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Theorem nsgbi 14858
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 14856 . . . 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 450 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  A. x  e.  X  A. y  e.  X  ( (
x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S
) )
5 oveq1 5988 . . . . . 6  |-  ( x  =  A  ->  (
x  .+  y )  =  ( A  .+  y ) )
65eleq1d 2432 . . . . 5  |-  ( x  =  A  ->  (
( x  .+  y
)  e.  S  <->  ( A  .+  y )  e.  S
) )
7 oveq2 5989 . . . . . 6  |-  ( x  =  A  ->  (
y  .+  x )  =  ( y  .+  A ) )
87eleq1d 2432 . . . . 5  |-  ( x  =  A  ->  (
( y  .+  x
)  e.  S  <->  ( y  .+  A )  e.  S
) )
96, 8bibi12d 312 . . . 4  |-  ( x  =  A  ->  (
( ( x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S )  <->  ( ( A  .+  y )  e.  S  <->  ( y  .+  A )  e.  S
) ) )
10 oveq2 5989 . . . . . 6  |-  ( y  =  B  ->  ( A  .+  y )  =  ( A  .+  B
) )
1110eleq1d 2432 . . . . 5  |-  ( y  =  B  ->  (
( A  .+  y
)  e.  S  <->  ( A  .+  B )  e.  S
) )
12 oveq1 5988 . . . . . 6  |-  ( y  =  B  ->  (
y  .+  A )  =  ( B  .+  A ) )
1312eleq1d 2432 . . . . 5  |-  ( y  =  B  ->  (
( y  .+  A
)  e.  S  <->  ( B  .+  A )  e.  S
) )
1411, 13bibi12d 312 . . . 4  |-  ( y  =  B  ->  (
( ( A  .+  y )  e.  S  <->  ( y  .+  A )  e.  S )  <->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S
) ) )
159, 14rspc2v 2975 . . 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 26 . 2  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( ( A  e.  X  /\  B  e.  X )  ->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S ) ) )
17163impib 1150 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 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   A.wral 2628   ` cfv 5358  (class class class)co 5981   Basecbs 13356   +g cplusg 13416  SubGrpcsubg 14825  NrmSGrpcnsg 14826
This theorem is referenced by:  nsgconj  14860  eqgcpbl  14881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fv 5366  df-ov 5984  df-subg 14828  df-nsg 14829
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