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Theorem ldualvsdi2 30016
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 2438 . . 3  |-  ( Base `  W )  =  (
Base `  W )
3 ldualvsdi2.r . . 3  |-  R  =  (Scalar `  W )
4 ldualvsdi2.k . . 3  |-  K  =  ( Base `  R
)
5 eqid 2438 . . 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 15966 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .+  Y )  e.  K )
138, 9, 10, 12syl3anc 1185 . . 3  |-  ( ph  ->  ( X  .+  Y
)  e.  K )
14 ldualvsdi2.g . . 3  |-  ( ph  ->  G  e.  F )
151, 2, 3, 4, 5, 6, 7, 8, 13, 14ldualvs 30009 . 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 29952 . 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 30011 . . . 4  |-  ( ph  ->  ( X  .x.  G
)  e.  F )
191, 3, 4, 6, 7, 8, 10, 14ldualvscl 30011 . . . 4  |-  ( ph  ->  ( Y  .x.  G
)  e.  F )
201, 3, 11, 6, 17, 8, 18, 19ldualvadd 30001 . . 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 30009 . . . 4  |-  ( ph  ->  ( X  .x.  G
)  =  ( G  o F ( .r
`  R ) ( ( Base `  W
)  X.  { X } ) ) )
221, 2, 3, 4, 5, 6, 7, 8, 10, 14ldualvs 30009 . . . 4  |-  ( ph  ->  ( Y  .x.  G
)  =  ( G  o F ( .r
`  R ) ( ( Base `  W
)  X.  { Y } ) ) )
2321, 22oveq12d 6102 . . 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 2471 . 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 2474 1  |-  ( ph  ->  ( ( X  .+  Y )  .x.  G
)  =  ( ( X  .x.  G ) 
.+b  ( Y  .x.  G ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   {csn 3816    X. cxp 4879   ` cfv 5457  (class class class)co 6084    o Fcof 6306   Basecbs 13474   +g cplusg 13534   .rcmulr 13535  Scalarcsca 13537   .scvsca 13538   LModclmod 15955  LFnlclfn 29929  LDualcld 29995
This theorem is referenced by:  lduallmodlem  30024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-plusg 13547  df-sca 13550  df-vsca 13551  df-mnd 14695  df-grp 14817  df-mgp 15654  df-rng 15668  df-lmod 15957  df-lfl 29930  df-ldual 29996
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