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Theorem lflvsdi2 32097
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  oF  .x.  ( ( V  X.  { X }
)  oF  .+  ( V  X.  { Y } ) ) )  =  ( ( G  oF  .x.  ( V  X.  { X }
) )  oF  .+  ( G  oF  .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 5859 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2486 . . 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 32081 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> K )
115, 6, 10syl2anc 659 . 2  |-  ( ph  ->  G : V --> K )
12 lfldi.x . . 3  |-  ( ph  ->  X  e.  K )
13 fconst6g 5757 . . 3  |-  ( X  e.  K  ->  ( V  X.  { X }
) : V --> K )
1412, 13syl 17 . 2  |-  ( ph  ->  ( V  X.  { X } ) : V --> K )
15 lfldi2.y . . 3  |-  ( ph  ->  Y  e.  K )
16 fconst6g 5757 . . 3  |-  ( Y  e.  K  ->  ( V  X.  { Y }
) : V --> K )
1715, 16syl 17 . 2  |-  ( ph  ->  ( V  X.  { Y } ) : V --> K )
187lmodring 17840 . . . 4  |-  ( W  e.  LMod  ->  R  e. 
Ring )
195, 18syl 17 . . 3  |-  ( ph  ->  R  e.  Ring )
20 lfldi.p . . . 4  |-  .+  =  ( +g  `  R )
21 lfldi.t . . . 4  |-  .x.  =  ( .r `  R )
228, 20, 21ringdi 17537 . . 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 469 . 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 6558 1  |-  ( ph  ->  ( G  oF  .x.  ( ( V  X.  { X }
)  oF  .+  ( V  X.  { Y } ) ) )  =  ( ( G  oF  .x.  ( V  X.  { X }
) )  oF  .+  ( G  oF  .x.  ( V  X.  { Y } ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    = wceq 1405    e. wcel 1842   _Vcvv 3059   {csn 3972    X. cxp 4821   -->wf 5565   ` cfv 5569  (class class class)co 6278    oFcof 6519   Basecbs 14841   +g cplusg 14909   .rcmulr 14910  Scalarcsca 14912   Ringcrg 17518   LModclmod 17832  LFnlclfn 32075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-map 7459  df-ring 17520  df-lmod 17834  df-lfl 32076
This theorem is referenced by:  lflvsdi2a  32098
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