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Theorem lflvsdi2a 29341
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 5646 . . . . . 6  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2436 . . . . 5  |-  V  e. 
_V
43a1i 10 . . . 4  |-  ( ph  ->  V  e.  _V )
5 lfldi.x . . . 4  |-  ( ph  ->  X  e.  K )
6 lfldi2.y . . . 4  |-  ( ph  ->  Y  e.  K )
74, 5, 6ofc12 6229 . . 3  |-  ( ph  ->  ( ( V  X.  { X } )  o F  .+  ( V  X.  { Y }
) )  =  ( V  X.  { ( X  .+  Y ) } ) )
87oveq2d 5997 . 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 29340 . 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 2400 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 1647    e. wcel 1715   _Vcvv 2873   {csn 3729    X. cxp 4790   ` cfv 5358  (class class class)co 5981    o Fcof 6203   Basecbs 13356   +g cplusg 13416   .rcmulr 13417  Scalarcsca 13419   LModclmod 15837  LFnlclfn 29318
This theorem is referenced by:  ldualvsdi2  29405
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-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-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  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-map 6917  df-rng 15550  df-lmod 15839  df-lfl 29319
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