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Theorem ablnsg 16435
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 2451 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2451 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
31, 2ablcom 16400 . . . . . 6  |-  ( ( G  e.  Abel  /\  y  e.  ( Base `  G
)  /\  z  e.  ( Base `  G )
)  ->  ( y
( +g  `  G ) z )  =  ( z ( +g  `  G
) y ) )
433expb 1189 . . . . 5  |-  ( ( G  e.  Abel  /\  (
y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( y ( +g  `  G ) z )  =  ( z ( +g  `  G ) y ) )
54eleq1d 2520 . . . 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 2906 . . 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 15814 . . . 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 898 . . 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 16 . 2  |-  ( G  e.  Abel  ->  ( x  e.  (NrmSGrp `  G
)  <->  x  e.  (SubGrp `  G ) ) )
109eqrdv 2448 1  |-  ( G  e.  Abel  ->  (NrmSGrp `  G
)  =  (SubGrp `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   ` cfv 5518  (class class class)co 6192   Basecbs 14278   +g cplusg 14342  SubGrpcsubg 15779  NrmSGrpcnsg 15780   Abelcabel 16384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fv 5526  df-ov 6195  df-subg 15782  df-nsg 15783  df-cmn 16385  df-abl 16386
This theorem is referenced by:  divsabl  16453  divs1  17425  divsrhm  17427
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