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Theorem lflvsdi1 29950
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 5745 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2508 . . 3  |-  V  e. 
_V
43a1i 11 . 2  |-  ( ph  ->  V  e.  _V )
5 lfldi.x . . 3  |-  ( ph  ->  X  e.  K )
6 fconst6g 5635 . . 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 29935 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> K )
148, 9, 13syl2anc 644 . 2  |-  ( ph  ->  G : V --> K )
15 lfldi1.h . . 3  |-  ( ph  ->  H  e.  F )
1610, 11, 1, 12lflf 29935 . . 3  |-  ( ( W  e.  LMod  /\  H  e.  F )  ->  H : V --> K )
178, 15, 16syl2anc 644 . 2  |-  ( ph  ->  H : V --> K )
1810lmodrng 15963 . . . 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 15688 . . 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 459 . 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 6344 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 937    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816    X. cxp 4879   -->wf 5453   ` cfv 5457  (class class class)co 6084    o Fcof 6306   Basecbs 13474   +g cplusg 13534   .rcmulr 13535  Scalarcsca 13537   Ringcrg 15665   LModclmod 15955  LFnlclfn 29929
This theorem is referenced by:  ldualvsdi1  30015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-map 7023  df-rng 15668  df-lmod 15957  df-lfl 29930
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