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Theorem lfl0sc 34541
Description: The (right vector space) scalar product of a functional with zero is the zero functional. Note that the first occurrence of  ( V  X.  {  .0.  }
) represents the zero scalar, and the second is the zero functional. (Contributed by NM, 7-Oct-2014.)
Hypotheses
Ref Expression
lfl0sc.v  |-  V  =  ( Base `  W
)
lfl0sc.d  |-  D  =  (Scalar `  W )
lfl0sc.f  |-  F  =  (LFnl `  W )
lfl0sc.k  |-  K  =  ( Base `  D
)
lfl0sc.t  |-  .x.  =  ( .r `  D )
lfl0sc.o  |-  .0.  =  ( 0g `  D )
lfl0sc.w  |-  ( ph  ->  W  e.  LMod )
lfl0sc.g  |-  ( ph  ->  G  e.  F )
Assertion
Ref Expression
lfl0sc  |-  ( ph  ->  ( G  oF  .x.  ( V  X.  {  .0.  } ) )  =  ( V  X.  {  .0.  } ) )

Proof of Theorem lfl0sc
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 lfl0sc.v . . . 4  |-  V  =  ( Base `  W
)
2 fvex 5866 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2527 . . 3  |-  V  e. 
_V
43a1i 11 . 2  |-  ( ph  ->  V  e.  _V )
5 lfl0sc.w . . 3  |-  ( ph  ->  W  e.  LMod )
6 lfl0sc.g . . 3  |-  ( ph  ->  G  e.  F )
7 lfl0sc.d . . . 4  |-  D  =  (Scalar `  W )
8 lfl0sc.k . . . 4  |-  K  =  ( Base `  D
)
9 lfl0sc.f . . . 4  |-  F  =  (LFnl `  W )
107, 8, 1, 9lflf 34522 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> K )
115, 6, 10syl2anc 661 . 2  |-  ( ph  ->  G : V --> K )
127lmodring 17394 . . . 4  |-  ( W  e.  LMod  ->  D  e. 
Ring )
135, 12syl 16 . . 3  |-  ( ph  ->  D  e.  Ring )
14 lfl0sc.o . . . 4  |-  .0.  =  ( 0g `  D )
158, 14ring0cl 17094 . . 3  |-  ( D  e.  Ring  ->  .0.  e.  K )
1613, 15syl 16 . 2  |-  ( ph  ->  .0.  e.  K )
17 lfl0sc.t . . . 4  |-  .x.  =  ( .r `  D )
188, 17, 14ringrz 17110 . . 3  |-  ( ( D  e.  Ring  /\  k  e.  K )  ->  (
k  .x.  .0.  )  =  .0.  )
1913, 18sylan 471 . 2  |-  ( (
ph  /\  k  e.  K )  ->  (
k  .x.  .0.  )  =  .0.  )
204, 11, 16, 16, 19caofid1 6555 1  |-  ( ph  ->  ( G  oF  .x.  ( V  X.  {  .0.  } ) )  =  ( V  X.  {  .0.  } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804   _Vcvv 3095   {csn 4014    X. cxp 4987   -->wf 5574   ` cfv 5578  (class class class)co 6281    oFcof 6523   Basecbs 14509   .rcmulr 14575  Scalarcsca 14577   0gc0g 14714   Ringcrg 17072   LModclmod 17386  LFnlclfn 34516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-recs 7044  df-rdg 7078  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-plusg 14587  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-mgp 17016  df-ring 17074  df-lmod 17388  df-lfl 34517
This theorem is referenced by:  lkrscss  34557  lfl1dim  34580  lfl1dim2N  34581
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