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Theorem lsssn0 18171
Description: The singleton of the zero vector is a subspace. (Contributed by NM, 13-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
lss0cl.z  |-  .0.  =  ( 0g `  W )
lss0cl.s  |-  S  =  ( LSubSp `  W )
Assertion
Ref Expression
lsssn0  |-  ( W  e.  LMod  ->  {  .0.  }  e.  S )

Proof of Theorem lsssn0
Dummy variables  x  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2452 . 2  |-  ( W  e.  LMod  ->  (Scalar `  W )  =  (Scalar `  W ) )
2 eqidd 2452 . 2  |-  ( W  e.  LMod  ->  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
) )
3 eqidd 2452 . 2  |-  ( W  e.  LMod  ->  ( Base `  W )  =  (
Base `  W )
)
4 eqidd 2452 . 2  |-  ( W  e.  LMod  ->  ( +g  `  W )  =  ( +g  `  W ) )
5 eqidd 2452 . 2  |-  ( W  e.  LMod  ->  ( .s
`  W )  =  ( .s `  W
) )
6 lss0cl.s . . 3  |-  S  =  ( LSubSp `  W )
76a1i 11 . 2  |-  ( W  e.  LMod  ->  S  =  ( LSubSp `  W )
)
8 eqid 2451 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
9 lss0cl.z . . . 4  |-  .0.  =  ( 0g `  W )
108, 9lmod0vcl 18120 . . 3  |-  ( W  e.  LMod  ->  .0.  e.  ( Base `  W )
)
1110snssd 4117 . 2  |-  ( W  e.  LMod  ->  {  .0.  } 
C_  ( Base `  W
) )
12 fvex 5875 . . . . 5  |-  ( 0g
`  W )  e. 
_V
139, 12eqeltri 2525 . . . 4  |-  .0.  e.  _V
1413snnz 4090 . . 3  |-  {  .0.  }  =/=  (/)
1514a1i 11 . 2  |-  ( W  e.  LMod  ->  {  .0.  }  =/=  (/) )
16 simpr2 1015 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  a  e.  {  .0.  } )
17 elsni 3993 . . . . . . . 8  |-  ( a  e.  {  .0.  }  ->  a  =  .0.  )
1816, 17syl 17 . . . . . . 7  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  a  =  .0.  )
1918oveq2d 6306 . . . . . 6  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  ( x
( .s `  W
) a )  =  ( x ( .s
`  W )  .0.  ) )
20 eqid 2451 . . . . . . . 8  |-  (Scalar `  W )  =  (Scalar `  W )
21 eqid 2451 . . . . . . . 8  |-  ( .s
`  W )  =  ( .s `  W
)
22 eqid 2451 . . . . . . . 8  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2320, 21, 22, 9lmodvs0 18125 . . . . . . 7  |-  ( ( W  e.  LMod  /\  x  e.  ( Base `  (Scalar `  W ) ) )  ->  ( x ( .s `  W )  .0.  )  =  .0.  )
24233ad2antr1 1173 . . . . . 6  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  ( x
( .s `  W
)  .0.  )  =  .0.  )
2519, 24eqtrd 2485 . . . . 5  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  ( x
( .s `  W
) a )  =  .0.  )
26 simpr3 1016 . . . . . 6  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  b  e.  {  .0.  } )
27 elsni 3993 . . . . . 6  |-  ( b  e.  {  .0.  }  ->  b  =  .0.  )
2826, 27syl 17 . . . . 5  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  b  =  .0.  )
2925, 28oveq12d 6308 . . . 4  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  ( (
x ( .s `  W ) a ) ( +g  `  W
) b )  =  (  .0.  ( +g  `  W )  .0.  )
)
30 eqid 2451 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  W )
318, 30, 9lmod0vlid 18121 . . . . . 6  |-  ( ( W  e.  LMod  /\  .0.  e.  ( Base `  W
) )  ->  (  .0.  ( +g  `  W
)  .0.  )  =  .0.  )
3210, 31mpdan 674 . . . . 5  |-  ( W  e.  LMod  ->  (  .0.  ( +g  `  W
)  .0.  )  =  .0.  )
3332adantr 467 . . . 4  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  (  .0.  ( +g  `  W )  .0.  )  =  .0.  )
3429, 33eqtrd 2485 . . 3  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  ( (
x ( .s `  W ) a ) ( +g  `  W
) b )  =  .0.  )
35 ovex 6318 . . . 4  |-  ( ( x ( .s `  W ) a ) ( +g  `  W
) b )  e. 
_V
3635elsnc 3992 . . 3  |-  ( ( ( x ( .s
`  W ) a ) ( +g  `  W
) b )  e. 
{  .0.  }  <->  ( (
x ( .s `  W ) a ) ( +g  `  W
) b )  =  .0.  )
3734, 36sylibr 216 . 2  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  ( (
x ( .s `  W ) a ) ( +g  `  W
) b )  e. 
{  .0.  } )
381, 2, 3, 4, 5, 7, 11, 15, 37islssd 18159 1  |-  ( W  e.  LMod  ->  {  .0.  }  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   _Vcvv 3045   (/)c0 3731   {csn 3968   ` cfv 5582  (class class class)co 6290   Basecbs 15121   +g cplusg 15190  Scalarcsca 15193   .scvsca 15194   0gc0g 15338   LModclmod 18091   LSubSpclss 18155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-plusg 15203  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-grp 16673  df-mgp 17724  df-ring 17782  df-lmod 18093  df-lss 18156
This theorem is referenced by:  lspsn0  18231  lsp0  18232  lmhmkerlss  18274  lidl0  18443  lsatcv0  32597  lsatcveq0  32598  lsat0cv  32599  lsatcv0eq  32613  dochsat  34951  mapd0  35233  mapdcnvatN  35234  mapdat  35235  mapdn0  35237  hdmapeq0  35415
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