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Theorem ldualvsdi1 29402
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 2358 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
3 ldualvsdi1.r . . . 4  |-  R  =  (Scalar `  W )
4 ldualvsdi1.k . . . 4  |-  K  =  ( Base `  R
)
5 eqid 2358 . . . 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 29396 . . 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 29396 . . 3  |-  ( ph  ->  ( X  .x.  H
)  =  ( H  o F ( .r
`  R ) ( ( Base `  W
)  X.  { X } ) ) )
1411, 13oveq12d 5963 . 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 2358 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
16 ldualvsdi1.p . . 3  |-  .+  =  ( +g  `  D )
171, 3, 4, 6, 7, 8, 9, 10ldualvscl 29398 . . 3  |-  ( ph  ->  ( X  .x.  G
)  e.  F )
181, 3, 4, 6, 7, 8, 9, 12ldualvscl 29398 . . 3  |-  ( ph  ->  ( X  .x.  H
)  e.  F )
191, 3, 15, 6, 16, 8, 17, 18ldualvadd 29388 . 2  |-  ( ph  ->  ( ( X  .x.  G )  .+  ( X  .x.  H ) )  =  ( ( X 
.x.  G )  o F ( +g  `  R
) ( X  .x.  H ) ) )
201, 6, 16, 8, 10, 12ldualvaddcl 29389 . . . 4  |-  ( ph  ->  ( G  .+  H
)  e.  F )
211, 2, 3, 4, 5, 6, 7, 8, 9, 20ldualvs 29396 . . 3  |-  ( ph  ->  ( X  .x.  ( G  .+  H ) )  =  ( ( G 
.+  H )  o F ( .r `  R ) ( (
Base `  W )  X.  { X } ) ) )
221, 3, 15, 6, 16, 8, 10, 12ldualvadd 29388 . . . 4  |-  ( ph  ->  ( G  .+  H
)  =  ( G  o F ( +g  `  R ) H ) )
2322oveq1d 5960 . . 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 29337 . . 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 2394 . 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 2401 1  |-  ( ph  ->  ( X  .x.  ( G  .+  H ) )  =  ( ( X 
.x.  G )  .+  ( X  .x.  H ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710   {csn 3716    X. cxp 4769   ` cfv 5337  (class class class)co 5945    o Fcof 6163   Basecbs 13245   +g cplusg 13305   .rcmulr 13306  Scalarcsca 13308   .scvsca 13309   LModclmod 15726  LFnlclfn 29316  LDualcld 29382
This theorem is referenced by:  lduallmodlem  29411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-n0 10058  df-z 10117  df-uz 10323  df-fz 10875  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-plusg 13318  df-sca 13321  df-vsca 13322  df-0g 13503  df-mnd 14466  df-grp 14588  df-minusg 14589  df-sbg 14590  df-cmn 15190  df-abl 15191  df-mgp 15425  df-rng 15439  df-ur 15441  df-lmod 15728  df-lfl 29317  df-ldual 29383
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