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Theorem lssatomic 29201
Description: The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 22938 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 15708 . . . 4  |-  ( U  e.  S  ->  ( U  =/=  {  .0.  }  <->  E. x  e.  U  x  =/=  .0.  ) )
62, 5syl 15 . . 3  |-  ( ph  ->  ( U  =/=  {  .0.  }  <->  E. x  e.  U  x  =/=  .0.  ) )
71, 6mpbid 201 . 2  |-  ( ph  ->  E. x  e.  U  x  =/=  .0.  )
8 lssatomic.w . . . . . 6  |-  ( ph  ->  W  e.  LMod )
983ad2ant1 976 . . . . 5  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  W  e.  LMod )
1023ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  U  e.  S
)
11 simp2 956 . . . . . 6  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  x  e.  U
)
12 eqid 2283 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
1312, 4lssel 15695 . . . . . 6  |-  ( ( U  e.  S  /\  x  e.  U )  ->  x  e.  ( Base `  W ) )
1410, 11, 13syl2anc 642 . . . . 5  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  x  e.  (
Base `  W )
)
15 simp3 957 . . . . 5  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  x  =/=  .0.  )
16 eqid 2283 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
17 lssatomic.a . . . . . 6  |-  A  =  (LSAtoms `  W )
1812, 16, 3, 17lsatlspsn2 29182 . . . . 5  |-  ( ( W  e.  LMod  /\  x  e.  ( Base `  W
)  /\  x  =/=  .0.  )  ->  ( (
LSpan `  W ) `  { x } )  e.  A )
199, 14, 15, 18syl3anc 1182 . . . 4  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  ( ( LSpan `  W ) `  {
x } )  e.  A )
204, 16, 9, 10, 11lspsnel5a 15753 . . . 4  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  ( ( LSpan `  W ) `  {
x } )  C_  U )
21 sseq1 3199 . . . . 5  |-  ( q  =  ( ( LSpan `  W ) `  {
x } )  -> 
( q  C_  U  <->  ( ( LSpan `  W ) `  { x } ) 
C_  U ) )
2221rspcev 2884 . . . 4  |-  ( ( ( ( LSpan `  W
) `  { x } )  e.  A  /\  ( ( LSpan `  W
) `  { x } )  C_  U
)  ->  E. q  e.  A  q  C_  U )
2319, 20, 22syl2anc 642 . . 3  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  E. q  e.  A  q  C_  U )
2423rexlimdv3a 2669 . 2  |-  ( ph  ->  ( E. x  e.  U  x  =/=  .0.  ->  E. q  e.  A  q  C_  U ) )
257, 24mpd 14 1  |-  ( ph  ->  E. q  e.  A  q  C_  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    C_ wss 3152   {csn 3640   ` cfv 5255   Basecbs 13148   0gc0g 13400   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728  LSAtomsclsa 29164
This theorem is referenced by:  lsatcvatlem  29239  dochexmidlem5  31654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lsatoms 29166
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