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Theorem 0subg 16786
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 2420 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
2 0subg.z . . . 4  |-  .0.  =  ( 0g `  G )
31, 2grpidcl 16638 . . 3  |-  ( G  e.  Grp  ->  .0.  e.  ( Base `  G
) )
43snssd 4139 . 2  |-  ( G  e.  Grp  ->  {  .0.  } 
C_  ( Base `  G
) )
5 fvex 5882 . . . . 5  |-  ( 0g
`  G )  e. 
_V
62, 5eqeltri 2504 . . . 4  |-  .0.  e.  _V
76snnz 4112 . . 3  |-  {  .0.  }  =/=  (/)
87a1i 11 . 2  |-  ( G  e.  Grp  ->  {  .0.  }  =/=  (/) )
9 eqid 2420 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
101, 9, 2grplid 16640 . . . . 5  |-  ( ( G  e.  Grp  /\  .0.  e.  ( Base `  G
) )  ->  (  .0.  ( +g  `  G
)  .0.  )  =  .0.  )
113, 10mpdan 672 . . . 4  |-  ( G  e.  Grp  ->  (  .0.  ( +g  `  G
)  .0.  )  =  .0.  )
12 ovex 6324 . . . . 5  |-  (  .0.  ( +g  `  G
)  .0.  )  e. 
_V
1312elsnc 4017 . . . 4  |-  ( (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  }  <->  (  .0.  ( +g  `  G )  .0.  )  =  .0.  )
1411, 13sylibr 215 . . 3  |-  ( G  e.  Grp  ->  (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  } )
15 eqid 2420 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
162, 15grpinvid 16661 . . . 4  |-  ( G  e.  Grp  ->  (
( invg `  G ) `  .0.  )  =  .0.  )
17 fvex 5882 . . . . 5  |-  ( ( invg `  G
) `  .0.  )  e.  _V
1817elsnc 4017 . . . 4  |-  ( ( ( invg `  G ) `  .0.  )  e.  {  .0.  }  <-> 
( ( invg `  G ) `  .0.  )  =  .0.  )
1916, 18sylibr 215 . . 3  |-  ( G  e.  Grp  ->  (
( invg `  G ) `  .0.  )  e.  {  .0.  } )
20 oveq1 6303 . . . . . . . 8  |-  ( a  =  .0.  ->  (
a ( +g  `  G
) b )  =  (  .0.  ( +g  `  G ) b ) )
2120eleq1d 2489 . . . . . . 7  |-  ( a  =  .0.  ->  (
( a ( +g  `  G ) b )  e.  {  .0.  }  <->  (  .0.  ( +g  `  G
) b )  e. 
{  .0.  } ) )
2221ralbidv 2862 . . . . . 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 6304 . . . . . . . 8  |-  ( b  =  .0.  ->  (  .0.  ( +g  `  G
) b )  =  (  .0.  ( +g  `  G )  .0.  )
)
2423eleq1d 2489 . . . . . . 7  |-  ( b  =  .0.  ->  (
(  .0.  ( +g  `  G ) b )  e.  {  .0.  }  <->  (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  } ) )
256, 24ralsn 4032 . . . . . 6  |-  ( A. b  e.  {  .0.  }  (  .0.  ( +g  `  G ) b )  e.  {  .0.  }  <->  (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  } )
2622, 25syl6bb 264 . . . . 5  |-  ( a  =  .0.  ->  ( A. b  e.  {  .0.  }  ( a ( +g  `  G ) b )  e.  {  .0.  }  <->  (  .0.  ( +g  `  G
)  .0.  )  e. 
{  .0.  } ) )
27 fveq2 5872 . . . . . 6  |-  ( a  =  .0.  ->  (
( invg `  G ) `  a
)  =  ( ( invg `  G
) `  .0.  )
)
2827eleq1d 2489 . . . . 5  |-  ( a  =  .0.  ->  (
( ( invg `  G ) `  a
)  e.  {  .0.  }  <-> 
( ( invg `  G ) `  .0.  )  e.  {  .0.  } ) )
2926, 28anbi12d 715 . . . 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 4032 . . 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 668 . 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 16776 . 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 1188 1  |-  ( G  e.  Grp  ->  {  .0.  }  e.  (SubGrp `  G
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616   A.wral 2773   _Vcvv 3078    C_ wss 3433   (/)c0 3758   {csn 3993   ` cfv 5592  (class class class)co 6296   Basecbs 15073   +g cplusg 15142   0gc0g 15290   Grpcgrp 16613   invgcminusg 16614  SubGrpcsubg 16755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-ress 15080  df-plusg 15155  df-0g 15292  df-mgm 16432  df-sgrp 16471  df-mnd 16481  df-grp 16617  df-minusg 16618  df-subg 16758
This theorem is referenced by:  0nsg  16806  idressubgsymg  16995  pgp0  17176  slwn0  17195  lsm01  17249  lsm02  17250  dprdz  17591  dprdsn  17597  pgpfac1lem5  17640  tgptsmscls  21088
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