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Theorem lsateln0 32269
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 2429 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2429 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 lsateln0.z . . . . . 6  |-  .0.  =  ( 0g `  W )
6 lsateln0.a . . . . . 6  |-  A  =  (LSAtoms `  W )
73, 4, 5, 6islsat 32265 . . . . 5  |-  ( W  e.  LMod  ->  ( U  e.  A  <->  E. v  e.  ( ( Base `  W
)  \  {  .0.  } ) U  =  ( ( LSpan `  W ) `  { v } ) ) )
82, 7syl 17 . . . 4  |-  ( ph  ->  ( U  e.  A  <->  E. v  e.  ( (
Base `  W )  \  {  .0.  } ) U  =  ( (
LSpan `  W ) `  { v } ) ) )
91, 8mpbid 213 . . 3  |-  ( ph  ->  E. v  e.  ( ( Base `  W
)  \  {  .0.  } ) U  =  ( ( LSpan `  W ) `  { v } ) )
10 eldifi 3593 . . . . . 6  |-  ( v  e.  ( ( Base `  W )  \  {  .0.  } )  ->  v  e.  ( Base `  W
) )
113, 4lspsnid 18151 . . . . . 6  |-  ( ( W  e.  LMod  /\  v  e.  ( Base `  W
) )  ->  v  e.  ( ( LSpan `  W
) `  { v } ) )
122, 10, 11syl2an 479 . . . . 5  |-  ( (
ph  /\  v  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  v  e.  ( ( LSpan `  W
) `  { v } ) )
13 eleq2 2502 . . . . 5  |-  ( U  =  ( ( LSpan `  W ) `  {
v } )  -> 
( v  e.  U  <->  v  e.  ( ( LSpan `  W ) `  {
v } ) ) )
1412, 13syl5ibrcom 225 . . . 4  |-  ( (
ph  /\  v  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  ( U  =  ( ( LSpan `  W ) `  { v } )  ->  v  e.  U
) )
1514reximdva 2907 . . 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 4128 . . . . . . 7  |-  ( v  e.  ( ( Base `  W )  \  {  .0.  } )  <->  ( v  e.  ( Base `  W
)  /\  v  =/=  .0.  ) )
1817anbi1i 699 . . . . . 6  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  <->  ( ( v  e.  ( Base `  W
)  /\  v  =/=  .0.  )  /\  v  e.  U ) )
19 anass 653 . . . . . 6  |-  ( ( ( v  e.  (
Base `  W )  /\  v  =/=  .0.  )  /\  v  e.  U
)  <->  ( v  e.  ( Base `  W
)  /\  ( v  =/=  .0.  /\  v  e.  U ) ) )
2018, 19bitri 252 . . . . 5  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  <->  ( v  e.  ( Base `  W
)  /\  ( v  =/=  .0.  /\  v  e.  U ) ) )
2120simprbi 465 . . . 4  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  ->  ( v  =/=  .0.  /\  v  e.  U ) )
2221ancomd 452 . . 3  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  ->  ( v  e.  U  /\  v  =/=  .0.  ) )
2322reximi2 2899 . 2  |-  ( E. v  e.  ( (
Base `  W )  \  {  .0.  } ) v  e.  U  ->  E. v  e.  U  v  =/=  .0.  )
2416, 23syl 17 1  |-  ( ph  ->  E. v  e.  U  v  =/=  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   E.wrex 2783    \ cdif 3439   {csn 4002   ` cfv 5601   Basecbs 15084   0gc0g 15297   LModclmod 18026   LSpanclspn 18129  LSAtomsclsa 32248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-lmod 18028  df-lss 18091  df-lsp 18130  df-lsatoms 32250
This theorem is referenced by:  dvh1dim  34718  dochkr1  34754  dochkr1OLDN  34755  lcfrlem40  34858
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