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Theorem lsateln0 29244
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 2366 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2366 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 lsateln0.z . . . . . 6  |-  .0.  =  ( 0g `  W )
6 lsateln0.a . . . . . 6  |-  A  =  (LSAtoms `  W )
73, 4, 5, 6islsat 29240 . . . . 5  |-  ( W  e.  LMod  ->  ( U  e.  A  <->  E. v  e.  ( ( Base `  W
)  \  {  .0.  } ) U  =  ( ( LSpan `  W ) `  { v } ) ) )
82, 7syl 15 . . . 4  |-  ( ph  ->  ( U  e.  A  <->  E. v  e.  ( (
Base `  W )  \  {  .0.  } ) U  =  ( (
LSpan `  W ) `  { v } ) ) )
91, 8mpbid 201 . . 3  |-  ( ph  ->  E. v  e.  ( ( Base `  W
)  \  {  .0.  } ) U  =  ( ( LSpan `  W ) `  { v } ) )
10 eldifi 3385 . . . . . 6  |-  ( v  e.  ( ( Base `  W )  \  {  .0.  } )  ->  v  e.  ( Base `  W
) )
113, 4lspsnid 15960 . . . . . 6  |-  ( ( W  e.  LMod  /\  v  e.  ( Base `  W
) )  ->  v  e.  ( ( LSpan `  W
) `  { v } ) )
122, 10, 11syl2an 463 . . . . 5  |-  ( (
ph  /\  v  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  v  e.  ( ( LSpan `  W
) `  { v } ) )
13 eleq2 2427 . . . . 5  |-  ( U  =  ( ( LSpan `  W ) `  {
v } )  -> 
( v  e.  U  <->  v  e.  ( ( LSpan `  W ) `  {
v } ) ) )
1412, 13syl5ibrcom 213 . . . 4  |-  ( (
ph  /\  v  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  ( U  =  ( ( LSpan `  W ) `  { v } )  ->  v  e.  U
) )
1514reximdva 2740 . . 3  |-  ( ph  ->  ( E. v  e.  ( ( Base `  W
)  \  {  .0.  } ) U  =  ( ( LSpan `  W ) `  { v } )  ->  E. v  e.  ( ( Base `  W
)  \  {  .0.  } ) v  e.  U
) )
169, 15mpd 14 . 2  |-  ( ph  ->  E. v  e.  ( ( Base `  W
)  \  {  .0.  } ) v  e.  U
)
17 eldifsn 3842 . . . . . . 7  |-  ( v  e.  ( ( Base `  W )  \  {  .0.  } )  <->  ( v  e.  ( Base `  W
)  /\  v  =/=  .0.  ) )
1817anbi1i 676 . . . . . 6  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  <->  ( ( v  e.  ( Base `  W
)  /\  v  =/=  .0.  )  /\  v  e.  U ) )
19 anass 630 . . . . . 6  |-  ( ( ( v  e.  (
Base `  W )  /\  v  =/=  .0.  )  /\  v  e.  U
)  <->  ( v  e.  ( Base `  W
)  /\  ( v  =/=  .0.  /\  v  e.  U ) ) )
2018, 19bitri 240 . . . . 5  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  <->  ( v  e.  ( Base `  W
)  /\  ( v  =/=  .0.  /\  v  e.  U ) ) )
2120simprbi 450 . . . 4  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  ->  ( v  =/=  .0.  /\  v  e.  U ) )
2221ancomd 438 . . 3  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  ->  ( v  e.  U  /\  v  =/=  .0.  ) )
2322reximi2 2734 . 2  |-  ( E. v  e.  ( (
Base `  W )  \  {  .0.  } ) v  e.  U  ->  E. v  e.  U  v  =/=  .0.  )
2416, 23syl 15 1  |-  ( ph  ->  E. v  e.  U  v  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715    =/= wne 2529   E.wrex 2629    \ cdif 3235   {csn 3729   ` cfv 5358   Basecbs 13356   0gc0g 13610   LModclmod 15837   LSpanclspn 15938  LSAtomsclsa 29223
This theorem is referenced by:  dvh1dim  31691  dochkr1  31727  dochkr1OLDN  31728  lcfrlem40  31831
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-riota 6446  df-0g 13614  df-mnd 14577  df-grp 14699  df-lmod 15839  df-lss 15900  df-lsp 15939  df-lsatoms 29225
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