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Theorem 0subg 16040
Description: The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.)
Hypothesis
Ref Expression
0subg.z  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
0subg  |-  ( G  e.  Grp  ->  {  .0.  }  e.  (SubGrp `  G
) )

Proof of Theorem 0subg
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
2 0subg.z . . . 4  |-  .0.  =  ( 0g `  G )
31, 2grpidcl 15892 . . 3  |-  ( G  e.  Grp  ->  .0.  e.  ( Base `  G
) )
43snssd 4172 . 2  |-  ( G  e.  Grp  ->  {  .0.  } 
C_  ( Base `  G
) )
5 fvex 5876 . . . . 5  |-  ( 0g
`  G )  e. 
_V
62, 5eqeltri 2551 . . . 4  |-  .0.  e.  _V
76snnz 4145 . . 3  |-  {  .0.  }  =/=  (/)
87a1i 11 . 2  |-  ( G  e.  Grp  ->  {  .0.  }  =/=  (/) )
9 eqid 2467 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
101, 9, 2grplid 15894 . . . . 5  |-  ( ( G  e.  Grp  /\  .0.  e.  ( Base `  G
) )  ->  (  .0.  ( +g  `  G
)  .0.  )  =  .0.  )
113, 10mpdan 668 . . . 4  |-  ( G  e.  Grp  ->  (  .0.  ( +g  `  G
)  .0.  )  =  .0.  )
12 ovex 6310 . . . . 5  |-  (  .0.  ( +g  `  G
)  .0.  )  e. 
_V
1312elsnc 4051 . . . 4  |-  ( (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  }  <->  (  .0.  ( +g  `  G )  .0.  )  =  .0.  )
1411, 13sylibr 212 . . 3  |-  ( G  e.  Grp  ->  (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  } )
15 eqid 2467 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
162, 15grpinvid 15915 . . . 4  |-  ( G  e.  Grp  ->  (
( invg `  G ) `  .0.  )  =  .0.  )
17 fvex 5876 . . . . 5  |-  ( ( invg `  G
) `  .0.  )  e.  _V
1817elsnc 4051 . . . 4  |-  ( ( ( invg `  G ) `  .0.  )  e.  {  .0.  }  <-> 
( ( invg `  G ) `  .0.  )  =  .0.  )
1916, 18sylibr 212 . . 3  |-  ( G  e.  Grp  ->  (
( invg `  G ) `  .0.  )  e.  {  .0.  } )
20 oveq1 6292 . . . . . . . 8  |-  ( a  =  .0.  ->  (
a ( +g  `  G
) b )  =  (  .0.  ( +g  `  G ) b ) )
2120eleq1d 2536 . . . . . . 7  |-  ( a  =  .0.  ->  (
( a ( +g  `  G ) b )  e.  {  .0.  }  <->  (  .0.  ( +g  `  G
) b )  e. 
{  .0.  } ) )
2221ralbidv 2903 . . . . . 6  |-  ( a  =  .0.  ->  ( A. b  e.  {  .0.  }  ( a ( +g  `  G ) b )  e.  {  .0.  }  <->  A. b  e.  {  .0.  }  (  .0.  ( +g  `  G ) b )  e.  {  .0.  }
) )
23 oveq2 6293 . . . . . . . 8  |-  ( b  =  .0.  ->  (  .0.  ( +g  `  G
) b )  =  (  .0.  ( +g  `  G )  .0.  )
)
2423eleq1d 2536 . . . . . . 7  |-  ( b  =  .0.  ->  (
(  .0.  ( +g  `  G ) b )  e.  {  .0.  }  <->  (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  } ) )
256, 24ralsn 4066 . . . . . 6  |-  ( A. b  e.  {  .0.  }  (  .0.  ( +g  `  G ) b )  e.  {  .0.  }  <->  (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  } )
2622, 25syl6bb 261 . . . . 5  |-  ( a  =  .0.  ->  ( A. b  e.  {  .0.  }  ( a ( +g  `  G ) b )  e.  {  .0.  }  <->  (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  } ) )
27 fveq2 5866 . . . . . 6  |-  ( a  =  .0.  ->  (
( invg `  G ) `  a
)  =  ( ( invg `  G
) `  .0.  )
)
2827eleq1d 2536 . . . . 5  |-  ( a  =  .0.  ->  (
( ( invg `  G ) `  a
)  e.  {  .0.  }  <-> 
( ( invg `  G ) `  .0.  )  e.  {  .0.  } ) )
2926, 28anbi12d 710 . . . 4  |-  ( a  =  .0.  ->  (
( A. b  e. 
{  .0.  }  (
a ( +g  `  G
) b )  e. 
{  .0.  }  /\  ( ( invg `  G ) `  a
)  e.  {  .0.  } )  <->  ( (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  }  /\  ( ( invg `  G ) `  .0.  )  e.  {  .0.  } ) ) )
306, 29ralsn 4066 . . 3  |-  ( A. a  e.  {  .0.  }  ( A. b  e. 
{  .0.  }  (
a ( +g  `  G
) b )  e. 
{  .0.  }  /\  ( ( invg `  G ) `  a
)  e.  {  .0.  } )  <->  ( (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  }  /\  ( ( invg `  G ) `  .0.  )  e.  {  .0.  } ) )
3114, 19, 30sylanbrc 664 . 2  |-  ( G  e.  Grp  ->  A. a  e.  {  .0.  }  ( A. b  e.  {  .0.  }  ( a ( +g  `  G ) b )  e.  {  .0.  }  /\  ( ( invg `  G ) `  a
)  e.  {  .0.  } ) )
321, 9, 15issubg2 16030 . 2  |-  ( G  e.  Grp  ->  ( {  .0.  }  e.  (SubGrp `  G )  <->  ( {  .0.  }  C_  ( Base `  G )  /\  {  .0.  }  =/=  (/)  /\  A. a  e.  {  .0.  }  ( A. b  e. 
{  .0.  }  (
a ( +g  `  G
) b )  e. 
{  .0.  }  /\  ( ( invg `  G ) `  a
)  e.  {  .0.  } ) ) ) )
334, 8, 31, 32mpbir3and 1179 1  |-  ( G  e.  Grp  ->  {  .0.  }  e.  (SubGrp `  G
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113    C_ wss 3476   (/)c0 3785   {csn 4027   ` cfv 5588  (class class class)co 6285   Basecbs 14493   +g cplusg 14558   0gc0g 14698   Grpcgrp 15730   invgcminusg 15731  SubGrpcsubg 16009
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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-0g 14700  df-mnd 15735  df-grp 15871  df-minusg 15872  df-subg 16012
This theorem is referenced by:  0nsg  16060  idressubgsymg  16249  pgp0  16431  slwn0  16450  lsm01  16504  lsm02  16505  dprdz  16891  dprdsn  16897  pgpfac1lem5  16944  tgptsmscls  20479
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