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Theorem nsgbi 16434
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 16432 . . . 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 462 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  A. x  e.  X  A. y  e.  X  ( (
x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S
) )
5 oveq1 6277 . . . . . 6  |-  ( x  =  A  ->  (
x  .+  y )  =  ( A  .+  y ) )
65eleq1d 2523 . . . . 5  |-  ( x  =  A  ->  (
( x  .+  y
)  e.  S  <->  ( A  .+  y )  e.  S
) )
7 oveq2 6278 . . . . . 6  |-  ( x  =  A  ->  (
y  .+  x )  =  ( y  .+  A ) )
87eleq1d 2523 . . . . 5  |-  ( x  =  A  ->  (
( y  .+  x
)  e.  S  <->  ( y  .+  A )  e.  S
) )
96, 8bibi12d 319 . . . 4  |-  ( x  =  A  ->  (
( ( x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S )  <->  ( ( A  .+  y )  e.  S  <->  ( y  .+  A )  e.  S
) ) )
10 oveq2 6278 . . . . . 6  |-  ( y  =  B  ->  ( A  .+  y )  =  ( A  .+  B
) )
1110eleq1d 2523 . . . . 5  |-  ( y  =  B  ->  (
( A  .+  y
)  e.  S  <->  ( A  .+  B )  e.  S
) )
12 oveq1 6277 . . . . . 6  |-  ( y  =  B  ->  (
y  .+  A )  =  ( B  .+  A ) )
1312eleq1d 2523 . . . . 5  |-  ( y  =  B  ->  (
( y  .+  A
)  e.  S  <->  ( B  .+  A )  e.  S
) )
1411, 13bibi12d 319 . . . 4  |-  ( y  =  B  ->  (
( ( A  .+  y )  e.  S  <->  ( y  .+  A )  e.  S )  <->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S
) ) )
159, 14rspc2v 3216 . . 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 1192 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787  SubGrpcsubg 16397  NrmSGrpcnsg 16398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-subg 16400  df-nsg 16401
This theorem is referenced by:  nsgconj  16436  eqgcpbl  16457
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