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Theorem lsatcv0 29903
Description: An atom covers the zero subspace. (atcv0 23850 analog.) (Contributed by NM, 7-Jan-2015.)
Hypotheses
Ref Expression
lsatcv0.o  |-  .0.  =  ( 0g `  W )
lsatcv0.a  |-  A  =  (LSAtoms `  W )
lsatcv0.c  |-  C  =  (  <oLL  `  W )
lsatcv0.w  |-  ( ph  ->  W  e.  LVec )
lsatcv0.q  |-  ( ph  ->  Q  e.  A )
Assertion
Ref Expression
lsatcv0  |-  ( ph  ->  {  .0.  } C Q )

Proof of Theorem lsatcv0
Dummy variables  x  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lsatcv0.w . . . . 5  |-  ( ph  ->  W  e.  LVec )
2 lveclmod 16183 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 16 . . . 4  |-  ( ph  ->  W  e.  LMod )
4 eqid 2438 . . . . 5  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
5 lsatcv0.a . . . . 5  |-  A  =  (LSAtoms `  W )
6 lsatcv0.q . . . . 5  |-  ( ph  ->  Q  e.  A )
74, 5, 3, 6lsatlssel 29869 . . . 4  |-  ( ph  ->  Q  e.  ( LSubSp `  W ) )
8 lsatcv0.o . . . . 5  |-  .0.  =  ( 0g `  W )
98, 4lss0ss 16030 . . . 4  |-  ( ( W  e.  LMod  /\  Q  e.  ( LSubSp `  W )
)  ->  {  .0.  } 
C_  Q )
103, 7, 9syl2anc 644 . . 3  |-  ( ph  ->  {  .0.  }  C_  Q )
118, 5, 3, 6lsatn0 29871 . . . 4  |-  ( ph  ->  Q  =/=  {  .0.  } )
1211necomd 2689 . . 3  |-  ( ph  ->  {  .0.  }  =/=  Q )
13 df-pss 3338 . . 3  |-  ( {  .0.  }  C.  Q  <->  ( {  .0.  }  C_  Q  /\  {  .0.  }  =/=  Q ) )
1410, 12, 13sylanbrc 647 . 2  |-  ( ph  ->  {  .0.  }  C.  Q )
15 eqid 2438 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
16 eqid 2438 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
1715, 16, 8, 5islsat 29863 . . . . 5  |-  ( W  e.  LMod  ->  ( Q  e.  A  <->  E. x  e.  ( ( Base `  W
)  \  {  .0.  } ) Q  =  ( ( LSpan `  W ) `  { x } ) ) )
183, 17syl 16 . . . 4  |-  ( ph  ->  ( Q  e.  A  <->  E. x  e.  ( (
Base `  W )  \  {  .0.  } ) Q  =  ( (
LSpan `  W ) `  { x } ) ) )
196, 18mpbid 203 . . 3  |-  ( ph  ->  E. x  e.  ( ( Base `  W
)  \  {  .0.  } ) Q  =  ( ( LSpan `  W ) `  { x } ) )
201adantr 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  W  e.  LVec )
21 eldifi 3471 . . . . . . . 8  |-  ( x  e.  ( ( Base `  W )  \  {  .0.  } )  ->  x  e.  ( Base `  W
) )
2221adantl 454 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  x  e.  ( Base `  W
) )
23 eldifsni 3930 . . . . . . . 8  |-  ( x  e.  ( ( Base `  W )  \  {  .0.  } )  ->  x  =/=  .0.  )
2423adantl 454 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  x  =/=  .0.  )
2515, 8, 4, 16, 20, 22, 24lspsncv0 16223 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  -.  E. s  e.  ( LSubSp `  W ) ( {  .0.  }  C.  s  /\  s  C.  ( (
LSpan `  W ) `  { x } ) ) )
2625ex 425 . . . . 5  |-  ( ph  ->  ( x  e.  ( ( Base `  W
)  \  {  .0.  } )  ->  -.  E. s  e.  ( LSubSp `  W )
( {  .0.  }  C.  s  /\  s  C.  ( ( LSpan `  W
) `  { x } ) ) ) )
27 psseq2 3437 . . . . . . . . 9  |-  ( Q  =  ( ( LSpan `  W ) `  {
x } )  -> 
( s  C.  Q  <->  s 
C.  ( ( LSpan `  W ) `  {
x } ) ) )
2827anbi2d 686 . . . . . . . 8  |-  ( Q  =  ( ( LSpan `  W ) `  {
x } )  -> 
( ( {  .0.  } 
C.  s  /\  s  C.  Q )  <->  ( {  .0.  }  C.  s  /\  s  C.  ( ( LSpan `  W ) `  {
x } ) ) ) )
2928rexbidv 2728 . . . . . . 7  |-  ( Q  =  ( ( LSpan `  W ) `  {
x } )  -> 
( E. s  e.  ( LSubSp `  W )
( {  .0.  }  C.  s  /\  s  C.  Q )  <->  E. s  e.  ( LSubSp `  W )
( {  .0.  }  C.  s  /\  s  C.  ( ( LSpan `  W
) `  { x } ) ) ) )
3029notbid 287 . . . . . 6  |-  ( Q  =  ( ( LSpan `  W ) `  {
x } )  -> 
( -.  E. s  e.  ( LSubSp `  W )
( {  .0.  }  C.  s  /\  s  C.  Q )  <->  -.  E. s  e.  ( LSubSp `  W )
( {  .0.  }  C.  s  /\  s  C.  ( ( LSpan `  W
) `  { x } ) ) ) )
3130biimprcd 218 . . . . 5  |-  ( -. 
