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Theorem lsatcv0 29221
Description: An atom covers the zero subspace. (atcv0 22922 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 15859 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 15 . . . 4  |-  ( ph  ->  W  e.  LMod )
4 eqid 2283 . . . . 5  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
5 lsatcv0.a . . . . 5  |-  A  =  (LSAtoms `  W )
6 lsatcv0.q . . . . 5  |-  ( ph  ->  Q  e.  A )
74, 5, 3, 6lsatlssel 29187 . . . 4  |-  ( ph  ->  Q  e.  ( LSubSp `  W ) )
8 lsatcv0.o . . . . 5  |-  .0.  =  ( 0g `  W )
98, 4lss0ss 15706 . . . 4  |-  ( ( W  e.  LMod  /\  Q  e.  ( LSubSp `  W )
)  ->  {  .0.  } 
C_  Q )
103, 7, 9syl2anc 642 . . 3  |-  ( ph  ->  {  .0.  }  C_  Q )
118, 5, 3, 6lsatn0 29189 . . . 4  |-  ( ph  ->  Q  =/=  {  .0.  } )
1211necomd 2529 . . 3  |-  ( ph  ->  {  .0.  }  =/=  Q )
13 df-pss 3168 . . 3  |-  ( {  .0.  }  C.  Q  <->  ( {  .0.  }  C_  Q  /\  {  .0.  }  =/=  Q ) )
1410, 12, 13sylanbrc 645 . 2  |-  ( ph  ->  {  .0.  }  C.  Q )
15 eqid 2283 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
16 eqid 2283 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
1715, 16, 8, 5islsat 29181 . . . . 5  |-  ( W  e.  LMod  ->  ( Q  e.  A  <->  E. x  e.  ( ( Base `  W
)  \  {  .0.  } ) Q  =  ( ( LSpan `  W ) `  { x } ) ) )
183, 17syl 15 . . . 4  |-  ( ph  ->  ( Q  e.  A  <->  E. x  e.  ( (
Base `  W )  \  {  .0.  } ) Q  =  ( (
LSpan `  W ) `  { x } ) ) )
196, 18mpbid 201 . . 3  |-  ( ph  ->  E. x  e.  ( ( Base `  W
)  \  {  .0.  } ) Q  =  ( ( LSpan `  W ) `  { x } ) )
201adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  W  e.  LVec )
21 eldifi 3298 . . . . . . . 8  |-  ( x  e.  ( ( Base `  W )  \  {  .0.  } )  ->  x  e.  ( Base `  W
) )
2221adantl 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  x  e.  ( Base `  W
) )
23 eldifsni 3750 . . . . . . . 8  |-  ( x  e.  ( ( Base `  W )  \  {  .0.  } )  ->  x  =/=  .0.  )
2423adantl 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  x  =/=  .0.  )
2515, 8, 4, 16, 20, 22, 24lspsncv0 15899 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  -.  E. s  e.  ( LSubSp `  W ) ( {  .0.  }  C.  s  /\  s  C.  ( (
LSpan `  W ) `  { x } ) ) )
2625ex 423 . . . . 5  |-  ( ph  ->  ( x  e.  ( ( Base `  W
)  \  {  .0.  } )  ->  -.  E. s  e.  ( LSubSp `  W )
( {  .0.  }  C.  s  /\  s  C.  ( ( LSpan `  W
) `  { x } ) ) ) )
27 psseq2 3264 . . . . . . . . 9  |-  ( Q  =  ( ( LSpan `  W ) `  {
x } )  -> 
( s  C.  Q  <->  s 
C.  ( ( LSpan `  W ) `  {
x } ) ) )
2827anbi2d 684 . . . . . . . 8  |-  ( Q  =  ( ( LSpan `  W ) `  {
x } )  -> 
( ( {  .0.  } 
C.  s  /\  s  C.  Q )  <->  ( {  .0.  }  C.  s  /\  s  C.  ( ( LSpan `  W ) `  {
x } ) ) ) )
2928rexbidv 2564 . . . . . . 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 285 . . . . . 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 216 . . . . 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 29 . . . 4  |-  ( ph  ->  ( x  e.  ( ( Base `  W
)  \  {  .0.  } )  ->  ( Q  =  ( ( LSpan `  W ) `  {
x } )  ->  -.  E. s  e.  (
LSubSp `  W ) ( {  .0.  }  C.  s  /\  s  C.  Q
) ) ) )
3332rexlimdv 2666 . . 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 14 . 2  |-  ( ph  ->  -.  E. s  e.  ( LSubSp `  W )
( {  .0.  }  C.  s  /\  s  C.  Q ) )
35 lsatcv0.c . . 3  |-  C  =  (  <oLL  `  W )
368, 4lsssn0 15705 . . . 4  |-  ( W  e.  LMod  ->  {  .0.  }  e.  ( LSubSp `  W
) )
373, 36syl 15 . . 3  |-  ( ph  ->  {  .0.  }  e.  ( LSubSp `  W )
)
384, 35, 1, 37, 7lcvbr 29211 . 2  |-  ( ph  ->  ( {  .0.  } C Q  <->  ( {  .0.  } 
C.  Q  /\  -.  E. s  e.  ( LSubSp `  W ) ( {  .0.  }  C.  s  /\  s  C.  Q ) ) ) )
3914, 34, 38mpbir2and 888 1  |-  ( ph  ->  {  .0.  } C Q )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    \ cdif 3149    C_ wss 3152    C. wpss 3153   {csn 3640   class class class wbr 4023   ` cfv 5255   Basecbs 13148   0gc0g 13400   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728   LVecclvec 15855  LSAtomsclsa 29164    <oLL clcv 29208
This theorem is referenced by:  lsatcveq0  29222  lsat0cv  29223  lsatcv0eq  29237  mapdcnvatN  31856  mapdat  31857
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  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2449  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 29166  df-lcv 29209
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