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Theorem lsatcv0 29147
Description: An atom covers the zero subspace. (atcv0 23694 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 16106 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 16 . . . 4  |-  ( ph  ->  W  e.  LMod )
4 eqid 2388 . . . . 5  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
5 lsatcv0.a . . . . 5  |-  A  =  (LSAtoms `  W )
6 lsatcv0.q . . . . 5  |-  ( ph  ->  Q  e.  A )
74, 5, 3, 6lsatlssel 29113 . . . 4  |-  ( ph  ->  Q  e.  ( LSubSp `  W ) )
8 lsatcv0.o . . . . 5  |-  .0.  =  ( 0g `  W )
98, 4lss0ss 15953 . . . 4  |-  ( ( W  e.  LMod  /\  Q  e.  ( LSubSp `  W )
)  ->  {  .0.  } 
C_  Q )
103, 7, 9syl2anc 643 . . 3  |-  ( ph  ->  {  .0.  }  C_  Q )
118, 5, 3, 6lsatn0 29115 . . . 4  |-  ( ph  ->  Q  =/=  {  .0.  } )
1211necomd 2634 . . 3  |-  ( ph  ->  {  .0.  }  =/=  Q )
13 df-pss 3280 . . 3  |-  ( {  .0.  }  C.  Q  <->  ( {  .0.  }  C_  Q  /\  {  .0.  }  =/=  Q ) )
1410, 12, 13sylanbrc 646 . 2  |-  ( ph  ->  {  .0.  }  C.  Q )
15 eqid 2388 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
16 eqid 2388 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
1715, 16, 8, 5islsat 29107 . . . . 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 202 . . 3  |-  ( ph  ->  E. x  e.  ( ( Base `  W
)  \  {  .0.  } ) Q  =  ( ( LSpan `  W ) `  { x } ) )
201adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  W  e.  LVec )
21 eldifi 3413 . . . . . . . 8  |-  ( x  e.  ( ( Base `  W )  \  {  .0.  } )  ->  x  e.  ( Base `  W
) )
2221adantl 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  x  e.  ( Base `  W
) )
23 eldifsni 3872 . . . . . . . 8  |-  ( x  e.  ( ( Base `  W )  \  {  .0.  } )  ->  x  =/=  .0.  )
2423adantl 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  x  =/=  .0.  )
2515, 8, 4, 16, 20, 22, 24lspsncv0 16146 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  -.  E. s  e.  ( LSubSp `  W ) ( {  .0.  }  C.  s  /\  s  C.  ( (
LSpan `  W ) `  { x } ) ) )
2625ex 424 . . . . 5  |-  ( ph  ->  ( x  e.  ( ( Base `  W
)  \  {  .0.  } )  ->  -.  E. s  e.  ( LSubSp `  W )
( {  .0.  }  C.  s  /\  s  C.  ( ( LSpan `  W
) `  { x } ) ) ) )
27 psseq2 3379 . . . . . . . . 9  |-  ( Q  =  ( ( LSpan `  W ) `  {
x } )  -> 
( s  C.  Q  <->  s 
C.  ( ( LSpan `  W ) `  {
x } ) ) )
2827anbi2d 685 . . . . . . . 8  |-  ( Q  =  ( ( LSpan `  W ) `  {
x } )  -> 
( ( {  .0.  } 
C.  s  /\  s  C.  Q )  <->  ( {  .0.  }  C.  s  /\  s  C.  ( ( LSpan `  W ) `  {
x } ) ) ) )
2928rexbidv 2671 . . . . . . 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 286 . . . . . 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 217 . . . . 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 31 . . . 4  |-  ( ph  ->  ( x  e.  ( ( Base `  W
)  \  {  .0.  } )  ->  ( Q  =  ( ( LSpan `  W ) `  {
x } )  ->  -.  E. s  e.  (
LSubSp `  W ) ( {  .0.  }  C.  s  /\  s  C.  Q
) ) ) )
3332rexlimdv 2773 . . 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 15952 . . . 4  |-  ( W  e.  LMod  ->  {  .0.  }  e.  ( LSubSp `  W
) )
373, 36syl 16 . . 3  |-  ( ph  ->  {  .0.  }  e.  ( LSubSp `  W )
)
384, 35, 1, 37, 7lcvbr 29137 . 2  |-  ( ph  ->  ( {  .0.  } C Q  <->  ( {  .0.  } 
C.  Q  /\  -.  E. s  e.  ( LSubSp `  W ) ( {  .0.  }  C.  s  /\  s  C.  Q ) ) ) )
3914, 34, 38mpbir2and 889 1  |-  ( ph  ->  {  .0.  } C Q )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   E.wrex 2651    \ cdif 3261    C_ wss 3264    C. wpss 3265   {csn 3758   class class class wbr 4154   ` cfv 5395   Basecbs 13397   0gc0g 13651   LModclmod 15878   LSubSpclss 15936   LSpanclspn 15975   LVecclvec 16102  LSAtomsclsa 29090    <oLL clcv 29134
This theorem is referenced by:  lsatcveq0  29148  lsat0cv  29149  lsatcv0eq  29163  mapdcnvatN  31782  mapdat  31783
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-tpos 6416  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-3 9992  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-0g 13655  df-mnd 14618  df-grp 14740  df-minusg 14741  df-sbg 14742  df-cmn 15342  df-abl 15343  df-mgp 15577  df-rng 15591  df-ur 15593  df-oppr 15656  df-dvdsr 15674  df-unit 15675  df-invr 15705  df-drng 15765  df-lmod 15880  df-lss 15937  df-lsp 15976  df-lvec 16103  df-lsatoms 29092  df-lcv 29135
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