, 6, 7, 8, 9, 12ldualvscl 29254 . . 3  |-  ( ph  ->  ( X  .x.  H
)  e.  F )
191, 3, 15, 6, 16, 8, 17, 18ldualvadd 29244 . 2  |-  ( ph  ->  ( ( X  .x.  G )  .+  ( X  .x.  H ) )  =  ( ( X 
.x.  G )  o F ( +g  `  R
) ( X  .x.  H ) ) )
201, 6, 16, 8, 10, 12ldualvaddcl 29245 . . . 4  |-  ( ph  ->  ( G  .+  H
)  e.  F )
211, 2, 3, 4, 5, 6, 7, 8, 9, 20ldualvs 29252 . . 3  |-  ( ph  ->  ( X  .x.  ( G  .+  H ) )  =  ( ( G 
.+  H )  o F ( .r `  R ) ( (
Base `  W )  X.  { X } ) ) )
221, 3, 15, 6, 16, 8, 10, 12ldualvadd 29244 . . . 4  |-  ( ph  ->  ( G  .+  H
)  =  ( G  o F ( +g  `  R ) H ) )
2322oveq1d 6035 . . 3  |-  ( ph  ->  ( ( G  .+  H )  o F ( .r `  R
) ( ( Base `  W )  X.  { X } ) )  =  ( ( G  o F ( +g  `  R
) H )  o F ( .r `  R ) ( (
Base `  W )  X.  { X } ) ) )
242, 3, 4, 15, 5, 1, 8, 9, 10, 12lflvsdi1 29193 . . 3  |-  ( ph  ->  ( ( G  o F ( +g  `  R
) H )  o F ( .r `  R ) ( (
Base `  W )  X.  { X } ) )  =  ( ( G  o F ( .r `  R ) ( ( Base `  W
)  X.  { X } ) )  o F ( +g  `  R
) ( H  o F ( .r `  R ) ( (
Base `  W )  X.  { X } ) ) ) )
2521, 23, 243eqtrd 2423 . 2  |-  ( ph  ->  ( X  .x.  ( G  .+  H ) )  =  ( ( G  o F ( .r
`  R ) ( ( Base `  W
)  X.  { X } ) )  o F ( +g  `  R
) ( H  o F ( .r `  R ) ( (
Base `  W )  X.  { X } ) ) ) )
2614, 19, 253eqtr4rd 2430 1  |-  ( ph  ->  ( X  .x.  ( G  .+  H ) )  =  ( ( X 
.x.  G )  .+  ( X  .x.  H ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   {csn 3757    X. cxp 4816   ` cfv 5394  (class class class)co 6020    o Fcof 6242   Basecbs 13396   +g cplusg 13456   .rcmulr 13457  Scalarcsca 13459   .scvsca 13460   LModclmod 15877  LFnlclfn 29172  LDualcld 29238
This theorem is referenced by:  lduallmodlem  29267
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-rep 4261  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-rmo 2657  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-map 6956  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-sets 13402  df-plusg 13469  df-sca 13472  df-vsca 13473  df-0g 13654  df-mnd 14617  df-grp 14739  df-minusg 14740  df-sbg 14741  df-cmn 15341  df-abl 15342  df-mgp 15576  df-rng 15590  df-ur 15592  df-lmod 15879  df-lfl 29173  df-ldual 29239
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