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Theorem 0subg 14642
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 2283 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
2 0subg.z . . . 4  |-  .0.  =  ( 0g `  G )
31, 2grpidcl 14510 . . 3  |-  ( G  e.  Grp  ->  .0.  e.  ( Base `  G
) )
43snssd 3760 . 2  |-  ( G  e.  Grp  ->  {  .0.  } 
C_  ( Base `  G
) )
5 fvex 5539 . . . . 5  |-  ( 0g
`  G )  e. 
_V
62, 5eqeltri 2353 . . . 4  |-  .0.  e.  _V
76snnz 3744 . . 3  |-  {  .0.  }  =/=  (/)
87a1i 10 . 2  |-  ( G  e.  Grp  ->  {  .0.  }  =/=  (/) )
9 eqid 2283 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
101, 9, 2grplid 14512 . . . . 5  |-  ( ( G  e.  Grp  /\  .0.  e.  ( Base `  G
) )  ->  (  .0.  ( +g  `  G
)  .0.  )  =  .0.  )
113, 10mpdan 649 . . . 4  |-  ( G  e.  Grp  ->  (  .0.  ( +g  `  G
)  .0.  )  =  .0.  )
12 ovex 5883 . . . . 5  |-  (  .0.  ( +g  `  G
)  .0.  )  e. 
_V
1312elsnc 3663 . . . 4  |-  ( (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  }  <->  (  .0.  ( +g  `  G )  .0.  )  =  .0.  )
1411, 13sylibr 203 . . 3  |-  ( G  e.  Grp  ->  (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  } )
15 eqid 2283 . . . . 5  |-  ( inv g `  G )  =  ( inv g `  G )
162, 15grpinvid 14533 . . . 4  |-  ( G  e.  Grp  ->  (
( inv g `  G ) `  .0.  )  =  .0.  )
17 fvex 5539 . . . . 5  |-  ( ( inv g `  G
) `  .0.  )  e.  _V
1817elsnc 3663 . . . 4  |-  ( ( ( inv g `  G ) `  .0.  )  e.  {  .0.  }  <-> 
( ( inv g `  G ) `  .0.  )  =  .0.  )
1916, 18sylibr 203 . . 3  |-  ( G  e.  Grp  ->  (
( inv g `  G ) `  .0.  )  e.  {  .0.  } )
20 oveq1 5865 . . . . . . . 8  |-  ( a  =  .0.  ->  (
a ( +g  `  G
) b )  =  (  .0.  ( +g  `  G ) b ) )
2120eleq1d 2349 . . . . . . 7  |-  ( a  =  .0.  ->  (
( a ( +g  `  G ) b )  e.  {  .0.  }  <->  (  .0.  ( +g  `  G
) b )  e. 
{  .0.  } ) )
2221ralbidv 2563 . . . . . 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 5866 . . . . . . . 8  |-  ( b  =  .0.  ->  (  .0.  ( +g  `  G
) b )  =  (  .0.  ( +g  `  G )  .0.  )
)
2423eleq1d 2349 . . . . . . 7  |-  ( b  =  .0.  ->  (
(  .0.  ( +g  `  G ) b )  e.  {  .0.  }  <->  (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  } ) )
256, 24ralsn 3674 . . . . . 6  |-  ( A. b  e.  {  .0.  }  (  .0.  ( +g  `  G ) b )  e.  {  .0.  }  <->  (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  } )
2622, 25syl6bb 252 . . . . 5  |-  ( a  =  .0.  ->  ( A. b  e.  {  .0.  }  ( a ( +g  `  G ) b )  e.  {  .0.  }  <->  (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  } ) )
27 fveq2 5525 . . . . . 6  |-  ( a  =  .0.  ->  (
( inv g `  G ) `  a
)  =  ( ( inv g `  G
) `  .0.  )
)
2827eleq1d 2349 . . . . 5  |-  ( a  =  .0.  ->  (
( ( inv g `  G ) `  a
)  e.  {  .0.  }  <-> 
( ( inv g `  G ) `  .0.  )  e.  {  .0.  } ) )
2926, 28anbi12d 691 . . . 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 3674 . . 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 645 . 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 14636 . 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 1135 1  |-  ( G  e.  Grp  ->  {  .0.  }  e.  (SubGrp `  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    C_ wss 3152   (/)c0 3455   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363  SubGrpcsubg 14615
This theorem is referenced by:  0nsg  14662  pgp0  14907  slwn0  14926  lsm01  14980  lsm02  14981  dprdz  15265  dprdsn  15271  pgpfac1lem5  15314  tgptsmscls  17832
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-subg 14618
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