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Theorem lsssn0 17029
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 2444 . 2  |-  ( W  e.  LMod  ->  (Scalar `  W )  =  (Scalar `  W ) )
2 eqidd 2444 . 2  |-  ( W  e.  LMod  ->  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
) )
3 eqidd 2444 . 2  |-  ( W  e.  LMod  ->  ( Base `  W )  =  (
Base `  W )
)
4 eqidd 2444 . 2  |-  ( W  e.  LMod  ->  ( +g  `  W )  =  ( +g  `  W ) )
5 eqidd 2444 . 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 2443 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
9 lss0cl.z . . . 4  |-  .0.  =  ( 0g `  W )
108, 9lmod0vcl 16977 . . 3  |-  ( W  e.  LMod  ->  .0.  e.  ( Base `  W )
)
1110snssd 4018 . 2  |-  ( W  e.  LMod  ->  {  .0.  } 
C_  ( Base `  W
) )
12 fvex 5701 . . . . 5  |-  ( 0g
`  W )  e. 
_V
139, 12eqeltri 2513 . . . 4  |-  .0.  e.  _V
1413snnz 3993 . . 3  |-  {  .0.  }  =/=  (/)
1514a1i 11 . 2  |-  ( W  e.  LMod  ->  {  .0.  }  =/=  (/) )
16 simpr2 995 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  a  e.  {  .0.  } )
17 elsni 3902 . . . . . . . 8  |-  ( a  e.  {  .0.  }  ->  a  =  .0.  )
1816, 17syl 16 . . . . . . 7  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  a  =  .0.  )
1918oveq2d 6107 . . . . . 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 2443 . . . . . . . 8  |-  (Scalar `  W )  =  (Scalar `  W )
21 eqid 2443 . . . . . . . 8  |-  ( .s
`  W )  =  ( .s `  W
)
22 eqid 2443 . . . . . . . 8  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2320, 21, 22, 9lmodvs0 16982 . . . . . . 7  |-  ( ( W  e.  LMod  /\  x  e.  ( Base `  (Scalar `  W ) ) )  ->  ( x ( .s `  W )  .0.  )  =  .0.  )
24233ad2antr1 1153 . . . . . 6  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  ( x
( .s `  W
)  .0.  )  =  .0.  )
2519, 24eqtrd 2475 . . . . 5  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  ( x
( .s `  W
) a )  =  .0.  )
26 simpr3 996 . . . . . 6  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  b  e.  {  .0.  } )
27 elsni 3902 . . . . . 6  |-  ( b  e.  {  .0.  }  ->  b  =  .0.  )
2826, 27syl 16 . . . . 5  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  b  =  .0.  )
2925, 28oveq12d 6109 . . . 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 2443 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  W )
318, 30, 9lmod0vlid 16978 . . . . . 6  |-  ( ( W  e.  LMod  /\  .0.  e.  ( Base `  W
) )  ->  (  .0.  ( +g  `  W
)  .0.  )  =  .0.  )
3210, 31mpdan 668 . . . . 5  |-  ( W  e.  LMod  ->  (  .0.  ( +g  `  W
)  .0.  )  =  .0.  )
3332adantr 465 . . . 4  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  (  .0.  ( +g  `  W )  .0.  )  =  .0.  )
3429, 33eqtrd 2475 . . 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 6116 . . . 4  |-  ( ( x ( .s `  W ) a ) ( +g  `  W
) b )  e. 
_V
3635elsnc 3901 . . 3  |-  ( ( ( x ( .s
`  W ) a ) ( +g  `  W
) b )  e. 
{  .0.  }  <->  ( (
x ( .s `  W ) a ) ( +g  `  W
) b )  =  .0.  )
3734, 36sylibr 212 . 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 17017 1  |-  ( W  e.  LMod  ->  {  .0.  }  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   _Vcvv 2972   (/)c0 3637   {csn 3877   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238  Scalarcsca 14241   .scvsca 14242   0gc0g 14378   LModclmod 16948   LSubSpclss 17013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-plusg 14251  df-0g 14380  df-mnd 15415  df-grp 15545  df-mgp 16592  df-rng 16647  df-lmod 16950  df-lss 17014
This theorem is referenced by:  lspsn0  17089  lsp0  17090  lmhmkerlss  17132  lidl0  17301  lsatcv0  32676  lsatcveq0  32677  lsat0cv  32678  lsatcv0eq  32692  dochsat  35028  mapd0  35310  mapdcnvatN  35311  mapdat  35312  mapdn0  35314  hdmapeq0  35492
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