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Theorem nsgbi 15710
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 15708 . . . 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 464 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  A. x  e.  X  A. y  e.  X  ( (
x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S
) )
5 oveq1 6096 . . . . . 6  |-  ( x  =  A  ->  (
x  .+  y )  =  ( A  .+  y ) )
65eleq1d 2507 . . . . 5  |-  ( x  =  A  ->  (
( x  .+  y
)  e.  S  <->  ( A  .+  y )  e.  S
) )
7 oveq2 6097 . . . . . 6  |-  ( x  =  A  ->  (
y  .+  x )  =  ( y  .+  A ) )
87eleq1d 2507 . . . . 5  |-  ( x  =  A  ->  (
( y  .+  x
)  e.  S  <->  ( y  .+  A )  e.  S
) )
96, 8bibi12d 321 . . . 4  |-  ( x  =  A  ->  (
( ( x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S )  <->  ( ( A  .+  y )  e.  S  <->  ( y  .+  A )  e.  S
) ) )
10 oveq2 6097 . . . . . 6  |-  ( y  =  B  ->  ( A  .+  y )  =  ( A  .+  B
) )
1110eleq1d 2507 . . . . 5  |-  ( y  =  B  ->  (
( A  .+  y
)  e.  S  <->  ( A  .+  B )  e.  S
) )
12 oveq1 6096 . . . . . 6  |-  ( y  =  B  ->  (
y  .+  A )  =  ( B  .+  A ) )
1312eleq1d 2507 . . . . 5  |-  ( y  =  B  ->  (
( y  .+  A
)  e.  S  <->  ( B  .+  A )  e.  S
) )
1411, 13bibi12d 321 . . . 4  |-  ( y  =  B  ->  (
( ( A  .+  y )  e.  S  <->  ( y  .+  A )  e.  S )  <->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S
) ) )
159, 14rspc2v 3077 . . 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 30 . 2  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( ( A  e.  X  /\  B  e.  X )  ->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S ) ) )
17163impib 1185 1  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2713   ` cfv 5416  (class class class)co 6089   Basecbs 14172   +g cplusg 14236  SubGrpcsubg 15673  NrmSGrpcnsg 15674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fv 5424  df-ov 6092  df-subg 15676  df-nsg 15677
This theorem is referenced by:  nsgconj  15712  eqgcpbl  15733
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