MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ablnsg Structured version   Unicode version

Theorem ablnsg 16639
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 2460 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2460 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
31, 2ablcom 16604 . . . . . 6  |-  ( ( G  e.  Abel  /\  y  e.  ( Base `  G
)  /\  z  e.  ( Base `  G )
)  ->  ( y
( +g  `  G ) z )  =  ( z ( +g  `  G
) y ) )
433expb 1192 . . . . 5  |-  ( ( G  e.  Abel  /\  (
y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( y ( +g  `  G ) z )  =  ( z ( +g  `  G ) y ) )
54eleq1d 2529 . . . 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 2878 . . 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 16018 . . . 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 900 . . 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 2457 1  |-  ( G  e.  Abel  ->  (NrmSGrp `  G
)  =  (SubGrp `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544  SubGrpcsubg 15983  NrmSGrpcnsg 15984   Abelcabel 16588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fv 5587  df-ov 6278  df-subg 15986  df-nsg 15987  df-cmn 16589  df-abl 16590
This theorem is referenced by:  divsabl  16657  divs1  17658  divsrhm  17660
  Copyright terms: Public domain W3C validator