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Theorem nsgbi 14926
Description: Defining property of a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
Hypotheses
Ref Expression
isnsg.1  |-  X  =  ( Base `  G
)
isnsg.2  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
nsgbi  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S
) )

Proof of Theorem nsgbi
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnsg.1 . . . . 5  |-  X  =  ( Base `  G
)
2 isnsg.2 . . . . 5  |-  .+  =  ( +g  `  G )
31, 2isnsg 14924 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  <->  ( S  e.  (SubGrp `  G )  /\  A. x  e.  X  A. y  e.  X  ( ( x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S ) ) )
43simprbi 451 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  A. x  e.  X  A. y  e.  X  ( (
x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S
) )
5 oveq1 6047 . . . . . 6  |-  ( x  =  A  ->  (
x  .+  y )  =  ( A  .+  y ) )
65eleq1d 2470 . . . . 5  |-  ( x  =  A  ->  (
( x  .+  y
)  e.  S  <->  ( A  .+  y )  e.  S
) )
7 oveq2 6048 . . . . . 6  |-  ( x  =  A  ->  (
y  .+  x )  =  ( y  .+  A ) )
87eleq1d 2470 . . . . 5  |-  ( x  =  A  ->  (
( y  .+  x
)  e.  S  <->  ( y  .+  A )  e.  S
) )
96, 8bibi12d 313 . . . 4  |-  ( x  =  A  ->  (
( ( x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S )  <->  ( ( A  .+  y )  e.  S  <->  ( y  .+  A )  e.  S
) ) )
10 oveq2 6048 . . . . . 6  |-  ( y  =  B  ->  ( A  .+  y )  =  ( A  .+  B
) )
1110eleq1d 2470 . . . . 5  |-  ( y  =  B  ->  (
( A  .+  y
)  e.  S  <->  ( A  .+  B )  e.  S
) )
12 oveq1 6047 . . . . . 6  |-  ( y  =  B  ->  (
y  .+  A )  =  ( B  .+  A ) )
1312eleq1d 2470 . . . . 5  |-  ( y  =  B  ->  (
( y  .+  A
)  e.  S  <->  ( B  .+  A )  e.  S
) )
1411, 13bibi12d 313 . . . 4  |-  ( y  =  B  ->  (
( ( A  .+  y )  e.  S  <->  ( y  .+  A )  e.  S )  <->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S
) ) )
159, 14rspc2v 3018 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  ( ( x 
.+  y )  e.  S  <->  ( y  .+  x )  e.  S
)  ->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S
) ) )
164, 15syl5com 28 . 2  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( ( A  e.  X  /\  B  e.  X )  ->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S ) ) )
17163impib 1151 1  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484  SubGrpcsubg 14893  NrmSGrpcnsg 14894
This theorem is referenced by:  nsgconj  14928  eqgcpbl  14949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-subg 14896  df-nsg 14897
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