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Theorem lspsnat 18001
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 26794 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 3745 . . . . . 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 998 . . . . . . . . . . 11  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  W  e.  LVec )
6 lveclmod 17962 . . . . . . . . . . 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 999 . . . . . . . . . 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 3423 . . . . . . . . . . . 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 17852 . . . . . . . . . . 11  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  ( N `  { x } ) 
C_  U )
12 0ss 3765 . . . . . . . . . . . . . 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 1000 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  X  e.  V
)
15 ssdif 3575 . . . . . . . . . . . . . . . 16  |-  ( U 
C_  ( N `  { X } )  -> 
( U  \  {  .0.  } )  C_  (
( N `  { X } )  \  {  .0.  } ) )
1615ad2antrl 726 . . . . . . . . . . . . . . 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 3440 . . . . . . . . . . . . . 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 3584 . . . . . . . . . . . . . . . . . 18  |-  ( (/)  u. 
{ X } )  =  ( { X }  u.  (/) )
19 un0 3761 . . . . . . . . . . . . . . . . . 18  |-  ( { X }  u.  (/) )  =  { X }
2018, 19eqtri 2429 . . . . . . . . . . . . . . . . 17  |-  ( (/)  u. 
{ X } )  =  { X }
2120fveq2i 5806 . . . . . . . . . . . . . . . 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 17865 . . . . . . . . . . . . . . . 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 3559 . . . . . . . . . . . . . 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 2491 . . . . . . . . . . . . 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 17999 . . . . . . . . . . . . 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 1230 . . . . . . . . . . . 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 3584 . . . . . . . . . . . . . 14  |-  ( (/)  u. 
{ x } )  =  ( { x }  u.  (/) )
32 un0 3761 . . . . . . . . . . . . . 14  |-  ( { x }  u.  (/) )  =  { x }
3331, 32eqtri 2429 . . . . . . . . . . . . 13  |-  ( (/)  u. 
{ x } )  =  { x }
3433fveq2i 5806 . . . . . . . . . . . 12  |-  ( N `
 ( (/)  u.  {
x } ) )  =  ( N `  { x } )
3530, 34syl6eleq 2498 . . . . . . . . . . 11  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  X  e.  ( N `  { x } ) )
3611, 35sseldd 3440 . . . . . . . . . 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 17852 . . . . . . . . 9  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  ( N `  { X } )  C_  U )
382, 37eqssd 3456 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  U  =  ( N `  { X } ) )
3938expr 613 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  (
x  e.  ( U 
\  {  .0.  }
)  ->  U  =  ( N `  { X } ) ) )
4039exlimdv 1743 . . . . . 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 2619 . . . 4  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( -.  U  =  ( N `  { X } )  ->  ( U  \  {  .0.  }
)  =  (/) ) )
43 ssdif0 3825 . . . 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 998 . . . . 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 999 . . . 4  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  U  e.  S )
4823, 3lssle0 17806 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( U  C_  {  .0.  }  <->  U  =  {  .0.  }
) )
4946, 47, 48syl2anc 659 . . 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 376 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 366    /\ wa 367    /\ w3a 972    = wceq 1403   E.wex 1631    e. wcel 1840    =/= wne 2596    \ cdif 3408    u. cun 3409    C_ wss 3411   (/)c0 3735   {csn 3969   ` cfv 5523   Basecbs 14731   0gc0g 14944   LModclmod 17722   LSubSpclss 17788   LSpanclspn 17827   LVecclvec 17958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-tpos 6910  df-recs 6997  df-rdg 7031  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-3 10554  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738  df-plusg 14812  df-mulr 14813  df-0g 14946  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-grp 16271  df-minusg 16272  df-sbg 16273  df-cmn 17014  df-abl 17015  df-mgp 17352  df-ur 17364  df-ring 17410  df-oppr 17482  df-dvdsr 17500  df-unit 17501  df-invr 17531  df-drng 17608  df-lmod 17724  df-lss 17789  df-lsp 17828  df-lvec 17959
This theorem is referenced by:  lspsncv0  18002  lsatcmp  31985  dihlspsnssN  34316  dihlspsnat  34317
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