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Theorem lssatomic 29871
Description: The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 23863 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 16029 . . . 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 203 . 2  |-  ( ph  ->  E. x  e.  U  x  =/=  .0.  )
8 lssatomic.w . . . . . 6  |-  ( ph  ->  W  e.  LMod )
983ad2ant1 979 . . . . 5  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  W  e.  LMod )
1023ad2ant1 979 . . . . . 6  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  U  e.  S
)
11 simp2 959 . . . . . 6  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  x  e.  U
)
12 eqid 2438 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
1312, 4lssel 16016 . . . . . 6  |-  ( ( U  e.  S  /\  x  e.  U )  ->  x  e.  ( Base `  W ) )
1410, 11, 13syl2anc 644 . . . . 5  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  x  e.  (
Base `  W )
)
15 simp3 960 . . . . 5  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  x  =/=  .0.  )
16 eqid 2438 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
17 lssatomic.a . . . . . 6  |-  A  =  (LSAtoms `  W )
1812, 16, 3, 17lsatlspsn2 29852 . . . . 5  |-  ( ( W  e.  LMod  /\  x  e.  ( Base `  W
)  /\  x  =/=  .0.  )  ->  ( (
LSpan `  W ) `  { x } )  e.  A )
199, 14, 15, 18syl3anc 1185 . . . 4  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  ( ( LSpan `  W ) `  {
x } )  e.  A )
204, 16, 9, 10, 11lspsnel5a 16074 . . . 4  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  ( ( LSpan `  W ) `  {
x } )  C_  U )
21 sseq1 3371 . . . . 5  |-  ( q  =  ( ( LSpan `  W ) `  {
x } )  -> 
( q  C_  U  <->  ( ( LSpan `  W ) `  { x } ) 
C_  U ) )
2221rspcev 3054 . . . 4  |-  ( ( ( ( LSpan `  W
) `  { x } )  e.  A  /\  ( ( LSpan `  W
) `  { x } )  C_  U
)  ->  E. q  e.  A  q  C_  U )
2319, 20, 22syl2anc 644 . . 3  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  E. q  e.  A  q  C_  U )
2423rexlimdv3a 2834 . 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    C_ wss 3322   {csn 3816   ` cfv 5456   Basecbs 13471   0gc0g 13725   LModclmod 15952   LSubSpclss 16010   LSpanclspn 16049  LSAtomsclsa 29834
This theorem is referenced by:  lsatcvatlem  29909  dochexmidlem5  32324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-riota 6551  df-0g 13729  df-mnd 14692  df-grp 14814  df-lmod 15954  df-lss 16011  df-lsp 16050  df-lsatoms 29836
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