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Theorem ldualvsdi1 29878
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 2435 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
3 ldualvsdi1.r . . . 4  |-  R  =  (Scalar `  W )
4 ldualvsdi1.k . . . 4  |-  K  =  ( Base `  R
)
5 eqid 2435 . . . 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 29872 . . 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 29872 . . 3  |-  ( ph  ->  ( X  .x.  H
)  =  ( H  o F ( .r
`  R ) ( ( Base `  W
)  X.  { X } ) ) )
1411, 13oveq12d 6091 . 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 2435 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
16 ldualvsdi1.p . . 3  |-  .+  =  ( +g  `  D )
171, 3, 4, 6, 7, 8, 9, 10ldualvscl 29874 . . 3  |-  ( ph  ->  ( X  .x.  G
)  e.  F )
181, 3, 4, 6, 7, 8, 9, 12ldualvscl 29874 . . 3  |-  ( ph  ->  ( X  .x.  H
)  e.  F )
191, 3, 15, 6, 16, 8, 17, 18ldualvadd 29864 . 2  |-  ( ph  ->  ( ( X  .x.  G )  .+  ( X  .x.  H ) )  =  ( ( X 
.x.  G )  o F ( +g  `  R
) ( X  .x.  H ) ) )
201, 6, 16, 8, 10, 12ldualvaddcl 29865 . . . 4  |-  ( ph  ->  ( G  .+  H
)  e.  F )
211, 2, 3, 4, 5, 6, 7, 8, 9, 20ldualvs 29872 . . 3  |-  ( ph  ->  ( X  .x.  ( G  .+  H ) )  =  ( ( G 
.+  H )  o F ( .r `  R ) ( (
Base `  W )  X.  { X } ) ) )
221, 3, 15, 6, 16, 8, 10, 12ldualvadd 29864 . . . 4  |-  ( ph  ->  ( G  .+  H
)  =  ( G  o F ( +g  `  R ) H ) )
2322oveq1d 6088 . . 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 29813 . . 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 2471 . 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 2478 1  |-  ( ph  ->  ( X  .x.  ( G  .+  H ) )  =  ( ( X 
.x.  G )  .+  ( X  .x.  H ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {csn 3806    X. cxp 4868   ` cfv 5446  (class class class)co 6073    o Fcof 6295   Basecbs 13461   +g cplusg 13521   .rcmulr 13522  Scalarcsca 13524   .scvsca 13525   LModclmod 15942  LFnlclfn 29792  LDualcld 29858
This theorem is referenced by:  lduallmodlem  29887
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-plusg 13534  df-sca 13537  df-vsca 13538  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-lmod 15944  df-lfl 29793  df-ldual 29859
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