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Theorem lflvsdi1 29890
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 5555 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2366 . . 3  |-  V  e. 
_V
43a1i 10 . 2  |-  ( ph  ->  V  e.  _V )
5 lfldi.x . . 3  |-  ( ph  ->  X  e.  K )
6 fconst6g 5446 . . 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 29875 . . 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 29875 . . 3  |-  ( ( W  e.  LMod  /\  H  e.  F )  ->  H : V --> K )
178, 15, 16syl2anc 642 . 2  |-  ( ph  ->  H : V --> K )
1810lmodrng 15651 . . . 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 15376 . . 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 6130 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 1632    e. wcel 1696   _Vcvv 2801   {csn 3653    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   Basecbs 13164   +g cplusg 13224   .rcmulr 13225  Scalarcsca 13227   Ringcrg 15353   LModclmod 15643  LFnlclfn 29869
This theorem is referenced by:  ldualvsdi1  29955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-map 6790  df-rng 15356  df-lmod 15645  df-lfl 29870
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