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Theorem ldualvaddval 29321
Description: The value of the value of vector addition in the dual of a vector space. (Contributed by NM, 7-Jan-2015.)
Hypotheses
Ref Expression
ldualvaddval.v  |-  V  =  ( Base `  W
)
ldualvaddval.r  |-  R  =  (Scalar `  W )
ldualvaddval.a  |-  .+  =  ( +g  `  R )
ldualvaddval.f  |-  F  =  (LFnl `  W )
ldualvaddval.d  |-  D  =  (LDual `  W )
ldualvaddval.p  |-  .+b  =  ( +g  `  D )
ldualvaddval.w  |-  ( ph  ->  W  e.  LMod )
ldualvaddval.g  |-  ( ph  ->  G  e.  F )
ldualvaddval.h  |-  ( ph  ->  H  e.  F )
ldualvaddval.x  |-  ( ph  ->  X  e.  V )
Assertion
Ref Expression
ldualvaddval  |-  ( ph  ->  ( ( G  .+b  H ) `  X )  =  ( ( G `
 X )  .+  ( H `  X ) ) )

Proof of Theorem ldualvaddval
StepHypRef Expression
1 ldualvaddval.f . . . 4  |-  F  =  (LFnl `  W )
2 ldualvaddval.r . . . 4  |-  R  =  (Scalar `  W )
3 ldualvaddval.a . . . 4  |-  .+  =  ( +g  `  R )
4 ldualvaddval.d . . . 4  |-  D  =  (LDual `  W )
5 ldualvaddval.p . . . 4  |-  .+b  =  ( +g  `  D )
6 ldualvaddval.w . . . 4  |-  ( ph  ->  W  e.  LMod )
7 ldualvaddval.g . . . 4  |-  ( ph  ->  G  e.  F )
8 ldualvaddval.h . . . 4  |-  ( ph  ->  H  e.  F )
91, 2, 3, 4, 5, 6, 7, 8ldualvadd 29319 . . 3  |-  ( ph  ->  ( G  .+b  H
)  =  ( G  o F  .+  H
) )
109fveq1d 5527 . 2  |-  ( ph  ->  ( ( G  .+b  H ) `  X )  =  ( ( G  o F  .+  H
) `  X )
)
11 ldualvaddval.x . . 3  |-  ( ph  ->  X  e.  V )
12 eqid 2283 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
13 ldualvaddval.v . . . . . . 7  |-  V  =  ( Base `  W
)
142, 12, 13, 1lflf 29253 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> ( Base `  R
) )
15 ffn 5389 . . . . . 6  |-  ( G : V --> ( Base `  R )  ->  G  Fn  V )
1614, 15syl 15 . . . . 5  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G  Fn  V )
176, 7, 16syl2anc 642 . . . 4  |-  ( ph  ->  G  Fn  V )
182, 12, 13, 1lflf 29253 . . . . . 6  |-  ( ( W  e.  LMod  /\  H  e.  F )  ->  H : V --> ( Base `  R
) )
19 ffn 5389 . . . . . 6  |-  ( H : V --> ( Base `  R )  ->  H  Fn  V )
2018, 19syl 15 . . . . 5  |-  ( ( W  e.  LMod  /\  H  e.  F )  ->  H  Fn  V )
216, 8, 20syl2anc 642 . . . 4  |-  ( ph  ->  H  Fn  V )
22 fvex 5539 . . . . . 6  |-  ( Base `  W )  e.  _V
2313, 22eqeltri 2353 . . . . 5  |-  V  e. 
_V
2423a1i 10 . . . 4  |-  ( ph  ->  V  e.  _V )
25 inidm 3378 . . . 4  |-  ( V  i^i  V )  =  V
26 eqidd 2284 . . . 4  |-  ( (
ph  /\  X  e.  V )  ->  ( G `  X )  =  ( G `  X ) )
27 eqidd 2284 . . . 4  |-  ( (
ph  /\  X  e.  V )  ->  ( H `  X )  =  ( H `  X ) )
2817, 21, 24, 24, 25, 26, 27ofval 6087 . . 3  |-  ( (
ph  /\  X  e.  V )  ->  (
( G  o F 
.+  H ) `  X )  =  ( ( G `  X
)  .+  ( H `  X ) ) )
2911, 28mpdan 649 . 2  |-  ( ph  ->  ( ( G  o F  .+  H ) `  X )  =  ( ( G `  X
)  .+  ( H `  X ) ) )
3010, 29eqtrd 2315 1  |-  ( ph  ->  ( ( G  .+b  H ) `  X )  =  ( ( G `
 X )  .+  ( H `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   LModclmod 15627  LFnlclfn 29247  LDualcld 29313
This theorem is referenced by:  ldualvsubval  29347  lkrin  29354  lcdvaddval  31788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-sca 13224  df-vsca 13225  df-lfl 29248  df-ldual 29314
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