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Theorem lspsnat 15898
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 22160 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 3464 . . . . . 6  |-  ( ( U  \  {  .0.  } )  =/=  (/)  <->  E. x  x  e.  ( U  \  {  .0.  } ) )
2 simprl 732 . . . . . . . . 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 958 . . . . . . . . . . 11  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  W  e.  LVec )
6 lveclmod 15859 . . . . . . . . . . 11  |-  ( W  e.  LVec  ->  W  e. 
LMod )
75, 6syl 15 . . . . . . . . . 10  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  W  e.  LMod )
8 simpl2 959 . . . . . . . . . 10  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  U  e.  S
)
9 simprr 733 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  x  e.  ( U  \  {  .0.  } ) )
10 eldifi 3298 . . . . . . . . . . . . 13  |-  ( x  e.  ( U  \  {  .0.  } )  ->  x  e.  U )
119, 10syl 15 . . . . . . . . . . . 12  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  x  e.  U
)
123, 4, 7, 8, 11lspsnel5a 15753 . . . . . . . . . . 11  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  ( N `  { x } ) 
C_  U )
13 0ss 3483 . . . . . . . . . . . . . 14  |-  (/)  C_  V
1413a1i 10 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  (/)  C_  V )
15 simpl3 960 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  X  e.  V
)
16 ssdif 3311 . . . . . . . . . . . . . . . 16  |-  ( U 
C_  ( N `  { X } )  -> 
( U  \  {  .0.  } )  C_  (
( N `  { X } )  \  {  .0.  } ) )
1716ad2antrl 708 . . . . . . . . . . . . . . 15  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  ( U  \  {  .0.  } )  C_  ( ( N `  { X } )  \  {  .0.  } ) )
1817, 9sseldd 3181 . . . . . . . . . . . . . 14  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  x  e.  ( ( N `  { X } )  \  {  .0.  } ) )
19 uncom 3319 . . . . . . . . . . . . . . . . . 18  |-  ( (/)  u. 
{ X } )  =  ( { X }  u.  (/) )
20 un0 3479 . . . . . . . . . . . . . . . . . 18  |-  ( { X }  u.  (/) )  =  { X }
2119, 20eqtri 2303 . . . . . . . . . . . . . . . . 17  |-  ( (/)  u. 
{ X } )  =  { X }
2221fveq2i 5528 . . . . . . . . . . . . . . . 16  |-  ( N `
 ( (/)  u.  { X } ) )  =  ( N `  { X } )
2322a1i 10 . . . . . . . . . . . . . . 15  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  ( N `  ( (/)  u.  { X } ) )  =  ( N `  { X } ) )
24 lspsnat.z . . . . . . . . . . . . . . . . 17  |-  .0.  =  ( 0g `  W )
2524, 4lsp0 15766 . . . . . . . . . . . . . . . 16  |-  ( W  e.  LMod  ->  ( N `
 (/) )  =  {  .0.  } )
267, 25syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  ( N `  (/) )  =  {  .0.  } )
2723, 26difeq12d 3295 . . . . . . . . . . . . . 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.  } ) )
2818, 27eleqtrrd 2360 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  x  e.  ( ( N `  ( (/) 
u.  { X }
) )  \  ( N `  (/) ) ) )
29 lspsnat.v . . . . . . . . . . . . . 14  |-  V  =  ( Base `  W
)
3029, 3, 4lspsolv 15896 . . . . . . . . . . . . 13  |-  ( ( W  e.  LVec  /\  ( (/)  C_  V  /\  X  e.  V  /\  x  e.  ( ( N `  ( (/)  u.  { X } ) )  \ 
( N `  (/) ) ) ) )  ->  X  e.  ( N `  ( (/) 
u.  { x }
) ) )
315, 14, 15, 28, 30syl13anc 1184 . . . . . . . . . . . 12  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  X  e.  ( N `  ( (/)  u. 
{ x } ) ) )
32 uncom 3319 . . . . . . . . . . . . . 14  |-  ( (/)  u. 
{ x } )  =  ( { x }  u.  (/) )
33 un0 3479 . . . . . . . . . . . . . 14  |-  ( { x }  u.  (/) )  =  { x }
3432, 33eqtri 2303 . . . . . . . . . . . . 13  |-  ( (/)  u. 
{ x } )  =  { x }
3534fveq2i 5528 . . . . . . . . . . . 12  |-  ( N `
 ( (/)  u.  {
x } ) )  =  ( N `  { x } )
3631, 35syl6eleq 2373 . . . . . . . . . . 11  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  X  e.  ( N `  { x } ) )
3712, 36sseldd 3181 . . . . . . . . . 10  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  X  e.  U
)
383, 4, 7, 8, 37lspsnel5a 15753 . . . . . . . . 9  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  ( N `  { X } )  C_  U )
392, 38eqssd 3196 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  ( U  C_  ( N `  { X } )  /\  x  e.  ( U  \  {  .0.  } ) ) )  ->  U  =  ( N `  { X } ) )
4039expr 598 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  (
x  e.  ( U 
\  {  .0.  }
)  ->  U  =  ( N `  { X } ) ) )
4140exlimdv 1664 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( E. x  x  e.  ( U  \  {  .0.  } )  ->  U  =  ( N `  { X } ) ) )
421, 41syl5bi 208 . . . . 5  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  (
( U  \  {  .0.  } )  =/=  (/)  ->  U  =  ( N `  { X } ) ) )
4342necon1bd 2514 . . . 4  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( -.  U  =  ( N `  { X } )  ->  ( U  \  {  .0.  }
)  =  (/) ) )
44 ssdif0 3513 . . . 4  |-  ( U 
C_  {  .0.  }  <->  ( U  \  {  .0.  } )  =  (/) )
4543, 44syl6ibr 218 . . 3  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( -.  U  =  ( N `  { X } )  ->  U  C_ 
{  .0.  } ) )
46 simpl1 958 . . . . 5  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  W  e.  LVec )
4746, 6syl 15 . . . 4  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  W  e.  LMod )
48 simpl2 959 . . . 4  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  U  e.  S )
4924, 3lssle0 15707 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( U  C_  {  .0.  }  <->  U  =  {  .0.  }
) )
5047, 48, 49syl2anc 642 . . 3  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( U  C_  {  .0.  }  <->  U  =  {  .0.  }
) )
5145, 50sylibd 205 . 2  |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `
 { X }
) )  ->  ( -.  U  =  ( N `  { X } )  ->  U  =  {  .0.  } ) )
5251orrd 367 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 176    \/ wo 357    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640   ` cfv 5255   Basecbs 13148   0gc0g 13400   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728   LVecclvec 15855
This theorem is referenced by:  lspsncv0  15899  lsatcmp  29193  dihlspsnssN  31522  dihlspsnat  31523
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856
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