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Theorem lsatspn0 29008
Description: The span of a vector is an atom iff the vector is nonzero. (Contributed by NM, 4-Feb-2015.)
Hypotheses
Ref Expression
lsatspn0.v  |-  V  =  ( Base `  W
)
lsatspn0.n  |-  N  =  ( LSpan `  W )
lsatspn0.o  |-  .0.  =  ( 0g `  W )
lsatspn0.a  |-  A  =  (LSAtoms `  W )
isateln0.w  |-  ( ph  ->  W  e.  LMod )
isateln0.x  |-  ( ph  ->  X  e.  V )
Assertion
Ref Expression
lsatspn0  |-  ( ph  ->  ( ( N `  { X } )  e.  A  <->  X  =/=  .0.  ) )

Proof of Theorem lsatspn0
StepHypRef Expression
1 lsatspn0.o . . . 4  |-  .0.  =  ( 0g `  W )
2 lsatspn0.a . . . 4  |-  A  =  (LSAtoms `  W )
3 isateln0.w . . . . 5  |-  ( ph  ->  W  e.  LMod )
43adantr 451 . . . 4  |-  ( (
ph  /\  ( N `  { X } )  e.  A )  ->  W  e.  LMod )
5 simpr 447 . . . 4  |-  ( (
ph  /\  ( N `  { X } )  e.  A )  -> 
( N `  { X } )  e.  A
)
61, 2, 4, 5lsatn0 29007 . . 3  |-  ( (
ph  /\  ( N `  { X } )  e.  A )  -> 
( N `  { X } )  =/=  {  .0.  } )
7 sneq 3685 . . . . . . . 8  |-  ( X  =  .0.  ->  { X }  =  {  .0.  } )
87fveq2d 5567 . . . . . . 7  |-  ( X  =  .0.  ->  ( N `  { X } )  =  ( N `  {  .0.  } ) )
98adantl 452 . . . . . 6  |-  ( ( ( ph  /\  ( N `  { X } )  e.  A
)  /\  X  =  .0.  )  ->  ( N `
 { X }
)  =  ( N `
 {  .0.  }
) )
104adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  ( N `  { X } )  e.  A
)  /\  X  =  .0.  )  ->  W  e. 
LMod )
11 lsatspn0.n . . . . . . . 8  |-  N  =  ( LSpan `  W )
121, 11lspsn0 15814 . . . . . . 7  |-  ( W  e.  LMod  ->  ( N `
 {  .0.  }
)  =  {  .0.  } )
1310, 12syl 15 . . . . . 6  |-  ( ( ( ph  /\  ( N `  { X } )  e.  A
)  /\  X  =  .0.  )  ->  ( N `
 {  .0.  }
)  =  {  .0.  } )
149, 13eqtrd 2348 . . . . 5  |-  ( ( ( ph  /\  ( N `  { X } )  e.  A
)  /\  X  =  .0.  )  ->  ( N `
 { X }
)  =  {  .0.  } )
1514ex 423 . . . 4  |-  ( (
ph  /\  ( N `  { X } )  e.  A )  -> 
( X  =  .0. 
->  ( N `  { X } )  =  {  .0.  } ) )
1615necon3d 2517 . . 3  |-  ( (
ph  /\  ( N `  { X } )  e.  A )  -> 
( ( N `  { X } )  =/= 
{  .0.  }  ->  X  =/=  .0.  ) )
176, 16mpd 14 . 2  |-  ( (
ph  /\  ( N `  { X } )  e.  A )  ->  X  =/=  .0.  )
18 lsatspn0.v . . 3  |-  V  =  ( Base `  W
)
193adantr 451 . . 3  |-  ( (
ph  /\  X  =/=  .0.  )  ->  W  e. 
LMod )
20 isateln0.x . . . . 5  |-  ( ph  ->  X  e.  V )
2120adantr 451 . . . 4  |-  ( (
ph  /\  X  =/=  .0.  )  ->  X  e.  V )
22 simpr 447 . . . 4  |-  ( (
ph  /\  X  =/=  .0.  )  ->  X  =/= 
.0.  )
23 eldifsn 3783 . . . 4  |-  ( X  e.  ( V  \  {  .0.  } )  <->  ( X  e.  V  /\  X  =/= 
.0.  ) )
2421, 22, 23sylanbrc 645 . . 3  |-  ( (
ph  /\  X  =/=  .0.  )  ->  X  e.  ( V  \  {  .0.  } ) )
2518, 11, 1, 2, 19, 24lsatlspsn 29001 . 2  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( N `
 { X }
)  e.  A )
2617, 25impbida 805 1  |-  ( ph  ->  ( ( N `  { X } )  e.  A  <->  X  =/=  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701    =/= wne 2479    \ cdif 3183   {csn 3674   ` cfv 5292   Basecbs 13195   0gc0g 13449   LModclmod 15676   LSpanclspn 15777  LSAtomsclsa 28982
This theorem is referenced by:  lsator0sp  29009  lcfl8b  31512  mapdpglem5N  31685  mapdpglem30a  31703  mapdpglem30b  31704
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-plusg 13268  df-0g 13453  df-mnd 14416  df-grp 14538  df-mgp 15375  df-rng 15389  df-lmod 15678  df-lss 15739  df-lsp 15778  df-lsatoms 28984
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