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Theorem ldualvsdi2 29393
Description: Reverse distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.)
Hypotheses
Ref Expression
ldualvsdi2.f  |-  F  =  (LFnl `  W )
ldualvsdi2.r  |-  R  =  (Scalar `  W )
ldualvsdi2.a  |-  .+  =  ( +g  `  R )
ldualvsdi2.k  |-  K  =  ( Base `  R
)
ldualvsdi2.d  |-  D  =  (LDual `  W )
ldualvsdi2.p  |-  .+b  =  ( +g  `  D )
ldualvsdi2.s  |-  .x.  =  ( .s `  D )
ldualvsdi2.w  |-  ( ph  ->  W  e.  LMod )
ldualvsdi2.x  |-  ( ph  ->  X  e.  K )
ldualvsdi2.y  |-  ( ph  ->  Y  e.  K )
ldualvsdi2.g  |-  ( ph  ->  G  e.  F )
Assertion
Ref Expression
ldualvsdi2  |-  ( ph  ->  ( ( X  .+  Y )  .x.  G
)  =  ( ( X  .x.  G ) 
.+b  ( Y  .x.  G ) ) )

Proof of Theorem ldualvsdi2
StepHypRef Expression
1 ldualvsdi2.f . . 3  |-  F  =  (LFnl `  W )
2 eqid 2366 . . 3  |-  ( Base `  W )  =  (
Base `  W )
3 ldualvsdi2.r . . 3  |-  R  =  (Scalar `  W )
4 ldualvsdi2.k . . 3  |-  K  =  ( Base `  R
)
5 eqid 2366 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
6 ldualvsdi2.d . . 3  |-  D  =  (LDual `  W )
7 ldualvsdi2.s . . 3  |-  .x.  =  ( .s `  D )
8 ldualvsdi2.w . . 3  |-  ( ph  ->  W  e.  LMod )
9 ldualvsdi2.x . . . 4  |-  ( ph  ->  X  e.  K )
10 ldualvsdi2.y . . . 4  |-  ( ph  ->  Y  e.  K )
11 ldualvsdi2.a . . . . 5  |-  .+  =  ( +g  `  R )
123, 4, 11lmodacl 15848 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .+  Y )  e.  K )
138, 9, 10, 12syl3anc 1183 . . 3  |-  ( ph  ->  ( X  .+  Y
)  e.  K )
14 ldualvsdi2.g . . 3  |-  ( ph  ->  G  e.  F )
151, 2, 3, 4, 5, 6, 7, 8, 13, 14ldualvs 29386 . 2  |-  ( ph  ->  ( ( X  .+  Y )  .x.  G
)  =  ( G  o F ( .r
`  R ) ( ( Base `  W
)  X.  { ( X  .+  Y ) } ) ) )
162, 3, 4, 11, 5, 1, 8, 9, 10, 14lflvsdi2a 29329 . 2  |-  ( ph  ->  ( G  o F ( .r `  R
) ( ( Base `  W )  X.  {
( X  .+  Y
) } ) )  =  ( ( G  o F ( .r
`  R ) ( ( Base `  W
)  X.  { X } ) )  o F  .+  ( G  o F ( .r
`  R ) ( ( Base `  W
)  X.  { Y } ) ) ) )
17 ldualvsdi2.p . . . 4  |-  .+b  =  ( +g  `  D )
181, 3, 4, 6, 7, 8, 9, 14ldualvscl 29388 . . . 4  |-  ( ph  ->  ( X  .x.  G
)  e.  F )
191, 3, 4, 6, 7, 8, 10, 14ldualvscl 29388 . . . 4  |-  ( ph  ->  ( Y  .x.  G
)  e.  F )
201, 3, 11, 6, 17, 8, 18, 19ldualvadd 29378 . . 3  |-  ( ph  ->  ( ( X  .x.  G )  .+b  ( Y  .x.  G ) )  =  ( ( X 
.x.  G )  o F  .+  ( Y 
.x.  G ) ) )
211, 2, 3, 4, 5, 6, 7, 8, 9, 14ldualvs 29386 . . . 4  |-  ( ph  ->  ( X  .x.  G
)  =  ( G  o F ( .r
`  R ) ( ( Base `  W
)  X.  { X } ) ) )
221, 2, 3, 4, 5, 6, 7, 8, 10, 14ldualvs 29386 . . . 4  |-  ( ph  ->  ( Y  .x.  G
)  =  ( G  o F ( .r
`  R ) ( ( Base `  W
)  X.  { Y } ) ) )
2321, 22oveq12d 5999 . . 3  |-  ( ph  ->  ( ( X  .x.  G )  o F 
.+  ( Y  .x.  G ) )  =  ( ( G  o F ( .r `  R ) ( (
Base `  W )  X.  { X } ) )  o F  .+  ( G  o F
( .r `  R
) ( ( Base `  W )  X.  { Y } ) ) ) )
2420, 23eqtr2d 2399 . 2  |-  ( ph  ->  ( ( G  o F ( .r `  R ) ( (
Base `  W )  X.  { X } ) )  o F  .+  ( G  o F
( .r `  R
) ( ( Base `  W )  X.  { Y } ) ) )  =  ( ( X 
.x.  G )  .+b  ( Y  .x.  G ) ) )
2515, 16, 243eqtrd 2402 1  |-  ( ph  ->  ( ( X  .+  Y )  .x.  G
)  =  ( ( X  .x.  G ) 
.+b  ( Y  .x.  G ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    e. wcel 1715   {csn 3729    X. cxp 4790   ` cfv 5358  (class class class)co 5981    o Fcof 6203   Basecbs 13356   +g cplusg 13416   .rcmulr 13417  Scalarcsca 13419   .scvsca 13420   LModclmod 15837  LFnlclfn 29306  LDualcld 29372
This theorem is referenced by:  lduallmodlem  29401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-plusg 13429  df-sca 13432  df-vsca 13433  df-mnd 14577  df-grp 14699  df-mgp 15536  df-rng 15550  df-lmod 15839  df-lfl 29307  df-ldual 29373
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