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Theorem lflvsdi2a 29270
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 5539 . . . . . 6  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2353 . . . . 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 6102 . . 3  |-  ( ph  ->  ( ( V  X.  { X } )  o F  .+  ( V  X.  { Y }
) )  =  ( V  X.  { ( X  .+  Y ) } ) )
87oveq2d 5874 . 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 29269 . 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 2317 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 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    X. cxp 4687   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   LModclmod 15627  LFnlclfn 29247
This theorem is referenced by:  ldualvsdi2  29334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-map 6774  df-rng 15340  df-lmod 15629  df-lfl 29248
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