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Theorem lfl0sc 33888
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 5875 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2551 . . 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 33869 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> K )
115, 6, 10syl2anc 661 . 2  |-  ( ph  ->  G : V --> K )
127lmodrng 17315 . . . 4  |-  ( W  e.  LMod  ->  D  e. 
Ring )
135, 12syl 16 . . 3  |-  ( ph  ->  D  e.  Ring )
14 lfl0sc.o . . . 4  |-  .0.  =  ( 0g `  D )
158, 14rng0cl 17016 . . 3  |-  ( D  e.  Ring  ->  .0.  e.  K )
1613, 15syl 16 . 2  |-  ( ph  ->  .0.  e.  K )
17 lfl0sc.t . . . 4  |-  .x.  =  ( .r `  D )
188, 17, 14rngrz 17032 . . 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 6553 1  |-  ( ph  ->  ( G  oF  .x.  ( V  X.  {  .0.  } ) )  =  ( V  X.  {  .0.  } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113   {csn 4027    X. cxp 4997   -->wf 5583   ` cfv 5587  (class class class)co 6283    oFcof 6521   Basecbs 14489   .rcmulr 14555  Scalarcsca 14557   0gc0g 14694   Ringcrg 16995   LModclmod 17307  LFnlclfn 33863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-recs 7042  df-rdg 7076  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-plusg 14567  df-0g 14696  df-mnd 15731  df-grp 15864  df-mgp 16941  df-rng 16997  df-lmod 17309  df-lfl 33864
This theorem is referenced by:  lkrscss  33904  lfl1dim  33927  lfl1dim2N  33928
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