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Theorem lspsnat 16218
Description: There is no subspace strictly between the zero subspace and the span of a vector (i.e. a 1-dimensional subspace is an atom). (h1datomi 23084 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)
Hypotheses
Ref Expression
lspsnat.v  |-  V  =  ( Base `  W
)
lspsnat.z  |-  .0.  =  ( 0g `  W )
lspsnat.s  |-  S  =  ( LSubSp `  W )
lspsnat.n  |-  N  =  ( LSpan `  W )
Assertion
Ref Expression
lspsnat  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( U  =  ( N `  { X } )  \/  U  =  {  .0.  } ) )

Proof of Theorem lspsnat
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 n0 3638 . . . . . 6  |-  ( ( U  \  {  .0.  } )  =/=  (/)  <->  E. x  x  e.  ( U  \  {  .0.  } ) )
2 simprl 734 . . . . . . . . 9  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  U  C_  ( N `  { X } ) )
3 lspsnat.s . . . . . . . . . 10  |-  S  =  ( LSubSp `  W )
4 lspsnat.n . . . . . . . . . 10  |-  N  =  ( LSpan `  W )
5 simpl1 961 . . . . . . . . . . 11  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  W  e.  LVec )
6 lveclmod 16179 . . . . . . . . . . 11  |-  ( W  e.  LVec  ->  W  e. 
LMod )
75, 6syl 16 . . . . . . . . . 10  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  W  e.  LMod )
8 simpl2 962 . . . . . . . . . 10  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  U  e.  S
)
9 simprr 735 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  x  e.  ( U  \  {  .0.  } ) )
109eldifad 3333 . . . . . . . . . . . 12  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  x  e.  U
)
113, 4, 7, 8, 10lspsnel5a 16073 . . . . . . . . . . 11  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  ( N `  { x } ) 
C_  U )
12 0ss 3657 . . . . . . . . . . . . . 14  |-  (/)  C_  V
1312a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  (/)  C_  V )
14 simpl3 963 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  X  e.  V
)
15 ssdif 3483 . . . . . . . . . . . . . . . 16  |-  ( U 
C_  ( N `  { X } )  -> 
( U  \  {  .0.  } )  C_  (
( N `  { X } )  \  {  .0.  } ) )
1615ad2antrl 710 . . . . . . . . . . . . . . 15  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  ( U  \  {  .0.  } )  C_  ( ( N `  { X } )  \  {  .0.  } ) )
1716, 9sseldd 3350 . . . . . . . . . . . . . 14  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  x  e.  ( ( N `  { X } )  \  {  .0.  } ) )
18 uncom 3492 . . . . . . . . . . . . . . . . . 18  |-  ( (/)  u. 
{ X } )  =  ( { X }  u.  (/) )
19 un0 3653 . . . . . . . . . . . . . . . . . 18  |-  ( { X }  u.  (/) )  =  { X }
2018, 19eqtri 2457 . . . . . . . . . . . . . . . . 17  |-  ( (/)  u. 
{ X } )  =  { X }
2120fveq2i 5732 . . . . . . . . . . . . . . . 16  |-  ( N `
 ( (/)  u.  { X } ) )  =  ( N `  { X } )
2221a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  ( N `  ( (/)  u.  { X } ) )  =  ( N `  { X } ) )
23 lspsnat.z . . . . . . . . . . . . . . . . 17  |-  .0.  =  ( 0g `  W )
2423, 4lsp0 16086 . . . . . . . . . . . . . . . 16  |-  ( W  e.  LMod  ->  ( N `
 (/) )  =  {  .0.  } )
257, 24syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  ( N `  (/) )  =  {  .0.  } )
2622, 25difeq12d 3467 . . . . . . . . . . . . . 14  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  ( ( N `
 ( (/)  u.  { X } ) )  \ 
( N `  (/) ) )  =  ( ( N `
 { X }
)  \  {  .0.  } ) )
2717, 26eleqtrrd 2514 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  x  e.  ( ( N `  ( (/) 
u.  { X }
) )  \  ( N `  (/) ) ) )
28 lspsnat.v . . . . . . . . . . . . . 14  |-  V  =  ( Base `  W
)
2928, 3, 4lspsolv 16216 . . . . . . . . . . . . 13  |-  ( ( W  e.  LVec  /\  ( (/)  C_  V  /\  X  e.  V  /\  x  e.  ( ( N `  ( (/)  u.  { X } ) )  \ 
( N `  (/) ) ) ) )  ->  X  e.  ( N `  ( (/) 
u.  { x }
) ) )
305, 13, 14, 27, 29syl13anc 1187 . . . . . . . . . . . 12  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  X  e.  ( N `  ( (/)  u. 
