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Theorem ldualvsdi1 29406
Description: Distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.)
Hypotheses
Ref Expression
ldualvsdi1.f  |-  F  =  (LFnl `  W )
ldualvsdi1.r  |-  R  =  (Scalar `  W )
ldualvsdi1.k  |-  K  =  ( Base `  R
)
ldualvsdi1.d  |-  D  =  (LDual `  W )
ldualvsdi1.p  |-  .+  =  ( +g  `  D )
ldualvsdi1.s  |-  .x.  =  ( .s `  D )
ldualvsdi1.w  |-  ( ph  ->  W  e.  LMod )
ldualvsdi1.x  |-  ( ph  ->  X  e.  K )
ldualvsdi1.g  |-  ( ph  ->  G  e.  F )
ldualvsdi1.h  |-  ( ph  ->  H  e.  F )
Assertion
Ref Expression
ldualvsdi1  |-  ( ph  ->  ( X  .x.  ( G  .+  H ) )  =  ( ( X 
.x.  G )  .+  ( X  .x.  H ) ) )

Proof of Theorem ldualvsdi1
StepHypRef Expression
1 ldualvsdi1.f . . . 4  |-  F  =  (LFnl `  W )
2 eqid 2285 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
3 ldualvsdi1.r . . . 4  |-  R  =  (Scalar `  W )
4 ldualvsdi1.k . . . 4  |-  K  =  ( Base `  R
)
5 eqid 2285 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
6 ldualvsdi1.d . . . 4  |-  D  =  (LDual `  W )
7 ldualvsdi1.s . . . 4  |-  .x.  =  ( .s `  D )
8 ldualvsdi1.w . . . 4  |-  ( ph  ->  W  e.  LMod )
9 ldualvsdi1.x . . . 4  |-  ( ph  ->  X  e.  K )
10 ldualvsdi1.g . . . 4  |-  ( ph  ->  G  e.  F )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10ldualvs 29400 . . 3  |-  ( ph  ->  ( X  .x.  G
)  =  ( G  o F ( .r
`  R ) ( ( Base `  W
)  X.  { X } ) ) )
12 ldualvsdi1.h . . . 4  |-  ( ph  ->  H  e.  F )
131, 2, 3, 4, 5, 6, 7, 8, 9, 12ldualvs 29400 . . 3  |-  ( ph  ->  ( X  .x.  H
)  =  ( H  o F ( .r
`  R ) ( ( Base `  W
)  X.  { X } ) ) )
1411, 13oveq12d 5878 . 2  |-  ( ph  ->  ( ( X  .x.  G )  o F ( +g  `  R
) ( X  .x.  H ) )  =  ( ( G  o F ( .r `  R ) ( (
Base `  W )  X.  { X } ) )  o F ( +g  `  R ) ( H  o F ( .r `  R
) ( ( Base `  W )  X.  { X } ) ) ) )
15 eqid 2285 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
16 ldualvsdi1.p . . 3  |-  .+  =  ( +g  `  D )
171, 3, 4, 6, 7, 8, 9, 10ldualvscl 29402 . . 3  |-  ( ph  ->  ( X  .x.  G
)  e.  F )
181, 3, 4, 6, 7, 8, 9, 12ldualvscl 29402 . . 3  |-  ( ph  ->  ( X  .x.  H
)  e.  F )
191, 3, 15, 6, 16, 8, 17, 18ldualvadd 29392 . 2  |-  ( ph  ->  ( ( X  .x.  G )  .+  ( X  .x.  H ) )  =  ( ( X 
.x.  G )  o F ( +g  `  R
) ( X  .x.  H ) ) )
201, 6, 16, 8, 10, 12ldualvaddcl 29393 . . . 4  |-  ( ph  ->  ( G  .+  H
)  e.  F )
211, 2, 3, 4, 5, 6, 7, 8, 9, 20ldualvs 29400 . . 3  |-  ( ph  ->  ( X  .x.  ( G  .+  H ) )  =  ( ( G 
.+  H )  o F ( .r `  R ) ( (
Base `  W )  X.  { X } ) ) )
221, 3, 15, 6, 16, 8, 10, 12ldualvadd 29392 . . . 4  |-  ( ph  ->  ( G  .+  H
)  =  ( G  o F ( +g  `  R ) H ) )
2322oveq1d 5875 . . 3  |-  ( ph  ->  ( ( G  .+  H )  o F ( .r `  R
) ( ( Base `  W )  X.  { X } ) )  =  ( ( G  o F ( +g  `  R
) H )  o F ( .r `  R ) ( (
Base `  W )  X.  { X } ) ) )
242, 3, 4, 15, 5, 1, 8, 9, 10, 12lflvsdi1 29341 . . 3  |-  ( ph  ->  ( ( G  o F ( +g  `  R
) H )  o F ( .r `  R ) ( (
Base `  W )  X.  { X } ) )  =  ( ( G  o F ( .r `  R ) ( ( Base `  W
)  X.  { X } ) )  o F ( +g  `  R
) ( H  o F ( .r `  R ) ( (
Base `  W )  X.  { X } ) ) ) )
2521, 23, 243eqtrd 2321 . 2  |-  ( ph  ->  ( X  .x.  ( G  .+  H ) )  =  ( ( G  o F ( .r
`  R ) ( ( Base `  W
)  X.  { X } ) )  o F ( +g  `  R
) ( H  o F ( .r `  R ) ( (
Base `  W )  X.  { X } ) ) ) )
2614, 19, 253eqtr4rd 2328 1  |-  ( ph  ->  ( X  .x.  ( G  .+  H ) )  =  ( ( X 
.x.  G )  .+  ( X  .x.  H ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1625    e. wcel 1686   {csn 3642    X. cxp 4689   ` cfv 5257  (class class class)co 5860    o Fcof 6078   Basecbs 13150   +g cplusg 13210   .rcmulr 13211  Scalarcsca 13213   .scvsca 13214   LModclmod 15629  LFnlclfn 29320  LDualcld 29386
This theorem is referenced by:  lduallmodlem  29415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-n0 9968  df-z 10027  df-uz 10233  df-fz 10785  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-plusg 13223  df-sca 13226  df-vsca 13227  df-0g 13406  df-mnd 14369  df-grp 14491  df-minusg 14492  df-sbg 14493  df-cmn 15093  df-abl 15094  df-mgp 15328  df-rng 15342  df-ur 15344  df-lmod 15631  df-lfl 29321  df-ldual 29387
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