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Theorem lflvsdi1 29194
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 5683 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2458 . . 3  |-  V  e. 
_V
43a1i 11 . 2  |-  ( ph  ->  V  e.  _V )
5 lfldi.x . . 3  |-  ( ph  ->  X  e.  K )
6 fconst6g 5573 . . 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 29179 . . 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 29179 . . 3  |-  ( ( W  e.  LMod  /\  H  e.  F )  ->  H : V --> K )
178, 15, 16syl2anc 643 . 2  |-  ( ph  ->  H : V --> K )
1810lmodrng 15886 . . . 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 15611 . . 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 6281 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 1649    e. wcel 1717   _Vcvv 2900   {csn 3758    X. cxp 4817   -->wf 5391   ` cfv 5395  (class class class)co 6021    o Fcof 6243   Basecbs 13397   +g cplusg 13457   .rcmulr 13458  Scalarcsca 13460   Ringcrg 15588   LModclmod 15878  LFnlclfn 29173
This theorem is referenced by:  ldualvsdi1  29259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-map 6957  df-rng 15591  df-lmod 15880  df-lfl 29174
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