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Theorem nmzbi 14985
Description: Defining property of the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
Hypothesis
Ref Expression
elnmz.1  |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S ) }
Assertion
Ref Expression
nmzbi  |-  ( ( A  e.  N  /\  B  e.  X )  ->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S ) )
Distinct variable groups:    x, A    x, y, S    x,  .+ , y    x, X, y
Allowed substitution hints:    A( y)    B( x, y)    N( x, y)

Proof of Theorem nmzbi
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elnmz.1 . . . 4  |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S ) }
21elnmz 14984 . . 3  |-  ( A  e.  N  <->  ( A  e.  X  /\  A. z  e.  X  ( ( A  .+  z )  e.  S  <->  ( z  .+  A )  e.  S
) ) )
32simprbi 452 . 2  |-  ( A  e.  N  ->  A. z  e.  X  ( ( A  .+  z )  e.  S  <->  ( z  .+  A )  e.  S
) )
4 oveq2 6092 . . . . 5  |-  ( z  =  B  ->  ( A  .+  z )  =  ( A  .+  B
) )
54eleq1d 2504 . . . 4  |-  ( z  =  B  ->  (
( A  .+  z
)  e.  S  <->  ( A  .+  B )  e.  S
) )
6 oveq1 6091 . . . . 5  |-  ( z  =  B  ->  (
z  .+  A )  =  ( B  .+  A ) )
76eleq1d 2504 . . . 4  |-  ( z  =  B  ->  (
( z  .+  A
)  e.  S  <->  ( B  .+  A )  e.  S
) )
85, 7bibi12d 314 . . 3  |-  ( z  =  B  ->  (
( ( A  .+  z )  e.  S  <->  ( z  .+  A )  e.  S )  <->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S
) ) )
98rspccva 3053 . 2  |-  ( ( A. z  e.  X  ( ( A  .+  z )  e.  S  <->  ( z  .+  A )  e.  S )  /\  B  e.  X )  ->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S ) )
103, 9sylan 459 1  |-  ( ( A  e.  N  /\  B  e.  X )  ->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711  (class class class)co 6084
This theorem is referenced by:  nmzsubg  14986  nmznsg  14989  conjnmz  15044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-ov 6087
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