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Theorem 0subg 14965
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 2436 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
2 0subg.z . . . 4  |-  .0.  =  ( 0g `  G )
31, 2grpidcl 14833 . . 3  |-  ( G  e.  Grp  ->  .0.  e.  ( Base `  G
) )
43snssd 3943 . 2  |-  ( G  e.  Grp  ->  {  .0.  } 
C_  ( Base `  G
) )
5 fvex 5742 . . . . 5  |-  ( 0g
`  G )  e. 
_V
62, 5eqeltri 2506 . . . 4  |-  .0.  e.  _V
76snnz 3922 . . 3  |-  {  .0.  }  =/=  (/)
87a1i 11 . 2  |-  ( G  e.  Grp  ->  {  .0.  }  =/=  (/) )
9 eqid 2436 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
101, 9, 2grplid 14835 . . . . 5  |-  ( ( G  e.  Grp  /\  .0.  e.  ( Base `  G
) )  ->  (  .0.  ( +g  `  G
)  .0.  )  =  .0.  )
113, 10mpdan 650 . . . 4  |-  ( G  e.  Grp  ->  (  .0.  ( +g  `  G
)  .0.  )  =  .0.  )
12 ovex 6106 . . . . 5  |-  (  .0.  ( +g  `  G
)  .0.  )  e. 
_V
1312elsnc 3837 . . . 4  |-  ( (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  }  <->  (  .0.  ( +g  `  G )  .0.  )  =  .0.  )
1411, 13sylibr 204 . . 3  |-  ( G  e.  Grp  ->  (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  } )
15 eqid 2436 . . . . 5  |-  ( inv g `  G )  =  ( inv g `  G )
162, 15grpinvid 14856 . . . 4  |-  ( G  e.  Grp  ->  (
( inv g `  G ) `  .0.  )  =  .0.  )
17 fvex 5742 . . . . 5  |-  ( ( inv g `  G
) `  .0.  )  e.  _V
1817elsnc 3837 . . . 4  |-  ( ( ( inv g `  G ) `  .0.  )  e.  {  .0.  }  <-> 
( ( inv g `  G ) `  .0.  )  =  .0.  )
1916, 18sylibr 204 . . 3  |-  ( G  e.  Grp  ->  (
( inv g `  G ) `  .0.  )  e.  {  .0.  } )
20 oveq1 6088 . . . . . . . 8  |-  ( a  =  .0.  ->  (
a ( +g  `  G
) b )  =  (  .0.  ( +g  `  G ) b ) )
2120eleq1d 2502 . . . . . . 7  |-  ( a  =  .0.  ->  (
( a ( +g  `  G ) b )  e.  {  .0.  }  <->  (  .0.  ( +g  `  G
) b )  e. 
{  .0.  } ) )
2221ralbidv 2725 . . . . . 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 6089 . . . . . . . 8  |-  ( b  =  .0.  ->  (  .0.  ( +g  `  G
) b )  =  (  .0.  ( +g  `  G )  .0.  )
)
2423eleq1d 2502 . . . . . . 7  |-  ( b  =  .0.  ->  (
(  .0.  ( +g  `  G ) b )  e.  {  .0.  }  <->  (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  } ) )
256, 24ralsn 3849 . . . . . 6  |-  ( A. b  e.  {  .0.  }  (  .0.  ( +g  `  G ) b )  e.  {  .0.  }  <->  (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  } )
2622, 25syl6bb 253 . . . . 5  |-  ( a  =  .0.  ->  ( A. b  e.  {  .0.  }  ( a ( +g  `  G ) b )  e.  {  .0.  }  <->  (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  } ) )
27 fveq2 5728 . . . . . 6  |-  ( a  =  .0.  ->  (
( inv g `  G ) `  a
)  =  ( ( inv g `  G
) `  .0.  )
)
2827eleq1d 2502 . . . . 5  |-  ( a  =  .0.  ->  (
( ( inv g `  G ) `  a
)  e.  {  .0.  }  <-> 
( ( inv g `  G ) `  .0.  )  e.  {  .0.  } ) )
2926, 28anbi12d 692 . . . 4  |-  ( a  =  .0.  ->  (
( A. b  e. 
{  .0.  }  (
a ( +g  `  G
) b )  e. 
{  .0.  }  /\  ( ( inv g `  G ) `  a
)  e.  {  .0.  } )  <->  ( (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  }  /\  ( ( inv g `  G ) `  .0.  )  e.  {  .0.  } ) ) )
306, 29ralsn 3849 . . 3  |-  ( A. a  e.  {  .0.  }  ( A. b  e. 
{  .0.  }  (
a ( +g  `  G
) b )  e. 
{  .0.  }  /\  ( ( inv g `  G ) `  a
)  e.  {  .0.  } )  <->  ( (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  }  /\  ( ( inv g `  G ) `  .0.  )  e.  {  .0.  } ) )
3114, 19, 30sylanbrc 646 . 2  |-  ( G  e.  Grp  ->  A. a  e.  {  .0.  }  ( A. b  e.  {  .0.  }  ( a ( +g  `  G ) b )  e.  {  .0.  }  /\  ( ( inv g `  G ) `  a
)  e.  {  .0.  } ) )
321, 9, 15issubg2 14959 . 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.  }  /\  ( ( inv g `  G ) `  a
)  e.  {  .0.  } ) ) ) )
334, 8, 31, 32mpbir3and 1137 1  |-  ( G  e.  Grp  ->  {  .0.  }  e.  (SubGrp `  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   _Vcvv 2956    C_ wss 3320   (/)c0 3628   {csn 3814   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Grpcgrp 14685   inv gcminusg 14686  SubGrpcsubg 14938
This theorem is referenced by:  0nsg  14985  pgp0  15230  slwn0  15249  lsm01  15303  lsm02  15304  dprdz  15588  dprdsn  15594  pgpfac1lem5  15637  tgptsmscls  18179
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-subg 14941
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