Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lsatcv0 Structured version   Unicode version

Theorem lsatcv0 29766
Description: An atom covers the zero subspace. (atcv0 23837 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 16170 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 16 . . . 4  |-  ( ph  ->  W  e.  LMod )
4 eqid 2435 . . . . 5  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
5 lsatcv0.a . . . . 5  |-  A  =  (LSAtoms `  W )
6 lsatcv0.q . . . . 5  |-  ( ph  ->  Q  e.  A )
74, 5, 3, 6lsatlssel 29732 . . . 4  |-  ( ph  ->  Q  e.  ( LSubSp `  W ) )
8 lsatcv0.o . . . . 5  |-  .0.  =  ( 0g `  W )
98, 4lss0ss 16017 . . . 4  |-  ( ( W  e.  LMod  /\  Q  e.  ( LSubSp `  W )
)  ->  {  .0.  } 
C_  Q )
103, 7, 9syl2anc 643 . . 3  |-  ( ph  ->  {  .0.  }  C_  Q )
118, 5, 3, 6lsatn0 29734 . . . 4  |-  ( ph  ->  Q  =/=  {  .0.  } )
1211necomd 2681 . . 3  |-  ( ph  ->  {  .0.  }  =/=  Q )
13 df-pss 3328 . . 3  |-  ( {  .0.  }  C.  Q  <->  ( {  .0.  }  C_  Q  /\  {  .0.  }  =/=  Q ) )
1410, 12, 13sylanbrc 646 . 2  |-  ( ph  ->  {  .0.  }  C.  Q )
15 eqid 2435 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
16 eqid 2435 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
1715, 16, 8, 5islsat 29726 . . . . 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 3461 . . . . . . . 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 3920 . . . . . . . 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 16210 . . . . . 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 3427 . . . . . . . . 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 2718 . . . . . . 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 2821 . . 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 16016 . . . 4  |-  ( W  e.  LMod  ->  {  .0.  }  e.  ( LSubSp `  W
) )
373, 36syl 16 . . 3  |-  ( ph  ->  {  .0.  }  e.  ( LSubSp `  W )
)
384, 35, 1, 37, 7lcvbr 29756 . 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 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    \ cdif 3309    C_ wss 3312    C. wpss 3313   {csn 3806   class class class wbr 4204   ` cfv 5446   Basecbs 13461   0gc0g 13715   LModclmod 15942   LSubSpclss 16000   LSpanclspn 16039   LVecclvec 16166  LSAtomsclsa 29709    <oLL clcv 29753
This theorem is referenced by:  lsatcveq0  29767  lsat0cv  29768  lsatcv0eq  29782  mapdcnvatN  32401  mapdat  32402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167  df-lsatoms 29711  df-lcv 29754
  Copyright terms: Public domain W3C validator