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Theorem lflvsdi1 29268
Description: 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 )
lfldi1.g  |-  ( ph  ->  G  e.  F )
lfldi1.h  |-  ( ph  ->  H  e.  F )
Assertion
Ref Expression
lflvsdi1  |-  ( ph  ->  ( ( G  o F  .+  H )  o F  .x.  ( V  X.  { X }
) )  =  ( ( G  o F 
.x.  ( V  X.  { X } ) )  o F  .+  ( H  o F  .x.  ( V  X.  { X }
) ) ) )

Proof of Theorem lflvsdi1
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 5539 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2353 . . 3  |-  V  e. 
_V
43a1i 10 . 2  |-  ( ph  ->  V  e.  _V )
5 lfldi.x . . 3  |-  ( ph  ->  X  e.  K )
6 fconst6g 5430 . . 3  |-  ( X  e.  K  ->  ( V  X.  { X }
) : V --> K )
75, 6syl 15 . 2  |-  ( ph  ->  ( V  X.  { X } ) : V --> K )
8 lfldi.w . . 3  |-  ( ph  ->  W  e.  LMod )
9 lfldi1.g . . 3  |-  ( ph  ->  G  e.  F )
10 lfldi.r . . . 4  |-  R  =  (Scalar `  W )
11 lfldi.k . . . 4  |-  K  =  ( Base `  R
)
12 lfldi.f . . . 4  |-  F  =  (LFnl `  W )
1310, 11, 1, 12lflf 29253 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> K )
148, 9, 13syl2anc 642 . 2  |-  ( ph  ->  G : V --> K )
15 lfldi1.h . . 3  |-  ( ph  ->  H  e.  F )
1610, 11, 1, 12lflf 29253 . . 3  |-  ( ( W  e.  LMod  /\  H  e.  F )  ->  H : V --> K )
178, 15, 16syl2anc 642 . 2  |-  ( ph  ->  H : V --> K )
1810lmodrng 15635 . . . 4  |-  ( W  e.  LMod  ->  R  e. 
Ring )
198, 18syl 15 . . 3  |-  ( ph  ->  R  e.  Ring )
20 lfldi.p . . . 4  |-  .+  =  ( +g  `  R )
21 lfldi.t . . . 4  |-  .x.  =  ( .r `  R )
2211, 20, 21rngdir 15360 . . 3  |-  ( ( R  e.  Ring  /\  (
x  e.  K  /\  y  e.  K  /\  z  e.  K )
)  ->  ( (
x  .+  y )  .x.  z )  =  ( ( x  .x.  z
)  .+  ( y  .x.  z ) ) )
2319, 22sylan 457 . 2  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K  /\  z  e.  K ) )  -> 
( ( x  .+  y )  .x.  z
)  =  ( ( x  .x.  z ) 
.+  ( y  .x.  z ) ) )
244, 7, 14, 17, 23caofdir 6114 1  |-  ( ph  ->  ( ( G  o F  .+  H )  o F  .x.  ( V  X.  { X }
) )  =  ( ( G  o F 
.x.  ( V  X.  { X } ) )  o F  .+  ( H  o F  .x.  ( V  X.  { X }
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   Ringcrg 15337   LModclmod 15627  LFnlclfn 29247
This theorem is referenced by:  ldualvsdi1  29333
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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