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Theorem ldualvsdi2 35012
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 2457 . . 3  |-  ( Base `  W )  =  (
Base `  W )
3 ldualvsdi2.r . . 3  |-  R  =  (Scalar `  W )
4 ldualvsdi2.k . . 3  |-  K  =  ( Base `  R
)
5 eqid 2457 . . 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 17650 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .+  Y )  e.  K )
138, 9, 10, 12syl3anc 1228 . . 3  |-  ( ph  ->  ( X  .+  Y
)  e.  K )
14 ldualvsdi2.g . . 3  |-  ( ph  ->  G  e.  F )
151, 2, 3, 4, 5, 6, 7, 8, 13, 14ldualvs 35005 . 2  |-  ( ph  ->  ( ( X  .+  Y )  .x.  G
)  =  ( G  oF ( .r
`  R ) ( ( Base `  W
)  X.  { ( X  .+  Y ) } ) ) )
162, 3, 4, 11, 5, 1, 8, 9, 10, 14lflvsdi2a 34948 . 2  |-  ( ph  ->  ( G  oF ( .r `  R
) ( ( Base `  W )  X.  {
( X  .+  Y
) } ) )  =  ( ( G  oF ( .r
`  R ) ( ( Base `  W
)  X.  { X } ) )  oF  .+  ( G  oF ( .r
`  R ) ( ( Base `  W
)  X.  { Y } ) ) ) )
17 ldualvsdi2.p . . . 4  |-  .+b  =  ( +g  `  D )
181, 3, 4, 6, 7, 8, 9, 14ldualvscl 35007 . . . 4  |-  ( ph  ->  ( X  .x.  G
)  e.  F )
191, 3, 4, 6, 7, 8, 10, 14ldualvscl 35007 . . . 4  |-  ( ph  ->  ( Y  .x.  G
)  e.  F )
201, 3, 11, 6, 17, 8, 18, 19ldualvadd 34997 . . 3  |-  ( ph  ->  ( ( X  .x.  G )  .+b  ( Y  .x.  G ) )  =  ( ( X 
.x.  G )  oF  .+  ( Y 
.x.  G ) ) )
211, 2, 3, 4, 5, 6, 7, 8, 9, 14ldualvs 35005 . . . 4  |-  ( ph  ->  ( X  .x.  G
)  =  ( G  oF ( .r
`  R ) ( ( Base `  W
)  X.  { X } ) ) )
221, 2, 3, 4, 5, 6, 7, 8, 10, 14ldualvs 35005 . . . 4  |-  ( ph  ->  ( Y  .x.  G
)  =  ( G  oF ( .r
`  R ) ( ( Base `  W
)  X.  { Y } ) ) )
2321, 22oveq12d 6314 . . 3  |-  ( ph  ->  ( ( X  .x.  G )  oF  .+  ( Y  .x.  G ) )  =  ( ( G  oF ( .r `  R ) ( (
Base `  W )  X.  { X } ) )  oF  .+  ( G  oF
( .r `  R
) ( ( Base `  W )  X.  { Y } ) ) ) )
2420, 23eqtr2d 2499 . 2  |-  ( ph  ->  ( ( G  oF ( .r `  R ) ( (
Base `  W )  X.  { X } ) )  oF  .+  ( G  oF
( .r `  R
) ( ( Base `  W )  X.  { Y } ) ) )  =  ( ( X 
.x.  G )  .+b  ( Y  .x.  G ) ) )
2515, 16, 243eqtrd 2502 1  |-  ( ph  ->  ( ( X  .+  Y )  .x.  G
)  =  ( ( X  .x.  G ) 
.+b  ( Y  .x.  G ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   {csn 4032    X. cxp 5006   ` cfv 5594  (class class class)co 6296    oFcof 6537   Basecbs 14644   +g cplusg 14712   .rcmulr 14713  Scalarcsca 14715   .scvsca 14716   LModclmod 17639  LFnlclfn 34925  LDualcld 34991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-plusg 14725  df-sca 14728  df-vsca 14729  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-mgp 17269  df-ring 17327  df-lmod 17641  df-lfl 34926  df-ldual 34992
This theorem is referenced by:  lduallmodlem  35020
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