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Theorem lflvsdi2a 29952
Description: Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 21-Oct-2014.)
Hypotheses
Ref Expression
lfldi.v  |-  V  =  ( Base `  W
)
lfldi.r  |-  R  =  (Scalar `  W )
lfldi.k  |-  K  =  ( Base `  R
)
lfldi.p  |-  .+  =  ( +g  `  R )
lfldi.t  |-  .x.  =  ( .r `  R )
lfldi.f  |-  F  =  (LFnl `  W )
lfldi.w  |-  ( ph  ->  W  e.  LMod )
lfldi.x  |-  ( ph  ->  X  e.  K )
lfldi2.y  |-  ( ph  ->  Y  e.  K )
lfldi2.g  |-  ( ph  ->  G  e.  F )
Assertion
Ref Expression
lflvsdi2a  |-  ( ph  ->  ( G  o F 
.x.  ( V  X.  { ( X  .+  Y ) } ) )  =  ( ( G  o F  .x.  ( V  X.  { X } ) )  o F  .+  ( G  o F  .x.  ( V  X.  { Y }
) ) ) )

Proof of Theorem lflvsdi2a
StepHypRef Expression
1 lfldi.v . . . . . 6  |-  V  =  ( Base `  W
)
2 fvex 5745 . . . . . 6  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2508 . . . . 5  |-  V  e. 
_V
43a1i 11 . . . 4  |-  ( ph  ->  V  e.  _V )
5 lfldi.x . . . 4  |-  ( ph  ->  X  e.  K )
6 lfldi2.y . . . 4  |-  ( ph  ->  Y  e.  K )
74, 5, 6ofc12 6332 . . 3  |-  ( ph  ->  ( ( V  X.  { X } )  o F  .+  ( V  X.  { Y }
) )  =  ( V  X.  { ( X  .+  Y ) } ) )
87oveq2d 6100 . 2  |-  ( ph  ->  ( G  o F 
.x.  ( ( V  X.  { X }
)  o F  .+  ( V  X.  { Y } ) ) )  =  ( G  o F  .x.  ( V  X.  { ( X  .+  Y ) } ) ) )
9 lfldi.r . . 3  |-  R  =  (Scalar `  W )
10 lfldi.k . . 3  |-  K  =  ( Base `  R
)
11 lfldi.p . . 3  |-  .+  =  ( +g  `  R )
12 lfldi.t . . 3  |-  .x.  =  ( .r `  R )
13 lfldi.f . . 3  |-  F  =  (LFnl `  W )
14 lfldi.w . . 3  |-  ( ph  ->  W  e.  LMod )
15 lfldi2.g . . 3  |-  ( ph  ->  G  e.  F )
161, 9, 10, 11, 12, 13, 14, 5, 6, 15lflvsdi2 29951 . 2  |-  ( ph  ->  ( G  o F 
.x.  ( ( V  X.  { X }
)  o F  .+  ( V  X.  { Y } ) ) )  =  ( ( G  o F  .x.  ( V  X.  { X }
) )  o F 
.+  ( G  o F  .x.  ( V  X.  { Y } ) ) ) )
178, 16eqtr3d 2472 1  |-  ( ph  ->  ( G  o F 
.x.  ( V  X.  { ( X  .+  Y ) } ) )  =  ( ( G  o F  .x.  ( V  X.  { X } ) )  o F  .+  ( G  o F  .x.  ( V  X.  { Y }
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816    X. cxp 4879   ` cfv 5457  (class class class)co 6084    o Fcof 6306   Basecbs 13474   +g cplusg 13534   .rcmulr 13535  Scalarcsca 13537   LModclmod 15955  LFnlclfn 29929
This theorem is referenced by:  ldualvsdi2  30016
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
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  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-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-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  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-map 7023  df-rng 15668  df-lmod 15957  df-lfl 29930
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