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Theorem nsgbi 16020
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 16018 . . . 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 6282 . . . . . 6  |-  ( x  =  A  ->  (
x  .+  y )  =  ( A  .+  y ) )
65eleq1d 2529 . . . . 5  |-  ( x  =  A  ->  (
( x  .+  y
)  e.  S  <->  ( A  .+  y )  e.  S
) )
7 oveq2 6283 . . . . . 6  |-  ( x  =  A  ->  (
y  .+  x )  =  ( y  .+  A ) )
87eleq1d 2529 . . . . 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 6283 . . . . . 6  |-  ( y  =  B  ->  ( A  .+  y )  =  ( A  .+  B
) )
1110eleq1d 2529 . . . . 5  |-  ( y  =  B  ->  (
( A  .+  y
)  e.  S  <->  ( A  .+  B )  e.  S
) )
12 oveq1 6282 . . . . . 6  |-  ( y  =  B  ->  (
y  .+  A )  =  ( B  .+  A ) )
1312eleq1d 2529 . . . . 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 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 1189 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 968    = wceq 1374    e. wcel 1762   A.wral 2807   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544  SubGrpcsubg 15983  NrmSGrpcnsg 15984
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
This theorem is referenced by:  nsgconj  16022  eqgcpbl  16043
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