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Theorem lssatomic 33683
Description: The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 26939 analog.) (Contributed by NM, 10-Jan-2015.)
Hypotheses
Ref Expression
lssatomic.s  |-  S  =  ( LSubSp `  W )
lssatomic.o  |-  .0.  =  ( 0g `  W )
lssatomic.a  |-  A  =  (LSAtoms `  W )
lssatomic.w  |-  ( ph  ->  W  e.  LMod )
lssatomic.u  |-  ( ph  ->  U  e.  S )
lssatomic.n  |-  ( ph  ->  U  =/=  {  .0.  } )
Assertion
Ref Expression
lssatomic  |-  ( ph  ->  E. q  e.  A  q  C_  U )
Distinct variable groups:    A, q    U, q    W, q
Allowed substitution hints:    ph( q)    S( q)    .0. ( q)

Proof of Theorem lssatomic
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lssatomic.n . . 3  |-  ( ph  ->  U  =/=  {  .0.  } )
2 lssatomic.u . . . 4  |-  ( ph  ->  U  e.  S )
3 lssatomic.o . . . . 5  |-  .0.  =  ( 0g `  W )
4 lssatomic.s . . . . 5  |-  S  =  ( LSubSp `  W )
53, 4lssne0 17373 . . . 4  |-  ( U  e.  S  ->  ( U  =/=  {  .0.  }  <->  E. x  e.  U  x  =/=  .0.  ) )
62, 5syl 16 . . 3  |-  ( ph  ->  ( U  =/=  {  .0.  }  <->  E. x  e.  U  x  =/=  .0.  ) )
71, 6mpbid 210 . 2  |-  ( ph  ->  E. x  e.  U  x  =/=  .0.  )
8 lssatomic.w . . . . . 6  |-  ( ph  ->  W  e.  LMod )
983ad2ant1 1012 . . . . 5  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  W  e.  LMod )
1023ad2ant1 1012 . . . . . 6  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  U  e.  S
)
11 simp2 992 . . . . . 6  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  x  e.  U
)
12 eqid 2460 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
1312, 4lssel 17360 . . . . . 6  |-  ( ( U  e.  S  /\  x  e.  U )  ->  x  e.  ( Base `  W ) )
1410, 11, 13syl2anc 661 . . . . 5  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  x  e.  (
Base `  W )
)
15 simp3 993 . . . . 5  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  x  =/=  .0.  )
16 eqid 2460 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
17 lssatomic.a . . . . . 6  |-  A  =  (LSAtoms `  W )
1812, 16, 3, 17lsatlspsn2 33664 . . . . 5  |-  ( ( W  e.  LMod  /\  x  e.  ( Base `  W
)  /\  x  =/=  .0.  )  ->  ( (
LSpan `  W ) `  { x } )  e.  A )
199, 14, 15, 18syl3anc 1223 . . . 4  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  ( ( LSpan `  W ) `  {
x } )  e.  A )
204, 16, 9, 10, 11lspsnel5a 17418 . . . 4  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  ( ( LSpan `  W ) `  {
x } )  C_  U )
21 sseq1 3518 . . . . 5  |-  ( q  =  ( ( LSpan `  W ) `  {
x } )  -> 
( q  C_  U  <->  ( ( LSpan `  W ) `  { x } ) 
C_  U ) )
2221rspcev 3207 . . . 4  |-  ( ( ( ( LSpan `  W
) `  { x } )  e.  A  /\  ( ( LSpan `  W
) `  { x } )  C_  U
)  ->  E. q  e.  A  q  C_  U )
2319, 20, 22syl2anc 661 . . 3  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  E. q  e.  A  q  C_  U )
2423rexlimdv3a 2950 . 2  |-  ( ph  ->  ( E. x  e.  U  x  =/=  .0.  ->  E. q  e.  A  q  C_  U ) )
257, 24mpd 15 1  |-  ( ph  ->  E. q  e.  A  q  C_  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   E.wrex 2808    C_ wss 3469   {csn 4020   ` cfv 5579   Basecbs 14479   0gc0g 14684   LModclmod 17288   LSubSpclss 17354   LSpanclspn 17393  LSAtomsclsa 33646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-0g 14686  df-mnd 15721  df-grp 15851  df-lmod 17290  df-lss 17355  df-lsp 17394  df-lsatoms 33648
This theorem is referenced by:  lsatcvatlem  33721  dochexmidlem5  36136
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