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Theorem lfl1 32548
Description: A non-zero functional has a value of 1 at some argument. (Contributed by NM, 16-Apr-2014.)
Hypotheses
Ref Expression
lfl1.d  |-  D  =  (Scalar `  W )
lfl1.o  |-  .0.  =  ( 0g `  D )
lfl1.u  |-  .1.  =  ( 1r `  D )
lfl1.v  |-  V  =  ( Base `  W
)
lfl1.f  |-  F  =  (LFnl `  W )
Assertion
Ref Expression
lfl1  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  ->  E. x  e.  V  ( G `  x )  =  .1.  )
Distinct variable groups:    x, D    x, G    x,  .1.    x, V   
x, W
Allowed substitution hints:    F( x)    .0. ( x)

Proof of Theorem lfl1
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nne 2605 . . . . . . 7  |-  ( -.  ( G `  z
)  =/=  .0.  <->  ( G `  z )  =  .0.  )
21ralbii 2796 . . . . . 6  |-  ( A. z  e.  V  -.  ( G `  z )  =/=  .0.  <->  A. z  e.  V  ( G `  z )  =  .0.  )
3 lfl1.d . . . . . . . . . 10  |-  D  =  (Scalar `  W )
4 eqid 2428 . . . . . . . . . 10  |-  ( Base `  D )  =  (
Base `  D )
5 lfl1.v . . . . . . . . . 10  |-  V  =  ( Base `  W
)
6 lfl1.f . . . . . . . . . 10  |-  F  =  (LFnl `  W )
73, 4, 5, 6lflf 32541 . . . . . . . . 9  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  G : V --> ( Base `  D
) )
8 ffn 5689 . . . . . . . . 9  |-  ( G : V --> ( Base `  D )  ->  G  Fn  V )
97, 8syl 17 . . . . . . . 8  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  G  Fn  V )
10 fconstfv 6085 . . . . . . . . 9  |-  ( G : V --> {  .0.  }  <-> 
( G  Fn  V  /\  A. z  e.  V  ( G `  z )  =  .0.  ) )
1110simplbi2 629 . . . . . . . 8  |-  ( G  Fn  V  ->  ( A. z  e.  V  ( G `  z )  =  .0.  ->  G : V --> {  .0.  }
) )
129, 11syl 17 . . . . . . 7  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( A. z  e.  V  ( G `  z )  =  .0.  ->  G : V --> {  .0.  }
) )
13 lfl1.o . . . . . . . . 9  |-  .0.  =  ( 0g `  D )
14 fvex 5835 . . . . . . . . 9  |-  ( 0g
`  D )  e. 
_V
1513, 14eqeltri 2502 . . . . . . . 8  |-  .0.  e.  _V
1615fconst2 6080 . . . . . . 7  |-  ( G : V --> {  .0.  }  <-> 
G  =  ( V  X.  {  .0.  }
) )
1712, 16syl6ib 229 . . . . . 6  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( A. z  e.  V  ( G `  z )  =  .0.  ->  G  =  ( V  X.  {  .0.  } ) ) )
182, 17syl5bi 220 . . . . 5  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( A. z  e.  V  -.  ( G `  z
)  =/=  .0.  ->  G  =  ( V  X.  {  .0.  } ) ) )
1918necon3ad 2614 . . . 4  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( G  =/=  ( V  X.  {  .0.  } )  ->  -.  A. z  e.  V  -.  ( G `  z
)  =/=  .0.  )
)
20 dfrex2 2815 . . . 4  |-  ( E. z  e.  V  ( G `  z )  =/=  .0.  <->  -.  A. z  e.  V  -.  ( G `  z )  =/=  .0.  )
2119, 20syl6ibr 230 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( G  =/=  ( V  X.  {  .0.  } )  ->  E. z  e.  V  ( G `  z )  =/=  .0.  ) )
22213impia 1202 . 2  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  ->  E. z  e.  V  ( G `  z )  =/=  .0.  )
23 simp1l 1029 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  W  e.  LVec )
24 lveclmod 18272 . . . . . . 7  |-  ( W  e.  LVec  ->  W  e. 
