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Theorem ablnsg 15139
Description: Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
Assertion
Ref Expression
ablnsg  |-  ( G  e.  Abel  ->  (NrmSGrp `  G
)  =  (SubGrp `  G ) )

Proof of Theorem ablnsg
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2283 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
31, 2ablcom 15106 . . . . . 6  |-  ( ( G  e.  Abel  /\  y  e.  ( Base `  G
)  /\  z  e.  ( Base `  G )
)  ->  ( y
( +g  `  G ) z )  =  ( z ( +g  `  G
) y ) )
433expb 1152 . . . . 5  |-  ( ( G  e.  Abel  /\  (
y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( y ( +g  `  G ) z )  =  ( z ( +g  `  G ) y ) )
54eleq1d 2349 . . . 4  |-  ( ( G  e.  Abel  /\  (
y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( ( y ( +g  `  G ) z )  e.  x  <->  ( z ( +g  `  G
) y )  e.  x ) )
65ralrimivva 2635 . . 3  |-  ( G  e.  Abel  ->  A. y  e.  ( Base `  G
) A. z  e.  ( Base `  G
) ( ( y ( +g  `  G
) z )  e.  x  <->  ( z ( +g  `  G ) y )  e.  x
) )
71, 2isnsg 14646 . . . 4  |-  ( x  e.  (NrmSGrp `  G
)  <->  ( x  e.  (SubGrp `  G )  /\  A. y  e.  (
Base `  G ) A. z  e.  ( Base `  G ) ( ( y ( +g  `  G ) z )  e.  x  <->  ( z
( +g  `  G ) y )  e.  x
) ) )
87rbaib 873 . . 3  |-  ( A. y  e.  ( Base `  G ) A. z  e.  ( Base `  G
) ( ( y ( +g  `  G
) z )  e.  x  <->  ( z ( +g  `  G ) y )  e.  x
)  ->  ( x  e.  (NrmSGrp `  G )  <->  x  e.  (SubGrp `  G
) ) )
96, 8syl 15 . 2  |-  ( G  e.  Abel  ->  ( x  e.  (NrmSGrp `  G
)  <->  x  e.  (SubGrp `  G ) ) )
109eqrdv 2281 1  |-  ( G  e.  Abel  ->  (NrmSGrp `  G
)  =  (SubGrp `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  SubGrpcsubg 14615  NrmSGrpcnsg 14616   Abelcabel 15090
This theorem is referenced by:  divsabl  15157  divs1  15987  divsrhm  15989
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-fv 5263  df-ov 5861  df-subg 14618  df-nsg 14619  df-cmn 15091  df-abl 15092
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