{ x } ) ) )
31 uncom 3492 . . . . . . . . . . . . . 14  |-  ( (/)  u. 
{ x } )  =  ( { x }  u.  (/) )
32 un0 3653 . . . . . . . . . . . . . 14  |-  ( { x }  u.  (/) )  =  { x }
3331, 32eqtri 2457 . . . . . . . . . . . . 13  |-  ( (/)  u. 
{ x } )  =  { x }
3433fveq2i 5732 . . . . . . . . . . . 12  |-  ( N `
 ( (/)  u.  {
x } ) )  =  ( N `  { x } )
3530, 34syl6eleq 2527 . . . . . . . . . . 11  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  X  e.  ( N `  { x } ) )
3611, 35sseldd 3350 . . . . . . . . . 10  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  X  e.  U
)
373, 4, 7, 8, 36lspsnel5a 16073 . . . . . . . . 9  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  ( N `  { X } )  C_  U )
382, 37eqssd 3366 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  U  =  ( N `  { X } ) )
3938expr 600 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  (
x  e.  ( U 
\  {  .0.  }
)  ->  U  =  ( N `  { X } ) ) )
4039exlimdv 1647 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( E. x  x  e.  ( U  \  {  .0.  } )  ->  U  =  ( N `  { X } ) ) )
411, 40syl5bi 210 . . . . 5  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  (
( U  \  {  .0.  } )  =/=  (/)  ->  U  =  ( N `  { X } ) ) )
4241necon1bd 2673 . . . 4  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( -.  U  =  ( N `  { X } )  ->  ( U  \  {  .0.  }
)  =  (/) ) )
43 ssdif0 3687 . . . 4  |-  ( U 
C_  {  .0.  }  <->  ( U  \  {  .0.  } )  =  (/) )
4442, 43syl6ibr 220 . . 3  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( -.  U  =  ( N `  { X } )  ->  U  C_ 
{  .0.  } ) )
45 simpl1 961 . . . . 5  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  W  e.  LVec )
4645, 6syl 16 . . . 4  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  W  e.  LMod )
47 simpl2 962 . . . 4  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  U  e.  S )
4823, 3lssle0 16027 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( U  C_  {  .0.  }  <->  U  =  {  .0.  }
) )
4946, 47, 48syl2anc 644 . . 3  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( U  C_  {  .0.  }  <->  U  =  {  .0.  }
) )
5044, 49sylibd 207 . 2  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( -.  U  =  ( N `  { X } )  ->  U  =  {  .0.  } ) )
5150orrd 369 1  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( U  =  ( N `  { X } )  \/  U  =  {  .0.  } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2600    \ cdif 3318    u. cun 3319    C_ wss 3321   (/)c0 3629   {csn 3815   ` cfv 5455   Basecbs 13470   0gc0g 13724   LModclmod 15951   LSubSpclss 16009   LSpanclspn 16048   LVecclvec 16175
This theorem is referenced by:  lspsncv0  16219  lsatcmp  29802  dihlspsnssN  32131  dihlspsnat  32132
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-tpos 6480  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-3 10060  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-mulr 13544  df-0g 13728  df-mnd 14691  df-grp 14813  df-minusg 14814  df-sbg 14815  df-cmn 15415  df-abl 15416  df-mgp 15650  df-rng 15664  df-ur 15666  df-oppr 15729  df-dvdsr 15747  df-unit 15748  df-invr 15778  df-drng 15838  df-lmod 15953  df-lss 16010  df-lsp 16049  df-lvec 16176
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