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Theorem lsateln0 29867
Description: A 1-dim subspace (atom) (of a left module or left vector space) contains a nonzero vector. (Contributed by NM, 2-Jan-2015.)
Hypotheses
Ref Expression
lsateln0.z  |-  .0.  =  ( 0g `  W )
lsateln0.a  |-  A  =  (LSAtoms `  W )
lsateln0.w  |-  ( ph  ->  W  e.  LMod )
lsateln0.u  |-  ( ph  ->  U  e.  A )
Assertion
Ref Expression
lsateln0  |-  ( ph  ->  E. v  e.  U  v  =/=  .0.  )
Distinct variable groups:    v, U    v, W    v,  .0.    ph, v
Allowed substitution hint:    A( v)

Proof of Theorem lsateln0
StepHypRef Expression
1 lsateln0.u . . . 4  |-  ( ph  ->  U  e.  A )
2 lsateln0.w . . . . 5  |-  ( ph  ->  W  e.  LMod )
3 eqid 2438 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2438 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 lsateln0.z . . . . . 6  |-  .0.  =  ( 0g `  W )
6 lsateln0.a . . . . . 6  |-  A  =  (LSAtoms `  W )
73, 4, 5, 6islsat 29863 . . . . 5  |-  ( W  e.  LMod  ->  ( U  e.  A  <->  E. v  e.  ( ( Base `  W
)  \  {  .0.  } ) U  =  ( ( LSpan `  W ) `  { v } ) ) )
82, 7syl 16 . . . 4  |-  ( ph  ->  ( U  e.  A  <->  E. v  e.  ( (
Base `  W )  \  {  .0.  } ) U  =  ( (
LSpan `  W ) `  { v } ) ) )
91, 8mpbid 203 . . 3  |-  ( ph  ->  E. v  e.  ( ( Base `  W
)  \  {  .0.  } ) U  =  ( ( LSpan `  W ) `  { v } ) )
10 eldifi 3471 . . . . . 6  |-  ( v  e.  ( ( Base `  W )  \  {  .0.  } )  ->  v  e.  ( Base `  W
) )
113, 4lspsnid 16074 . . . . . 6  |-  ( ( W  e.  LMod  /\  v  e.  ( Base `  W
) )  ->  v  e.  ( ( LSpan `  W
) `  { v } ) )
122, 10, 11syl2an 465 . . . . 5  |-  ( (
ph  /\  v  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  v  e.  ( ( LSpan `  W
) `  { v } ) )
13 eleq2 2499 . . . . 5  |-  ( U  =  ( ( LSpan `  W ) `  {
v } )  -> 
( v  e.  U  <->  v  e.  ( ( LSpan `  W ) `  {
v } ) ) )
1412, 13syl5ibrcom 215 . . . 4  |-  ( (
ph  /\  v  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  ( U  =  ( ( LSpan `  W ) `  { v } )  ->  v  e.  U
) )
1514reximdva 2820 . . 3  |-  ( ph  ->  ( E. v  e.  ( ( Base `  W
)  \  {  .0.  } ) U  =  ( ( LSpan `  W ) `  { v } )  ->  E. v  e.  ( ( Base `  W
)  \  {  .0.  } ) v  e.  U
) )
169, 15mpd 15 . 2  |-  ( ph  ->  E. v  e.  ( ( Base `  W
)  \  {  .0.  } ) v  e.  U
)
17 eldifsn 3929 . . . . . . 7  |-  ( v  e.  ( ( Base `  W )  \  {  .0.  } )  <->  ( v  e.  ( Base `  W
)  /\  v  =/=  .0.  ) )
1817anbi1i 678 . . . . . 6  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  <->  ( ( v  e.  ( Base `  W
)  /\  v  =/=  .0.  )  /\  v  e.  U ) )
19 anass 632 . . . . . 6  |-  ( ( ( v  e.  (
Base `  W )  /\  v  =/=  .0.  )  /\  v  e.  U
)  <->  ( v  e.  ( Base `  W
)  /\  ( v  =/=  .0.  /\  v  e.  U ) ) )
2018, 19bitri 242 . . . . 5  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  <->  ( v  e.  ( Base `  W
)  /\  ( v  =/=  .0.  /\  v  e.  U ) ) )
2120simprbi 452 . . . 4  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  ->  ( v  =/=  .0.  /\  v  e.  U ) )
2221ancomd 440 . . 3  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  ->  ( v  e.  U  /\  v  =/=  .0.  ) )
2322reximi2 2814 . 2  |-  ( E. v  e.  ( (
Base `  W )  \  {  .0.  } ) v  e.  U  ->  E. v  e.  U  v  =/=  .0.  )
2416, 23syl 16 1  |-  ( ph  ->  E. v  e.  U  v  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    \ cdif 3319   {csn 3816   ` cfv 5457   Basecbs 13474   0gc0g 13728   LModclmod 15955   LSpanclspn 16052  LSAtomsclsa 29846
This theorem is referenced by:  dvh1dim  32314  dochkr1  32350  dochkr1OLDN  32351  lcfrlem40  32454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-riota 6552  df-0g 13732  df-mnd 14695  df-grp 14817  df-lmod 15957  df-lss 16014  df-lsp 16053  df-lsatoms 29848
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