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Theorem lsatn0 29241
Description: A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 23033 analog.) (Contributed by NM, 25-Aug-2014.)
Hypotheses
Ref Expression
lsatn0.o  |-  .0.  =  ( 0g `  W )
lsatn0.a  |-  A  =  (LSAtoms `  W )
lsatn0.w  |-  ( ph  ->  W  e.  LMod )
lsatn0.u  |-  ( ph  ->  U  e.  A )
Assertion
Ref Expression
lsatn0  |-  ( ph  ->  U  =/=  {  .0.  } )

Proof of Theorem lsatn0
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 lsatn0.u . . 3  |-  ( ph  ->  U  e.  A )
2 lsatn0.w . . . 4  |-  ( ph  ->  W  e.  LMod )
3 eqid 2358 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2358 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 lsatn0.o . . . . 5  |-  .0.  =  ( 0g `  W )
6 lsatn0.a . . . . 5  |-  A  =  (LSAtoms `  W )
73, 4, 5, 6islsat 29233 . . . 4  |-  ( W  e.  LMod  ->  ( U  e.  A  <->  E. v  e.  ( ( Base `  W
)  \  {  .0.  } ) U  =  ( ( LSpan `  W ) `  { v } ) ) )
82, 7syl 15 . . 3  |-  ( ph  ->  ( U  e.  A  <->  E. v  e.  ( (
Base `  W )  \  {  .0.  } ) U  =  ( (
LSpan `  W ) `  { v } ) ) )
91, 8mpbid 201 . 2  |-  ( ph  ->  E. v  e.  ( ( Base `  W
)  \  {  .0.  } ) U  =  ( ( LSpan `  W ) `  { v } ) )
10 eldifsn 3825 . . . . 5  |-  ( v  e.  ( ( Base `  W )  \  {  .0.  } )  <->  ( v  e.  ( Base `  W
)  /\  v  =/=  .0.  ) )
113, 5, 4lspsneq0 15862 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  v  e.  ( Base `  W
) )  ->  (
( ( LSpan `  W
) `  { v } )  =  {  .0.  }  <->  v  =  .0.  ) )
122, 11sylan 457 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  ( (
( LSpan `  W ) `  { v } )  =  {  .0.  }  <->  v  =  .0.  ) )
1312biimpd 198 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  ( (
( LSpan `  W ) `  { v } )  =  {  .0.  }  ->  v  =  .0.  )
)
1413necon3d 2559 . . . . . 6  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  ( v  =/=  .0.  ->  ( ( LSpan `  W ) `  { v } )  =/=  {  .0.  }
) )
1514expimpd 586 . . . . 5  |-  ( ph  ->  ( ( v  e.  ( Base `  W
)  /\  v  =/=  .0.  )  ->  ( (
LSpan `  W ) `  { v } )  =/=  {  .0.  }
) )
1610, 15syl5bi 208 . . . 4  |-  ( ph  ->  ( v  e.  ( ( Base `  W
)  \  {  .0.  } )  ->  ( ( LSpan `  W ) `  { v } )  =/=  {  .0.  }
) )
17 neeq1 2529 . . . . 5  |-  ( U  =  ( ( LSpan `  W ) `  {
v } )  -> 
( U  =/=  {  .0.  }  <->  ( ( LSpan `  W ) `  {
v } )  =/= 
{  .0.  } ) )
1817biimprcd 216 . . . 4  |-  ( ( ( LSpan `  W ) `  { v } )  =/=  {  .0.  }  ->  ( U  =  ( ( LSpan `  W ) `  { v } )  ->  U  =/=  {  .0.  } ) )
1916, 18syl6 29 . . 3  |-  ( ph  ->  ( v  e.  ( ( Base `  W
)  \  {  .0.  } )  ->  ( U  =  ( ( LSpan `  W ) `  {
v } )  ->  U  =/=  {  .0.  }
) ) )
2019rexlimdv 2742 . 2  |-  ( ph  ->  ( E. v  e.  ( ( Base `  W
)  \  {  .0.  } ) U  =  ( ( LSpan `  W ) `  { v } )  ->  U  =/=  {  .0.  } ) )
219, 20mpd 14 1  |-  ( ph  ->  U  =/=  {  .0.  } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   E.wrex 2620    \ cdif 3225   {csn 3716   ` cfv 5334   Basecbs 13239   0gc0g 13493   LModclmod 15720   LSpanclspn 15821  LSAtomsclsa 29216
This theorem is referenced by:  lsatspn0  29242  lsatssn0  29244  lsatcmp  29245  lsatcv0  29273  dochsat  31625  dochsatshp  31693  dochshpsat  31696  dochexmidlem1  31702
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-2 9891  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-plusg 13312  df-0g 13497  df-mnd 14460  df-grp 14582  df-mgp 15419  df-rng 15433  df-lmod 15722  df-lss 15783  df-lsp 15822  df-lsatoms 29218
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