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Theorem lssatomic 29019
Description: The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 22993 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 15757 . . . 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 2316 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
1312, 4lssel 15744 . . . . . 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 2316 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
17 lssatomic.a . . . . . 6  |-  A  =  (LSAtoms `  W )
1812, 16, 3, 17lsatlspsn2 29000 . . . . 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 15802 . . . 4  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  ( ( LSpan `  W ) `  {
x } )  C_  U )
21 sseq1 3233 . . . . 5  |-  ( q  =  ( ( LSpan `  W ) `  {
x } )  -> 
( q  C_  U  <->  ( ( LSpan `  W ) `  { x } ) 
C_  U ) )
2221rspcev 2918 . . . 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 2703 . 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 1633    e. wcel 1701    =/= wne 2479   E.wrex 2578    C_ wss 3186   {csn 3674   ` cfv 5292   Basecbs 13195   0gc0g 13449   LModclmod 15676   LSubSpclss 15738   LSpanclspn 15777  LSAtomsclsa 28982
This theorem is referenced by:  lsatcvatlem  29057  dochexmidlem5  31472
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-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-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  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-riota 6346  df-0g 13453  df-mnd 14416  df-grp 14538  df-lmod 15678  df-lss 15739  df-lsp 15778  df-lsatoms 28984
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