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Theorem lfl1 29260
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 2450 . . . . . . 7  |-  ( -.  ( G `  z
)  =/=  .0.  <->  ( G `  z )  =  .0.  )
21ralbii 2567 . . . . . 6  |-  ( A. z  e.  V  -.  ( G `  z )  =/=  .0.  <->  A. z  e.  V  ( G `  z )  =  .0.  )
3 lfl1.d . . . . . . . . . 10  |-  D  =  (Scalar `  W )
4 eqid 2283 . . . . . . . . . 10  |-  ( Base `  D )  =  (
Base `  D )
5 lfl1.v . . . . . . . . . 10  |-  V  =  ( Base `  W
)
6 lfl1.f . . . . . . . . . 10  |-  F  =  (LFnl `  W )
73, 4, 5, 6lflf 29253 . . . . . . . . 9  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  G : V --> ( Base `  D
) )
8 ffn 5389 . . . . . . . . 9  |-  ( G : V --> ( Base `  D )  ->  G  Fn  V )
97, 8syl 15 . . . . . . . 8  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  G  Fn  V )
10 fconstfv 5734 . . . . . . . . 9  |-  ( G : V --> {  .0.  }  <-> 
( G  Fn  V  /\  A. z  e.  V  ( G `  z )  =  .0.  ) )
1110simplbi2 608 . . . . . . . 8  |-  ( G  Fn  V  ->  ( A. z  e.  V  ( G `  z )  =  .0.  ->  G : V --> {  .0.  }
) )
129, 11syl 15 . . . . . . 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 5539 . . . . . . . . 9  |-  ( 0g
`  D )  e. 
_V
1513, 14eqeltri 2353 . . . . . . . 8  |-  .0.  e.  _V
1615fconst2 5730 . . . . . . 7  |-  ( G : V --> {  .0.  }  <-> 
G  =  ( V  X.  {  .0.  }
) )
1712, 16syl6ib 217 . . . . . 6  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( A. z  e.  V  ( G `  z )  =  .0.  ->  G  =  ( V  X.  {  .0.  } ) ) )
182, 17syl5bi 208 . . . . 5  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( A. z  e.  V  -.  ( G `  z
)  =/=  .0.  ->  G  =  ( V  X.  {  .0.  } ) ) )
1918necon3ad 2482 . . . 4  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( G  =/=  ( V  X.  {  .0.  } )  ->  -.  A. z  e.  V  -.  ( G `  z
)  =/=  .0.  )
)
20 dfrex2 2556 . . . 4  |-  ( E. z  e.  V  ( G `  z )  =/=  .0.  <->  -.  A. z  e.  V  -.  ( G `  z )  =/=  .0.  )
2119, 20syl6ibr 218 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( G  =/=  ( V  X.  {  .0.  } )  ->  E. z  e.  V  ( G `  z )  =/=  .0.  ) )
22213impia 1148 . 2  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  ->  E. z  e.  V  ( G `  z )  =/=  .0.  )
23 simp1l 979 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  W  e.  LVec )
24 lveclmod 15859 . . . . . . 7  |-  ( W  e.  LVec  ->  W  e. 
LMod )
2523, 24syl 15 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  W  e.  LMod )
263lvecdrng 15858 . . . . . . . 8  |-  ( W  e.  LVec  ->  D  e.  DivRing )
2723, 26syl 15 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  D  e.  DivRing )
28 simp1r 980 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  G  e.  F )
29 simp2 956 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  z  e.  V )
303, 4, 5, 6lflcl 29254 . . . . . . . 8  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  z  e.  V )  ->  ( G `  z )  e.  ( Base `  D
) )
3123, 28, 29, 30syl3anc 1182 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( G `  z
)  e.  ( Base `  D ) )
32 simp3 957 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( G `  z
)  =/=  .0.  )
33 eqid 2283 . . . . . . . 8  |-  ( invr `  D )  =  (
invr `  D )
344, 13, 33drnginvrcl 15529 . . . . . . 7  |-  ( ( D  e.  DivRing  /\  ( G `  z )  e.  ( Base `  D
)  /\  ( G `  z )  =/=  .0.  )  ->  ( ( invr `  D ) `  ( G `  z )
)  e.  ( Base `  D ) )
3527, 31, 32, 34syl3anc 1182 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( ( invr `  D
) `  ( G `  z ) )  e.  ( Base `  D
) )
36 eqid 2283 . . . . . . 7  |-  ( .s
`  W )  =  ( .s `  W
)
375, 3, 36, 4lmodvscl 15644 . . . . . 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 1182 . . . . 5  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( ( ( invr `  D ) `  ( G `  z )
) ( .s `  W ) z )  e.  V )
39 eqid 2283 . . . . . . . 8  |-  ( .r
`  D )  =  ( .r `  D
)
403, 4, 39, 5, 36, 6lflmul 29258 . . . . . . 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 1186 . . . . . 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 15531 . . . . . . 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 1182 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( ( ( invr `  D ) `  ( G `  z )
) ( .r `  D ) ( G `
 z ) )  =  .1.  )
4541, 44eqtrd 2315 . . . . 5  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  ( G `  (
( ( invr `  D
) `  ( G `  z ) ) ( .s `  W ) z ) )  =  .1.  )
46 fveq2 5525 . . . . . . 7  |-  ( x  =  ( ( (
invr `  D ) `  ( G `  z
) ) ( .s
`  W ) z )  ->  ( G `  x )  =  ( G `  ( ( ( invr `  D
) `  ( G `  z ) ) ( .s `  W ) z ) ) )
4746eqeq1d 2291 . . . . . 6  |-  ( x  =  ( ( (
invr `  D ) `  ( G `  z
) ) ( .s
`  W ) z )  ->  ( ( G `  x )  =  .1.  <->  ( G `  ( ( ( invr `  D ) `  ( G `  z )
) ( .s `  W ) z ) )  =  .1.  )
)
4847rspcev 2884 . . . . 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 642 . . . 4  |-  ( ( ( W  e.  LVec  /\  G  e.  F )  /\  z  e.  V  /\  ( G `  z
)  =/=  .0.  )  ->  E. x  e.  V  ( G `  x )  =  .1.  )
5049rexlimdv3a 2669 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F )  ->  ( E. z  e.  V  ( G `  z )  =/=  .0.  ->  E. x  e.  V  ( G `  x )  =  .1.  ) )
51503adant3 975 . 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 14 1  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  ->  E. x  e.  V  ( G `  x )  =  .1.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788   {csn 3640    X. cxp 4687    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   1rcur 15339   invrcinvr 15453   DivRingcdr 15512   LModclmod 15627   LVecclvec 15855  LFnlclfn 29247
This theorem is referenced by:  eqlkr  29289  lkrshp  29295
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lvec 15856  df-lfl 29248
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