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Theorem lsssn0 17374
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 2468 . 2  |-  ( W  e.  LMod  ->  (Scalar `  W )  =  (Scalar `  W ) )
2 eqidd 2468 . 2  |-  ( W  e.  LMod  ->  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
) )
3 eqidd 2468 . 2  |-  ( W  e.  LMod  ->  ( Base `  W )  =  (
Base `  W )
)
4 eqidd 2468 . 2  |-  ( W  e.  LMod  ->  ( +g  `  W )  =  ( +g  `  W ) )
5 eqidd 2468 . 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 2467 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
9 lss0cl.z . . . 4  |-  .0.  =  ( 0g `  W )
108, 9lmod0vcl 17321 . . 3  |-  ( W  e.  LMod  ->  .0.  e.  ( Base `  W )
)
1110snssd 4172 . 2  |-  ( W  e.  LMod  ->  {  .0.  } 
C_  ( Base `  W
) )
12 fvex 5874 . . . . 5  |-  ( 0g
`  W )  e. 
_V
139, 12eqeltri 2551 . . . 4  |-  .0.  e.  _V
1413snnz 4145 . . 3  |-  {  .0.  }  =/=  (/)
1514a1i 11 . 2  |-  ( W  e.  LMod  ->  {  .0.  }  =/=  (/) )
16 simpr2 1003 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  a  e.  {  .0.  } )
17 elsni 4052 . . . . . . . 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 6298 . . . . . 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 2467 . . . . . . . 8  |-  (Scalar `  W )  =  (Scalar `  W )
21 eqid 2467 . . . . . . . 8  |-  ( .s
`  W )  =  ( .s `  W
)
22 eqid 2467 . . . . . . . 8  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2320, 21, 22, 9lmodvs0 17326 . . . . . . 7  |-  ( ( W  e.  LMod  /\  x  e.  ( Base `  (Scalar `  W ) ) )  ->  ( x ( .s `  W )  .0.  )  =  .0.  )
24233ad2antr1 1161 . . . . . 6  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  ( x
( .s `  W
)  .0.  )  =  .0.  )
2519, 24eqtrd 2508 . . . . 5  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  ( x
( .s `  W
) a )  =  .0.  )
26 simpr3 1004 . . . . . 6  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  {  .0.  }  /\  b  e.  {  .0.  } ) )  ->  b  e.  {  .0.  } )
27 elsni 4052 . . . . . 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 6300 . . . 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 2467 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  W )
318, 30, 9lmod0vlid 17322 . . . . . 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 2508 . . 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 6307 . . . 4  |-  ( ( x ( .s `  W ) a ) ( +g  `  W
) b )  e. 
_V
3635elsnc 4051 . . 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 17362 1  |-  ( W  e.  LMod  ->  {  .0.  }  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   (/)c0 3785   {csn 4027   ` cfv 5586  (class class class)co 6282   Basecbs 14483   +g cplusg 14548  Scalarcsca 14551   .scvsca 14552   0gc0g 14688   LModclmod 17292   LSubSpclss 17358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-plusg 14561  df-0g 14690  df-mnd 15725  df-grp 15855  df-mgp 16929  df-rng 16985  df-lmod 17294  df-lss 17359
This theorem is referenced by:  lspsn0  17434  lsp0  17435  lmhmkerlss  17477  lidl0  17646  lsatcv0  33828  lsatcveq0  33829  lsat0cv  33830  lsatcv0eq  33844  dochsat  36180  mapd0  36462  mapdcnvatN  36463  mapdat  36464  mapdn0  36466  hdmapeq0  36644
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