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Theorem 0subg 14658
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 2296 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
2 0subg.z . . . 4  |-  .0.  =  ( 0g `  G )
31, 2grpidcl 14526 . . 3  |-  ( G  e.  Grp  ->  .0.  e.  ( Base `  G
) )
43snssd 3776 . 2  |-  ( G  e.  Grp  ->  {  .0.  } 
C_  ( Base `  G
) )
5 fvex 5555 . . . . 5  |-  ( 0g
`  G )  e. 
_V
62, 5eqeltri 2366 . . . 4  |-  .0.  e.  _V
76snnz 3757 . . 3  |-  {  .0.  }  =/=  (/)
87a1i 10 . 2  |-  ( G  e.  Grp  ->  {  .0.  }  =/=  (/) )
9 eqid 2296 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
101, 9, 2grplid 14528 . . . . 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 5899 . . . . 5  |-  (  .0.  ( +g  `  G
)  .0.  )  e. 
_V
1312elsnc 3676 . . . 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 2296 . . . . 5  |-  ( inv g `  G )  =  ( inv g `  G )
162, 15grpinvid 14549 . . . 4  |-  ( G  e.  Grp  ->  (
( inv g `  G ) `  .0.  )  =  .0.  )
17 fvex 5555 . . . . 5  |-  ( ( inv g `  G
) `  .0.  )  e.  _V
1817elsnc 3676 . . . 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 5881 . . . . . . . 8  |-  ( a  =  .0.  ->  (
a ( +g  `  G
) b )  =  (  .0.  ( +g  `  G ) b ) )
2120eleq1d 2362 . . . . . . 7  |-  ( a  =  .0.  ->  (
( a ( +g  `  G ) b )  e.  {  .0.  }  <->  (  .0.  ( +g  `  G
) b )  e. 
{  .0.  } ) )
2221ralbidv 2576 . . . . . 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 5882 . . . . . . . 8  |-  ( b  =  .0.  ->  (  .0.  ( +g  `  G
) b )  =  (  .0.  ( +g  `  G )  .0.  )
)
2423eleq1d 2362 . . . . . . 7  |-  ( b  =  .0.  ->  (
(  .0.  ( +g  `  G ) b )  e.  {  .0.  }  <->  (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  } ) )
256, 24ralsn 3687 . . . . . 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 5541 . . . . . 6  |-  ( a  =  .0.  ->  (
( inv g `  G ) `  a
)  =  ( ( inv g `  G
) `  .0.  )
)
2827eleq1d 2362 . . . . 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 3687 . . 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 14652 . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801    C_ wss 3165   (/)c0 3468   {csn 3653   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Grpcgrp 14378   inv gcminusg 14379  SubGrpcsubg 14631
This theorem is referenced by:  0nsg  14678  pgp0  14923  slwn0  14942  lsm01  14996  lsm02  14997  dprdz  15281  dprdsn  15287  pgpfac1lem5  15330  tgptsmscls  17848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-subg 14634
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