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Theorem lflvsdi2a 29575
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 5709 . . . . . 6  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2482 . . . . 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 6296 . . 3  |-  ( ph  ->  ( ( V  X.  { X } )  o F  .+  ( V  X.  { Y }
) )  =  ( V  X.  { ( X  .+  Y ) } ) )
87oveq2d 6064 . 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 29574 . 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 2446 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 1649    e. wcel 1721   _Vcvv 2924   {csn 3782    X. cxp 4843   ` cfv 5421  (class class class)co 6048    o Fcof 6270   Basecbs 13432   +g cplusg 13492   .rcmulr 13493  Scalarcsca 13495   LModclmod 15913  LFnlclfn 29552
This theorem is referenced by:  ldualvsdi2  29639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-map 6987  df-rng 15626  df-lmod 15915  df-lfl 29553
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