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Theorem lsateln0 29482
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 2408 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2408 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 lsateln0.z . . . . . 6  |-  .0.  =  ( 0g `  W )
6 lsateln0.a . . . . . 6  |-  A  =  (LSAtoms `  W )
73, 4, 5, 6islsat 29478 . . . . 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 202 . . 3  |-  ( ph  ->  E. v  e.  ( ( Base `  W
)  \  {  .0.  } ) U  =  ( ( LSpan `  W ) `  { v } ) )
10 eldifi 3433 . . . . . 6  |-  ( v  e.  ( ( Base `  W )  \  {  .0.  } )  ->  v  e.  ( Base `  W
) )
113, 4lspsnid 16028 . . . . . 6  |-  ( ( W  e.  LMod  /\  v  e.  ( Base `  W
) )  ->  v  e.  ( ( LSpan `  W
) `  { v } ) )
122, 10, 11syl2an 464 . . . . 5  |-  ( (
ph  /\  v  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  v  e.  ( ( LSpan `  W
) `  { v } ) )
13 eleq2 2469 . . . . 5  |-  ( U  =  ( ( LSpan `  W ) `  {
v } )  -> 
( v  e.  U  <->  v  e.  ( ( LSpan `  W ) `  {
v } ) ) )
1412, 13syl5ibrcom 214 . . . 4  |-  ( (
ph  /\  v  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  ( U  =  ( ( LSpan `  W ) `  { v } )  ->  v  e.  U
) )
1514reximdva 2782 . . 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 3891 . . . . . . 7  |-  ( v  e.  ( ( Base `  W )  \  {  .0.  } )  <->  ( v  e.  ( Base `  W
)  /\  v  =/=  .0.  ) )
1817anbi1i 677 . . . . . 6  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  <->  ( ( v  e.  ( Base `  W
)  /\  v  =/=  .0.  )  /\  v  e.  U ) )
19 anass 631 . . . . . 6  |-  ( ( ( v  e.  (
Base `  W )  /\  v  =/=  .0.  )  /\  v  e.  U
)  <->  ( v  e.  ( Base `  W
)  /\  ( v  =/=  .0.  /\  v  e.  U ) ) )
2018, 19bitri 241 . . . . 5  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  <->  ( v  e.  ( Base `  W
)  /\  ( v  =/=  .0.  /\  v  e.  U ) ) )
2120simprbi 451 . . . 4  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  ->  ( v  =/=  .0.  /\  v  e.  U ) )
2221ancomd 439 . . 3  |-  ( ( v  e.  ( (
Base `  W )  \  {  .0.  } )  /\  v  e.  U
)  ->  ( v  e.  U  /\  v  =/=  .0.  ) )
2322reximi2 2776 . 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2571   E.wrex 2671    \ cdif 3281   {csn 3778   ` cfv 5417   Basecbs 13428   0gc0g 13682   LModclmod 15909   LSpanclspn 16006  LSAtomsclsa 29461
This theorem is referenced by:  dvh1dim  31929  dochkr1  31965  dochkr1OLDN  31966  lcfrlem40  32069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-riota 6512  df-0g 13686  df-mnd 14649  df-grp 14771  df-lmod 15911  df-lss 15968  df-lsp 16007  df-lsatoms 29463
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