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Theorem lspsnat 18423
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 27290 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 3753 . . . . . 6  |-  ( ( U  \  {  .0.  } )  =/=  (/)  <->  E. x  x  e.  ( U  \  {  .0.  } ) )
2 simprl 769 . . . . . . . . 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 1017 . . . . . . . . . . 11  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  W  e.  LVec )
6 lveclmod 18384 . . . . . . . . . . 11  |-  ( W  e.  LVec  ->  W  e. 
LMod )
75, 6syl 17 . . . . . . . . . 10  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  W  e.  LMod )
8 simpl2 1018 . . . . . . . . . 10  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  U  e.  S
)
9 simprr 771 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  x  e.  ( U  \  {  .0.  } ) )
109eldifad 3428 . . . . . . . . . . . 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 18274 . . . . . . . . . . 11  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  ( N `  { x } ) 
C_  U )
12 0ss 3775 . . . . . . . . . . . . . 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 1019 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  X  e.  V
)
15 ssdif 3580 . . . . . . . . . . . . . . . 16  |-  ( U 
C_  ( N `  { X } )  -> 
( U  \  {  .0.  } )  C_  (
( N `  { X } )  \  {  .0.  } ) )
1615ad2antrl 739 . . . . . . . . . . . . . . 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 3445 . . . . . . . . . . . . . 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 3590 . . . . . . . . . . . . . . . . . 18  |-  ( (/)  u. 
{ X } )  =  ( { X }  u.  (/) )
19 un0 3771 . . . . . . . . . . . . . . . . . 18  |-  ( { X }  u.  (/) )  =  { X }
2018, 19eqtri 2484 . . . . . . . . . . . . . . . . 17  |-  ( (/)  u. 
{ X } )  =  { X }
2120fveq2i 5895 . . . . . . . . . . . . . . . 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 18287 . . . . . . . . . . . . . . . 16  |-  ( W  e.  LMod  ->  ( N `
 (/) )  =  {  .0.  } )
257, 24syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  ( N `  (/) )  =  {  .0.  } )
2622, 25difeq12d 3564 . . . . . . . . . . . . . 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 2543 . . . . . . . . . . . . 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 18421 . . . . . . . . . . . . 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 1278 . . . . . . . . . . . 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 3590 . . . . . . . . . . . . . 14  |-  ( (/)  u. 
{ x } )  =  ( { x }  u.  (/) )
32 un0 3771 . . . . . . . . . . . . . 14  |-  ( { x }  u.  (/) )  =  { x }
3331, 32eqtri 2484 . . . . . . . . . . . . 13  |-  ( (/)  u. 
{ x } )  =  { x }
3433fveq2i 5895 . . . . . . . . . . . 12  |-  ( N `
 ( (/)  u.  {
x } ) )  =  ( N `  { x } )
3530, 34syl6eleq 2550 . . . . . . . . . . 11  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  X  e.  ( N `  { x } ) )
3611, 35sseldd 3445 . . . . . . . . . 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 18274 . . . . . . . . 9  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  ( N `  { X } )  C_  U )
382, 37eqssd 3461 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  U  =  ( N `  { X } ) )
3938expr 624 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  (
x  e.  ( U 
\  {  .0.  }
)  ->  U  =  ( N `  { X } ) ) )
4039exlimdv 1790 . . . . . 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 225 . . . . 5  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  (
( U  \  {  .0.  } )  =/=  (/)  ->  U  =  ( N `  { X } ) ) )
4241necon1bd 2654 . . . 4  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( -.  U  =  ( N `  { X } )  ->  ( U  \  {  .0.  }
)  =  (/) ) )
43 ssdif0 3835 . . . 4  |-  ( U 
C_  {  .0.  }  <->  ( U  \  {  .0.  } )  =  (/) )
4442, 43syl6ibr 235 . . 3  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( -.  U  =  ( N `  { X } )  ->  U  C_ 
{  .0.  } ) )
45 simpl1 1017 . . . . 5  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  W  e.  LVec )
4645, 6syl 17 . . . 4  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  W  e.  LMod )
47 simpl2 1018 . . . 4  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  U  e.  S )
4823, 3lssle0 18228 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( U  C_  {  .0.  }  <->  U  =  {  .0.  }
) )
4946, 47, 48syl2anc 671 . . 3  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( U  C_  {  .0.  }  <->  U  =  {  .0.  }
) )
5044, 49sylibd 222 . 2  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( -.  U  =  ( N `  { X } )  ->  U  =  {  .0.  } ) )
5150orrd 384 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 189    \/ wo 374    /\ wa 375    /\ w3a 991    = wceq 1455   E.wex 1674    e. wcel 1898    =/= wne 2633    \ cdif 3413    u. cun 3414    C_ wss 3416   (/)c0 3743   {csn 3980   ` cfv 5605   Basecbs 15176   0gc0g 15393   LModclmod 18146   LSubSpclss 18210   LSpanclspn 18249   LVecclvec 18380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-tpos 7004  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-er 7394  df-en 7601  df-dom 7602  df-sdom 7603  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-nn 10643  df-2 10701  df-3 10702  df-ndx 15179  df-slot 15180  df-base 15181  df-sets 15182  df-ress 15183  df-plusg 15258  df-mulr 15259  df-0g 15395  df-mgm 16543  df-sgrp 16582  df-mnd 16592  df-grp 16728  df-minusg 16729  df-sbg 16730  df-cmn 17487  df-abl 17488  df-mgp 17779  df-ur 17791  df-ring 17837  df-oppr 17906  df-dvdsr 17924  df-unit 17925  df-invr 17955  df-drng 18032  df-lmod 18148  df-lss 18211  df-lsp 18250  df-lvec 18381
This theorem is referenced by:  lspsncv0  18424  lsatcmp  32615  dihlspsnssN  34946  dihlspsnat  34947
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