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Theorem lspsnat 17352
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 25156 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 3757 . . . . . 6  |-  ( ( U  \  {  .0.  } )  =/=  (/)  <->  E. x  x  e.  ( U  \  {  .0.  } ) )
2 simprl 755 . . . . . . . . 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 991 . . . . . . . . . . 11  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  W  e.  LVec )
6 lveclmod 17313 . . . . . . . . . . 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 992 . . . . . . . . . 10  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  U  e.  S
)
9 simprr 756 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  x  e.  ( U  \  {  .0.  } ) )
109eldifad 3451 . . . . . . . . . . . 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 17203 . . . . . . . . . . 11  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  ( N `  { x } ) 
C_  U )
12 0ss 3777 . . . . . . . . . . . . . 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 993 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  X  e.  V
)
15 ssdif 3602 . . . . . . . . . . . . . . . 16  |-  ( U 
C_  ( N `  { X } )  -> 
( U  \  {  .0.  } )  C_  (
( N `  { X } )  \  {  .0.  } ) )
1615ad2antrl 727 . . . . . . . . . . . . . . 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 3468 . . . . . . . . . . . . . 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 3611 . . . . . . . . . . . . . . . . . 18  |-  ( (/)  u. 
{ X } )  =  ( { X }  u.  (/) )
19 un0 3773 . . . . . . . . . . . . . . . . . 18  |-  ( { X }  u.  (/) )  =  { X }
2018, 19eqtri 2483 . . . . . . . . . . . . . . . . 17  |-  ( (/)  u. 
{ X } )  =  { X }
2120fveq2i 5805 . . . . . . . . . . . . . . . 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 17216 . . . . . . . . . . . . . . . 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 3586 . . . . . . . . . . . . . 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 2545 . . . . . . . . . . . . 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 17350 . . . . . . . . . . . . 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 1221 . . . . . . . . . . . 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 3611 . . . . . . . . . . . . . 14  |-  ( (/)  u. 
{ x } )  =  ( { x }  u.  (/) )
32 un0 3773 . . . . . . . . . . . . . 14  |-  ( { x }  u.  (/) )  =  { x }
3331, 32eqtri 2483 . . . . . . . . . . . . 13  |-  ( (/)  u. 
{ x } )  =  { x }
3433fveq2i 5805 . . . . . . . . . . . 12  |-  ( N `
 ( (/)  u.  {
x } ) )  =  ( N `  { x } )
3530, 34syl6eleq 2552 . . . . . . . . . . 11  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  X  e.  ( N `  { x } ) )
3611, 35sseldd 3468 . . . . . . . . . 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 17203 . . . . . . . . 9  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  ( N `  { X } )  C_  U )
382, 37eqssd 3484 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  U  =  ( N `  { X } ) )
3938expr 615 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  (
x  e.  ( U 
\  {  .0.  }
)  ->  U  =  ( N `  { X } ) ) )
4039exlimdv 1691 . . . . . 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 217 . . . . 5  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  (
( U  \  {  .0.  } )  =/=  (/)  ->  U  =  ( N `  { X } ) ) )
4241necon1bd 2670 . . . 4  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( -.  U  =  ( N `  { X } )  ->  ( U  \  {  .0.  }
)  =  (/) ) )
43 ssdif0 3848 . . . 4  |-  ( U 
C_  {  .0.  }  <->  ( U  \  {  .0.  } )  =  (/) )
4442, 43syl6ibr 227 . . 3  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( -.  U  =  ( N `  { X } )  ->  U  C_ 
{  .0.  } ) )
45 simpl1 991 . . . . 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 992 . . . 4  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  U  e.  S )
4823, 3lssle0 17157 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( U  C_  {  .0.  }  <->  U  =  {  .0.  }
) )
4946, 47, 48syl2anc 661 . . 3  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( U  C_  {  .0.  }  <->  U  =  {  .0.  }
) )
5044, 49sylibd 214 . 2  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( -.  U  =  ( N `  { X } )  ->  U  =  {  .0.  } ) )
5150orrd 378 1  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( U  =  ( N `  { X } )  \/  U  =  {  .0.  } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2648    \ cdif 3436    u. cun 3437    C_ wss 3439   (/)c0 3748   {csn 3988   ` cfv 5529   Basecbs 14295   0gc0g 14500   LModclmod 17074   LSubSpclss 17139   LSpanclspn 17178   LVecclvec 17309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-tpos 6858  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-plusg 14373  df-mulr 14374  df-0g 14502  df-mnd 15537  df-grp 15667  df-minusg 15668  df-sbg 15669  df-cmn 16403  df-abl 16404  df-mgp 16717  df-ur 16729  df-rng 16773  df-oppr 16841  df-dvdsr 16859  df-unit 16860  df-invr 16890  df-drng 16960  df-lmod 17076  df-lss 17140  df-lsp 17179  df-lvec 17310
This theorem is referenced by:  lspsncv0  17353  lsatcmp  33006  dihlspsnssN  35335  dihlspsnat  35336
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