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Theorem lflvsdi2 33753
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 5869 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2546 . . 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 33737 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> K )
115, 6, 10syl2anc 661 . 2  |-  ( ph  ->  G : V --> K )
12 lfldi.x . . 3  |-  ( ph  ->  X  e.  K )
13 fconst6g 5767 . . 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 5767 . . 3  |-  ( Y  e.  K  ->  ( V  X.  { Y }
) : V --> K )
1715, 16syl 16 . 2  |-  ( ph  ->  ( V  X.  { Y } ) : V --> K )
187lmodrng 17298 . . . 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 16999 . . 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 471 . 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 6553 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 968    = wceq 1374    e. wcel 1762   _Vcvv 3108   {csn 4022    X. cxp 4992   -->wf 5577   ` cfv 5581  (class class class)co 6277    oFcof 6515   Basecbs 14481   +g cplusg 14546   .rcmulr 14547  Scalarcsca 14549   Ringcrg 16981   LModclmod 17290  LFnlclfn 33731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-map 7414  df-rng 16983  df-lmod 17292  df-lfl 33732
This theorem is referenced by:  lflvsdi2a  33754
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