E. s  e.  (
LSubSp `  W ) ( {  .0.  }  C.  s  /\  s  C.  (
( LSpan `  W ) `  { x } ) )  ->  ( Q  =  ( ( LSpan `  W ) `  {
x } )  ->  -.  E. s  e.  (
LSubSp `  W ) ( {  .0.  }  C.  s  /\  s  C.  Q
) ) )
3226, 31syl6 32 . . . 4  |-  ( ph  ->  ( x  e.  ( ( Base `  W
)  \  {  .0.  } )  ->  ( Q  =  ( ( LSpan `  W ) `  {
x } )  ->  -.  E. s  e.  (
LSubSp `  W ) ( {  .0.  }  C.  s  /\  s  C.  Q
) ) ) )
3332rexlimdv 2831 . . 3  |-  ( ph  ->  ( E. x  e.  ( ( Base `  W
)  \  {  .0.  } ) Q  =  ( ( LSpan `  W ) `  { x } )  ->  -.  E. s  e.  ( LSubSp `  W )
( {  .0.  }  C.  s  /\  s  C.  Q ) ) )
3419, 33mpd 15 . 2  |-  ( ph  ->  -.  E. s  e.  ( LSubSp `  W )
( {  .0.  }  C.  s  /\  s  C.  Q ) )
35 lsatcv0.c . . 3  |-  C  =  (  <oLL  `  W )
368, 4lsssn0 16029 . . . 4  |-  ( W  e.  LMod  ->  {  .0.  }  e.  ( LSubSp `  W
) )
373, 36syl 16 . . 3  |-  ( ph  ->  {  .0.  }  e.  ( LSubSp `  W )
)
384, 35, 1, 37, 7lcvbr 29893 . 2  |-  ( ph  ->  ( {  .0.  } C Q  <->  ( {  .0.  } 
C.  Q  /\  -.  E. s  e.  ( LSubSp `  W ) ( {  .0.  }  C.  s  /\  s  C.  Q ) ) ) )
3914, 34, 38mpbir2and 890 1  |-  ( ph  ->  {  .0.  } C Q )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    \ cdif 3319    C_ wss 3322    C. wpss 3323   {csn 3816   class class class wbr 4215   ` cfv 5457   Basecbs 13474   0gc0g 13728   LModclmod 15955   LSubSpclss 16013   LSpanclspn 16052   LVecclvec 16179  LSAtomsclsa 29846    <oLL clcv 29890
This theorem is referenced by:  lsatcveq0  29904  lsat0cv  29905  lsatcv0eq  29919  mapdcnvatN  32538  mapdat  32539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  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-nel 2604  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-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-tpos 6482  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-0g 13732  df-mnd 14695  df-grp 14817  df-minusg 14818  df-sbg 14819  df-cmn 15419  df-abl 15420  df-mgp 15654  df-rng 15668  df-ur 15670  df-oppr 15733  df-dvdsr 15751  df-unit 15752  df-invr 15782  df-drng 15842  df-lmod 15957  df-lss 16014  df-lsp 16053  df-lvec 16180  df-lsatoms 29848  df-lcv 29891
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