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Theorem lsateln0 33810
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 2467 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2467 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 lsateln0.z . . . . . 6  |-  .0.  =  ( 0g `  W )
6 lsateln0.a . . . . . 6  |-  A  =  (LSAtoms `  W )
73, 4, 5, 6islsat 33806 . . . . 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 210 . . 3  |-  ( ph  ->  E. v  e.  ( ( Base `  W
)  \  {  .0.  } ) U  =  ( ( LSpan `  W ) `  { v } ) )
10 eldifi 3626 . . . . . 6  |-  ( v  e.  ( ( Base `  W )  \  {  .0.  } )  ->  v  e.  ( Base `  W
) )
113, 4lspsnid 17439 . . . . . 6  |-  ( ( W  e.  LMod  /\  v  e.  ( Base `  W
) )  ->  v  e.  ( ( LSpan `  W
) `  { v } ) )
122, 10, 11syl2an 477 . . . . 5  |-  ( (
ph  /\  v  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  v  e.  ( ( LSpan `  W
) `  { v } ) )
13 eleq2 2540 . . . . 5  |-  ( U  =  ( ( LSpan `  W ) `  {
v } )  -> 
( v  e.  U  <->  v  e.  ( ( LSpan `  W ) `  {
v } ) ) )
1412, 13syl5ibrcom 222 . . . 4  |-  ( (
ph  /\  v  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  ( U  =  ( ( LSpan `  W ) `  { v } )  ->  v  e.  U
) )
1514reximdva 2938 . . 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 4152 . . . . . . 7  |-  ( v  e.  ( ( Base `  W )  \  {  .0.  } )  <->  ( v  e.  ( Base `  W
)  /\  v  =/=  .0.  ) )
1817anbi1i 695 . . . . . 6  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  <->  ( ( v  e.  ( Base `  W
)  /\  v  =/=  .0.  )  /\  v  e.  U ) )
19 anass 649 . . . . . 6  |-  ( ( ( v  e.  (
Base `  W )  /\  v  =/=  .0.  )  /\  v  e.  U
)  <->  ( v  e.  ( Base `  W
)  /\  ( v  =/=  .0.  /\  v  e.  U ) ) )
2018, 19bitri 249 . . . . 5  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  <->  ( v  e.  ( Base `  W
)  /\  ( v  =/=  .0.  /\  v  e.  U ) ) )
2120simprbi 464 . . . 4  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  ->  ( v  =/=  .0.  /\  v  e.  U ) )
2221ancomd 451 . . 3  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  ->  ( v  e.  U  /\  v  =/=  .0.  ) )
2322reximi2 2931 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    \ cdif 3473   {csn 4027   ` cfv 5588   Basecbs 14490   0gc0g 14695   LModclmod 17312   LSpanclspn 17417  LSAtomsclsa 33789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-0g 14697  df-mnd 15732  df-grp 15867  df-lmod 17314  df-lss 17379  df-lsp 17418  df-lsatoms 33791
This theorem is referenced by:  dvh1dim  36257  dochkr1  36293  dochkr1OLDN  36294  lcfrlem40  36397
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