LMod )
2523, 24syl 17 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  W  e.  LMod )
263lvecdrng 18271 . . . . . . . 8  |-  ( W  e.  LVec  ->  D  e.  DivRing )
2723, 26syl 17 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  D  e.  DivRing )
28 simp1r 1030 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  G  e.  F )
29 simp2 1006 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  z  e.  V )
303, 4, 5, 6lflcl 32542 . . . . . . . 8  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  z  e.  V )  ->  ( G `  z )  e.  ( Base `  D
) )
3123, 28, 29, 30syl3anc 1264 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( G `  z
)  e.  ( Base `  D ) )
32 simp3 1007 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( G `  z
)  =/=  .0.  )
33 eqid 2428 . . . . . . . 8  |-  ( invr `  D )  =  (
invr `  D )
344, 13, 33drnginvrcl 17935 . . . . . . 7  |-  ( ( D  e.  DivRing  /\  ( G `  z )  e.  ( Base `  D
)  /\  ( G `  z )  =/=  .0.  )  ->  ( ( invr `  D ) `  ( G `  z )
)  e.  ( Base `  D ) )
3527, 31, 32, 34syl3anc 1264 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( ( invr `  D
) `  ( G `  z ) )  e.  ( Base `  D
) )
36 eqid 2428 . . . . . . 7  |-  ( .s
`  W )  =  ( .s `  W
)
375, 3, 36, 4lmodvscl 18051 . . . . . 6  |-  ( ( W  e.  LMod  /\  (
( invr `  D ) `  ( G `  z
) )  e.  (
Base `  D )  /\  z  e.  V
)  ->  ( (
( invr `  D ) `  ( G `  z
) ) ( .s
`  W ) z )  e.  V )
3825, 35, 29, 37syl3anc 1264 . . . . 5  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( ( ( invr `  D ) `  ( G `  z )
) ( .s `  W ) z )  e.  V )
39 eqid 2428 . . . . . . . 8  |-  ( .r
`  D )  =  ( .r `  D
)
403, 4, 39, 5, 36, 6lflmul 32546 . . . . . . 7  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  (
( ( invr `  D
) `  ( G `  z ) )  e.  ( Base `  D
)  /\  z  e.  V ) )  -> 
( G `  (
( ( invr `  D
) `  ( G `  z ) ) ( .s `  W ) z ) )  =  ( ( ( invr `  D ) `  ( G `  z )
) ( .r `  D ) ( G `
 z ) ) )
4125, 28, 35, 29, 40syl112anc 1268 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( G `  (
( ( invr `  D
) `  ( G `  z ) ) ( .s `  W ) z ) )  =  ( ( ( invr `  D ) `  ( G `  z )
) ( .r `  D ) ( G `
 z ) ) )
42 lfl1.u . . . . . . . 8  |-  .1.  =  ( 1r `  D )
434, 13, 39, 42, 33drnginvrl 17937 . . . . . . 7  |-  ( ( D  e.  DivRing  /\  ( G `  z )  e.  ( Base `  D
)  /\  ( G `  z )  =/=  .0.  )  ->  ( ( (
invr `  D ) `  ( G `  z
) ) ( .r
`  D ) ( G `  z ) )  =  .1.  )
4427, 31, 32, 43syl3anc 1264 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( ( ( invr `  D ) `  ( G `  z )
) ( .r `  D ) ( G `
 z ) )  =  .1.  )
4541, 44eqtrd 2462 . . . . 5  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( G `  (
( ( invr `  D
) `  ( G `  z ) ) ( .s `  W ) z ) )  =  .1.  )
46 fveq2 5825 . . . . . . 7  |-  ( x  =  ( ( (
invr `  D ) `  ( G `  z
) ) ( .s
`  W ) z )  ->  ( G `  x )  =  ( G `  ( ( ( invr `  D
) `  ( G `  z ) ) ( .s `  W ) z ) ) )
4746eqeq1d 2430 . . . . . 6  |-  ( x  =  ( ( (
invr `  D ) `  ( G `  z
) ) ( .s
`  W ) z )  ->  ( ( G `  x )  =  .1.  <->  ( G `  ( ( ( invr `  D ) `  ( G `  z )
) ( .s `  W ) z ) )  =  .1.  )
)
4847rspcev 3125 . . . . 5  |-  ( ( ( ( ( invr `  D ) `  ( G `  z )
) ( .s `  W ) z )  e.  V  /\  ( G `  ( (
( invr `  D ) `  ( G `  z
) ) ( .s
`  W ) z ) )  =  .1.  )  ->  E. x  e.  V  ( G `  x )  =  .1.  )
4938, 45, 48syl2anc 665 . . . 4  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  E. x  e.  V  ( G `  x )  =  .1.  )
5049rexlimdv3a 2858 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( E. z  e.  V  ( G `  z )  =/=  .0.  ->  E. x  e.  V  ( G `  x )  =  .1.  ) )
51503adant3 1025 . 2  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  -> 
( E. z  e.  V  ( G `  z )  =/=  .0.  ->  E. x  e.  V  ( G `  x )  =  .1.  ) )
5222, 51mpd 15 1  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  ->  E. x  e.  V  ( G `  x )  =  .1.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2599   A.wral 2714   E.wrex 2715   _Vcvv 3022   {csn 3941    X. cxp 4794    Fn wfn 5539   -->wf 5540   ` cfv 5544  (class class class)co 6249   Basecbs 15064   .rcmulr 15134  Scalarcsca 15136   .scvsca 15137   0gc0g 15281   1rcur 17678   invrcinvr 17842   DivRingcdr 17918   LModclmod 18034   LVecclvec 18268  LFnlclfn 32535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-tpos 6928  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-map 7429  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-2 10619  df-3 10620  df-ndx 15067  df-slot 15068  df-base 15069  df-sets 15070  df-ress 15071  df-plusg 15146  df-mulr 15147  df-0g 15283  df-mgm 16431  df-sgrp 16470  df-mnd 16480  df-grp 16616  df-minusg 16617  df-sbg 16618  df-mgp 17667  df-ur 17679  df-ring 17725  df-oppr 17794  df-dvdsr 17812  df-unit 17813  df-invr 17843  df-drng 17920  df-lmod 18036  df-lvec 18269  df-lfl 32536
This theorem is referenced by:  eqlkr  32577  lkrshp  32583
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