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Theorem lflvsdi2 29877
Description: Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-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
lflvsdi2  |-  ( 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 } ) ) ) )

Proof of Theorem lflvsdi2
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lfldi.v . . . 4  |-  V  =  ( Base `  W
)
2 fvex 5742 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2506 . . 3  |-  V  e. 
_V
43a1i 11 . 2  |-  ( ph  ->  V  e.  _V )
5 lfldi.w . . 3  |-  ( ph  ->  W  e.  LMod )
6 lfldi2.g . . 3  |-  ( ph  ->  G  e.  F )
7 lfldi.r . . . 4  |-  R  =  (Scalar `  W )
8 lfldi.k . . . 4  |-  K  =  ( Base `  R
)
9 lfldi.f . . . 4  |-  F  =  (LFnl `  W )
107, 8, 1, 9lflf 29861 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> K )
115, 6, 10syl2anc 643 . 2  |-  ( ph  ->  G : V --> K )
12 lfldi.x . . 3  |-  ( ph  ->  X  e.  K )
13 fconst6g 5632 . . 3  |-  ( X  e.  K  ->  ( V  X.  { X }
) : V --> K )
1412, 13syl 16 . 2  |-  ( ph  ->  ( V  X.  { X } ) : V --> K )
15 lfldi2.y . . 3  |-  ( ph  ->  Y  e.  K )
16 fconst6g 5632 . . 3  |-  ( Y  e.  K  ->  ( V  X.  { Y }
) : V --> K )
1715, 16syl 16 . 2  |-  ( ph  ->  ( V  X.  { Y } ) : V --> K )
187lmodrng 15958 . . . 4  |-  ( W  e.  LMod  ->  R  e. 
Ring )
195, 18syl 16 . . 3  |-  ( ph  ->  R  e.  Ring )
20 lfldi.p . . . 4  |-  .+  =  ( +g  `  R )
21 lfldi.t . . . 4  |-  .x.  =  ( .r `  R )
228, 20, 21rngdi 15682 . . 3  |-  ( ( R  e.  Ring  /\  (
x  e.  K  /\  y  e.  K  /\  z  e.  K )
)  ->  ( x  .x.  ( y  .+  z
) )  =  ( ( x  .x.  y
)  .+  ( x  .x.  z ) ) )
2319, 22sylan 458 . 2  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K  /\  z  e.  K ) )  -> 
( x  .x.  (
y  .+  z )
)  =  ( ( x  .x.  y ) 
.+  ( x  .x.  z ) ) )
244, 11, 14, 17, 23caofdi 6340 1  |-  ( 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 } ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2956   {csn 3814    X. cxp 4876   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303   Basecbs 13469   +g cplusg 13529   .rcmulr 13530  Scalarcsca 13532   Ringcrg 15660   LModclmod 15950  LFnlclfn 29855
This theorem is referenced by:  lflvsdi2a  29878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-map 7020  df-rng 15663  df-lmod 15952  df-lfl 29856
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