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Theorem lssatomic 34879
Description: The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 27404 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 17724 . . . 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 1017 . . . . 5  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  W  e.  LMod )
1023ad2ant1 1017 . . . . . 6  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  U  e.  S
)
11 simp2 997 . . . . . 6  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  x  e.  U
)
12 eqid 2457 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
1312, 4lssel 17711 . . . . . 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 998 . . . . 5  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  x  =/=  .0.  )
16 eqid 2457 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
17 lssatomic.a . . . . . 6  |-  A  =  (LSAtoms `  W )
1812, 16, 3, 17lsatlspsn2 34860 . . . . 5  |-  ( ( W  e.  LMod  /\  x  e.  ( Base `  W
)  /\  x  =/=  .0.  )  ->  ( (
LSpan `  W ) `  { x } )  e.  A )
199, 14, 15, 18syl3anc 1228 . . . 4  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  ( ( LSpan `  W ) `  {
x } )  e.  A )
204, 16, 9, 10, 11lspsnel5a 17769 . . . 4  |-  ( (
ph  /\  x  e.  U  /\  x  =/=  .0.  )  ->  ( ( LSpan `  W ) `  {
x } )  C_  U )
21 sseq1 3520 . . . . 5  |-  ( q  =  ( ( LSpan `  W ) `  {
x } )  -> 
( q  C_  U  <->  ( ( LSpan `  W ) `  { x } ) 
C_  U ) )
2221rspcev 3210 . . . 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 2951 . 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808    C_ wss 3471   {csn 4032   ` cfv 5594   Basecbs 14644   0gc0g 14857   LModclmod 17639   LSubSpclss 17705   LSpanclspn 17744  LSAtomsclsa 34842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-lmod 17641  df-lss 17706  df-lsp 17745  df-lsatoms 34844
This theorem is referenced by:  lsatcvatlem  34917  dochexmidlem5  37334
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