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Theorem lfl1 35192
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 2655 . . . . . . 7  |-  ( -.  ( G `  z
)  =/=  .0.  <->  ( G `  z )  =  .0.  )
21ralbii 2885 . . . . . 6  |-  ( A. z  e.  V  -.  ( G `  z )  =/=  .0.  <->  A. z  e.  V  ( G `  z )  =  .0.  )
3 lfl1.d . . . . . . . . . 10  |-  D  =  (Scalar `  W )
4 eqid 2454 . . . . . . . . . 10  |-  ( Base `  D )  =  (
Base `  D )
5 lfl1.v . . . . . . . . . 10  |-  V  =  ( Base `  W
)
6 lfl1.f . . . . . . . . . 10  |-  F  =  (LFnl `  W )
73, 4, 5, 6lflf 35185 . . . . . . . . 9  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  G : V --> ( Base `  D
) )
8 ffn 5713 . . . . . . . . 9  |-  ( G : V --> ( Base `  D )  ->  G  Fn  V )
97, 8syl 16 . . . . . . . 8  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  G  Fn  V )
10 fconstfv 6108 . . . . . . . . 9  |-  ( G : V --> {  .0.  }  <-> 
( G  Fn  V  /\  A. z  e.  V  ( G `  z )  =  .0.  ) )
1110simplbi2 623 . . . . . . . 8  |-  ( G  Fn  V  ->  ( A. z  e.  V  ( G `  z )  =  .0.  ->  G : V --> {  .0.  }
) )
129, 11syl 16 . . . . . . 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 5858 . . . . . . . . 9  |-  ( 0g
`  D )  e. 
_V
1513, 14eqeltri 2538 . . . . . . . 8  |-  .0.  e.  _V
1615fconst2 6104 . . . . . . 7  |-  ( G : V --> {  .0.  }  <-> 
G  =  ( V  X.  {  .0.  }
) )
1712, 16syl6ib 226 . . . . . 6  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( A. z  e.  V  ( G `  z )  =  .0.  ->  G  =  ( V  X.  {  .0.  } ) ) )
182, 17syl5bi 217 . . . . 5  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( A. z  e.  V  -.  ( G `  z
)  =/=  .0.  ->  G  =  ( V  X.  {  .0.  } ) ) )
1918necon3ad 2664 . . . 4  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( G  =/=  ( V  X.  {  .0.  } )  ->  -.  A. z  e.  V  -.  ( G `  z
)  =/=  .0.  )
)
20 dfrex2 2905 . . . 4  |-  ( E. z  e.  V  ( G `  z )  =/=  .0.  <->  -.  A. z  e.  V  -.  ( G `  z )  =/=  .0.  )
2119, 20syl6ibr 227 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( G  =/=  ( V  X.  {  .0.  } )  ->  E. z  e.  V  ( G `  z )  =/=  .0.  ) )
22213impia 1191 . 2  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  ->  E. z  e.  V  ( G `  z )  =/=  .0.  )
23 simp1l 1018 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  W  e.  LVec )
24 lveclmod 17947 . . . . . . 7  |-  ( W  e.  LVec  ->  W  e. 
LMod )
2523, 24syl 16 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  W  e.  LMod )
263lvecdrng 17946 . . . . . . . 8  |-  ( W  e.  LVec  ->  D  e.  DivRing )
2723, 26syl 16 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  D  e.  DivRing )
28 simp1r 1019 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  G  e.  F )
29 simp2 995 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  z  e.  V )
303, 4, 5, 6lflcl 35186 . . . . . . . 8  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  z  e.  V )  ->  ( G `  z )  e.  ( Base `  D
) )
3123, 28, 29, 30syl3anc 1226 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( G `  z
)  e.  ( Base `  D ) )
32 simp3 996 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( G `  z
)  =/=  .0.  )
33 eqid 2454 . . . . . . . 8  |-  ( invr `  D )  =  (
invr `  D )
344, 13, 33drnginvrcl 17608 . . . . . . 7  |-  ( ( D  e.  DivRing  /\  ( G `  z )  e.  ( Base `  D
)  /\  ( G `  z )  =/=  .0.  )  ->  ( ( invr `  D ) `  ( G `  z )
)  e.  ( Base `  D ) )
3527, 31, 32, 34syl3anc 1226 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( ( invr `  D
) `  ( G `  z ) )  e.  ( Base `  D
) )
36 eqid 2454 . . . . . . 7  |-  ( .s
`  W )  =  ( .s `  W
)
375, 3, 36, 4lmodvscl 17724 . . . . . 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 1226 . . . . 5  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( ( ( invr `  D ) `  ( G `  z )
) ( .s `  W ) z )  e.  V )
39 eqid 2454 . . . . . . . 8  |-  ( .r
`  D )  =  ( .r `  D
)
403, 4, 39, 5, 36, 6lflmul 35190 . . . . . . 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 1230 . . . . . 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 17610 . . . . . . 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 1226 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( ( ( invr `  D ) `  ( G `  z )
) ( .r `  D ) ( G `
 z ) )  =  .1.  )
4541, 44eqtrd 2495 . . . . 5  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( G `  (
( ( invr `  D
) `  ( G `  z ) ) ( .s `  W ) z ) )  =  .1.  )
46 fveq2 5848 . . . . . . 7  |-  ( x  =  ( ( (
invr `  D ) `  ( G `  z
) ) ( .s
`  W ) z )  ->  ( G `  x )  =  ( G `  ( ( ( invr `  D
) `  ( G `  z ) ) ( .s `  W ) z ) ) )
4746eqeq1d 2456 . . . . . 6  |-  ( x  =  ( ( (
invr `  D ) `  ( G `  z
) ) ( .s
`  W ) z )  ->  ( ( G `  x )  =  .1.  <->  ( G `  ( ( ( invr `  D ) `  ( G `  z )
) ( .s `  W ) z ) )  =  .1.  )
)
4847rspcev 3207 . . . . 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 659 . . . 4  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  E. x  e.  V  ( G `  x )  =  .1.  )
5049rexlimdv3a 2948 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( E. z  e.  V  ( G `  z )  =/=  .0.  ->  E. x  e.  V  ( G `  x )  =  .1.  ) )
51503adant3 1014 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   _Vcvv 3106   {csn 4016    X. cxp 4986    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   Basecbs 14716   .rcmulr 14785  Scalarcsca 14787   .scvsca 14788   0gc0g 14929   1rcur 17348   invrcinvr 17515   DivRingcdr 17591   LModclmod 17707   LVecclvec 17943  LFnlclfn 35179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-sbg 16258  df-mgp 17337  df-ur 17349  df-ring 17395  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-drng 17593  df-lmod 17709  df-lvec 17944  df-lfl 35180
This theorem is referenced by:  eqlkr  35221  lkrshp  35227
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