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Theorem lfl1 32707
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 2647 . . . . . . 7  |-  ( -.  ( G `  z
)  =/=  .0.  <->  ( G `  z )  =  .0.  )
21ralbii 2823 . . . . . 6  |-  ( A. z  e.  V  -.  ( G `  z )  =/=  .0.  <->  A. z  e.  V  ( G `  z )  =  .0.  )
3 lfl1.d . . . . . . . . . 10  |-  D  =  (Scalar `  W )
4 eqid 2471 . . . . . . . . . 10  |-  ( Base `  D )  =  (
Base `  D )
5 lfl1.v . . . . . . . . . 10  |-  V  =  ( Base `  W
)
6 lfl1.f . . . . . . . . . 10  |-  F  =  (LFnl `  W )
73, 4, 5, 6lflf 32700 . . . . . . . . 9  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  G : V --> ( Base `  D
) )
8 ffn 5739 . . . . . . . . 9  |-  ( G : V --> ( Base `  D )  ->  G  Fn  V )
97, 8syl 17 . . . . . . . 8  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  G  Fn  V )
10 fconstfv 6143 . . . . . . . . 9  |-  ( G : V --> {  .0.  }  <-> 
( G  Fn  V  /\  A. z  e.  V  ( G `  z )  =  .0.  ) )
1110simplbi2 637 . . . . . . . 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 5889 . . . . . . . . 9  |-  ( 0g
`  D )  e. 
_V
1513, 14eqeltri 2545 . . . . . . . 8  |-  .0.  e.  _V
1615fconst2 6137 . . . . . . 7  |-  ( G : V --> {  .0.  }  <-> 
G  =  ( V  X.  {  .0.  }
) )
1712, 16syl6ib 234 . . . . . 6  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( A. z  e.  V  ( G `  z )  =  .0.  ->  G  =  ( V  X.  {  .0.  } ) ) )
182, 17syl5bi 225 . . . . 5  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( A. z  e.  V  -.  ( G `  z
)  =/=  .0.  ->  G  =  ( V  X.  {  .0.  } ) ) )
1918necon3ad 2656 . . . 4  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( G  =/=  ( V  X.  {  .0.  } )  ->  -.  A. z  e.  V  -.  ( G `  z
)  =/=  .0.  )
)
20 dfrex2 2837 . . . 4  |-  ( E. z  e.  V  ( G `  z )  =/=  .0.  <->  -.  A. z  e.  V  -.  ( G `  z )  =/=  .0.  )
2119, 20syl6ibr 235 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( G  =/=  ( V  X.  {  .0.  } )  ->  E. z  e.  V  ( G `  z )  =/=  .0.  ) )
22213impia 1228 . 2  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  ->  E. z  e.  V  ( G `  z )  =/=  .0.  )
23 simp1l 1054 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  W  e.  LVec )
24 lveclmod 18407 . . . . . . 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 18406 . . . . . . . 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 1055 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  G  e.  F )
29 simp2 1031 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  z  e.  V )
303, 4, 5, 6lflcl 32701 . . . . . . . 8  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  z  e.  V )  ->  ( G `  z )  e.  ( Base `  D
) )
3123, 28, 29, 30syl3anc 1292 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( G `  z
)  e.  ( Base `  D ) )
32 simp3 1032 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( G `  z
)  =/=  .0.  )
33 eqid 2471 . . . . . . . 8  |-  ( invr `  D )  =  (
invr `  D )
344, 13, 33drnginvrcl 18070 . . . . . . 7  |-  ( ( D  e.  DivRing  /\  ( G `  z )  e.  ( Base `  D
)  /\  ( G `  z )  =/=  .0.  )  ->  ( ( invr `  D ) `  ( G `  z )
)  e.  ( Base `  D ) )
3527, 31, 32, 34syl3anc 1292 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( ( invr `  D
) `  ( G `  z ) )  e.  ( Base `  D
) )
36 eqid 2471 . . . . . . 7  |-  ( .s
`  W )  =  ( .s `  W
)
375, 3, 36, 4lmodvscl 18186 . . . . . 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 1292 . . . . 5  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( ( ( invr `  D ) `  ( G `  z )
) ( .s `  W ) z )  e.  V )
39 eqid 2471 . . . . . . . 8  |-  ( .r
`  D )  =  ( .r `  D
)
403, 4, 39, 5, 36, 6lflmul 32705 . . . . . . 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 1296 . . . . . 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 18072 . . . . . . 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 1292 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( ( ( invr `  D ) `  ( G `  z )
) ( .r `  D ) ( G `
 z ) )  =  .1.  )
4541, 44eqtrd 2505 . . . . 5  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( G `  (
( ( invr `  D
) `  ( G `  z ) ) ( .s `  W ) z ) )  =  .1.  )
46 fveq2 5879 . . . . . . 7  |-  ( x  =  ( ( (
invr `  D ) `  ( G `  z
) ) ( .s
`  W ) z )  ->  ( G `  x )  =  ( G `  ( ( ( invr `  D
) `  ( G `  z ) ) ( .s `  W ) z ) ) )
4746eqeq1d 2473 . . . . . 6  |-  ( x  =  ( ( (
invr `  D ) `  ( G `  z
) ) ( .s
`  W ) z )  ->  ( ( G `  x )  =  .1.  <->  ( G `  ( ( ( invr `  D ) `  ( G `  z )
) ( .s `  W ) z ) )  =  .1.  )
)
4847rspcev 3136 . . . . 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 673 . . . 4  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  E. x  e.  V  ( G `  x )  =  .1.  )
5049rexlimdv3a 2873 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( E. z  e.  V  ( G `  z )  =/=  .0.  ->  E. x  e.  V  ( G `  x )  =  .1.  ) )
51503adant3 1050 . 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 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   _Vcvv 3031   {csn 3959    X. cxp 4837    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   Basecbs 15199   .rcmulr 15269  Scalarcsca 15271   .scvsca 15272   0gc0g 15416   1rcur 17813   invrcinvr 17977   DivRingcdr 18053   LModclmod 18169   LVecclvec 18403  LFnlclfn 32694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mgp 17802  df-ur 17814  df-ring 17860  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-drng 18055  df-lmod 18171  df-lvec 18404  df-lfl 32695
This theorem is referenced by:  eqlkr  32736  lkrshp  32742
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