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Theorem lflvsdi1 29813
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 5734 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2505 . . 3  |-  V  e. 
_V
43a1i 11 . 2  |-  ( ph  ->  V  e.  _V )
5 lfldi.x . . 3  |-  ( ph  ->  X  e.  K )
6 fconst6g 5624 . . 3  |-  ( X  e.  K  ->  ( V  X.  { X }
) : V --> K )
75, 6syl 16 . 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 29798 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> K )
148, 9, 13syl2anc 643 . 2  |-  ( ph  ->  G : V --> K )
15 lfldi1.h . . 3  |-  ( ph  ->  H  e.  F )
1610, 11, 1, 12lflf 29798 . . 3  |-  ( ( W  e.  LMod  /\  H  e.  F )  ->  H : V --> K )
178, 15, 16syl2anc 643 . 2  |-  ( ph  ->  H : V --> K )
1810lmodrng 15950 . . . 4  |-  ( W  e.  LMod  ->  R  e. 
Ring )
198, 18syl 16 . . 3  |-  ( ph  ->  R  e.  Ring )
20 lfldi.p . . . 4  |-  .+  =  ( +g  `  R )
21 lfldi.t . . . 4  |-  .x.  =  ( .r `  R )
2211, 20, 21rngdir 15675 . . 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 458 . 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 6333 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 936    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806    X. cxp 4868   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295   Basecbs 13461   +g cplusg 13521   .rcmulr 13522  Scalarcsca 13524   Ringcrg 15652   LModclmod 15942  LFnlclfn 29792
This theorem is referenced by:  ldualvsdi1  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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-map 7012  df-rng 15655  df-lmod 15944  df-lfl 29793
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