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Theorem ldualvsdi2 29627
Description: Reverse distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.)
Hypotheses
Ref Expression
ldualvsdi2.f  |-  F  =  (LFnl `  W )
ldualvsdi2.r  |-  R  =  (Scalar `  W )
ldualvsdi2.a  |-  .+  =  ( +g  `  R )
ldualvsdi2.k  |-  K  =  ( Base `  R
)
ldualvsdi2.d  |-  D  =  (LDual `  W )
ldualvsdi2.p  |-  .+b  =  ( +g  `  D )
ldualvsdi2.s  |-  .x.  =  ( .s `  D )
ldualvsdi2.w  |-  ( ph  ->  W  e.  LMod )
ldualvsdi2.x  |-  ( ph  ->  X  e.  K )
ldualvsdi2.y  |-  ( ph  ->  Y  e.  K )
ldualvsdi2.g  |-  ( ph  ->  G  e.  F )
Assertion
Ref Expression
ldualvsdi2  |-  ( ph  ->  ( ( X  .+  Y )  .x.  G
)  =  ( ( X  .x.  G ) 
.+b  ( Y  .x.  G ) ) )

Proof of Theorem ldualvsdi2
StepHypRef Expression
1 ldualvsdi2.f . . 3  |-  F  =  (LFnl `  W )
2 eqid 2404 . . 3  |-  ( Base `  W )  =  (
Base `  W )
3 ldualvsdi2.r . . 3  |-  R  =  (Scalar `  W )
4 ldualvsdi2.k . . 3  |-  K  =  ( Base `  R
)
5 eqid 2404 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
6 ldualvsdi2.d . . 3  |-  D  =  (LDual `  W )
7 ldualvsdi2.s . . 3  |-  .x.  =  ( .s `  D )
8 ldualvsdi2.w . . 3  |-  ( ph  ->  W  e.  LMod )
9 ldualvsdi2.x . . . 4  |-  ( ph  ->  X  e.  K )
10 ldualvsdi2.y . . . 4  |-  ( ph  ->  Y  e.  K )
11 ldualvsdi2.a . . . . 5  |-  .+  =  ( +g  `  R )
123, 4, 11lmodacl 15916 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .+  Y )  e.  K )
138, 9, 10, 12syl3anc 1184 . . 3  |-  ( ph  ->  ( X  .+  Y
)  e.  K )
14 ldualvsdi2.g . . 3  |-  ( ph  ->  G  e.  F )
151, 2, 3, 4, 5, 6, 7, 8, 13, 14ldualvs 29620 . 2  |-  ( ph  ->  ( ( X  .+  Y )  .x.  G
)  =  ( G  o F ( .r
`  R ) ( ( Base `  W
)  X.  { ( X  .+  Y ) } ) ) )
162, 3, 4, 11, 5, 1, 8, 9, 10, 14lflvsdi2a 29563 . 2  |-  ( ph  ->  ( G  o F ( .r `  R
) ( ( Base `  W )  X.  {
( X  .+  Y
) } ) )  =  ( ( G  o F ( .r
`  R ) ( ( Base `  W
)  X.  { X } ) )  o F  .+  ( G  o F ( .r
`  R ) ( ( Base `  W
)  X.  { Y } ) ) ) )
17 ldualvsdi2.p . . . 4  |-  .+b  =  ( +g  `  D )
181, 3, 4, 6, 7, 8, 9, 14ldualvscl 29622 . . . 4  |-  ( ph  ->  ( X  .x.  G
)  e.  F )
191, 3, 4, 6, 7, 8, 10, 14ldualvscl 29622 . . . 4  |-  ( ph  ->  ( Y  .x.  G
)  e.  F )
201, 3, 11, 6, 17, 8, 18, 19ldualvadd 29612 . . 3  |-  ( ph  ->  ( ( X  .x.  G )  .+b  ( Y  .x.  G ) )  =  ( ( X 
.x.  G )  o F  .+  ( Y 
.x.  G ) ) )
211, 2, 3, 4, 5, 6, 7, 8, 9, 14ldualvs 29620 . . . 4  |-  ( ph  ->  ( X  .x.  G
)  =  ( G  o F ( .r
`  R ) ( ( Base `  W
)  X.  { X } ) ) )
221, 2, 3, 4, 5, 6, 7, 8, 10, 14ldualvs 29620 . . . 4  |-  ( ph  ->  ( Y  .x.  G
)  =  ( G  o F ( .r
`  R ) ( ( Base `  W
)  X.  { Y } ) ) )
2321, 22oveq12d 6058 . . 3  |-  ( ph  ->  ( ( X  .x.  G )  o F 
.+  ( Y  .x.  G ) )  =  ( ( G  o F ( .r `  R ) ( (
Base `  W )  X.  { X } ) )  o F  .+  ( G  o F
( .r `  R
) ( ( Base `  W )  X.  { Y } ) ) ) )
2420, 23eqtr2d 2437 . 2  |-  ( ph  ->  ( ( G  o F ( .r `  R ) ( (
Base `  W )  X.  { X } ) )  o F  .+  ( G  o F
( .r `  R
) ( ( Base `  W )  X.  { Y } ) ) )  =  ( ( X 
.x.  G )  .+b  ( Y  .x.  G ) ) )
2515, 16, 243eqtrd 2440 1  |-  ( ph  ->  ( ( X  .+  Y )  .x.  G
)  =  ( ( X  .x.  G ) 
.+b  ( Y  .x.  G ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   {csn 3774    X. cxp 4835   ` cfv 5413  (class class class)co 6040    o Fcof 6262   Basecbs 13424   +g cplusg 13484   .rcmulr 13485  Scalarcsca 13487   .scvsca 13488   LModclmod 15905  LFnlclfn 29540  LDualcld 29606
This theorem is referenced by:  lduallmodlem  29635
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-plusg 13497  df-sca 13500  df-vsca 13501  df-mnd 14645  df-grp 14767  df-mgp 15604  df-rng 15618  df-lmod 15907  df-lfl 29541  df-ldual 29607
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