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Theorem nmzbi 14943
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 14942 . . 3  |-  ( A  e.  N  <->  ( A  e.  X  /\  A. z  e.  X  ( ( A  .+  z )  e.  S  <->  ( z  .+  A )  e.  S
) ) )
32simprbi 451 . 2  |-  ( A  e.  N  ->  A. z  e.  X  ( ( A  .+  z )  e.  S  <->  ( z  .+  A )  e.  S
) )
4 oveq2 6056 . . . . 5  |-  ( z  =  B  ->  ( A  .+  z )  =  ( A  .+  B
) )
54eleq1d 2478 . . . 4  |-  ( z  =  B  ->  (
( A  .+  z
)  e.  S  <->  ( A  .+  B )  e.  S
) )
6 oveq1 6055 . . . . 5  |-  ( z  =  B  ->  (
z  .+  A )  =  ( B  .+  A ) )
76eleq1d 2478 . . . 4  |-  ( z  =  B  ->  (
( z  .+  A
)  e.  S  <->  ( B  .+  A )  e.  S
) )
85, 7bibi12d 313 . . 3  |-  ( z  =  B  ->  (
( ( A  .+  z )  e.  S  <->  ( z  .+  A )  e.  S )  <->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S
) ) )
98rspccva 3019 . 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 458 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   {crab 2678  (class class class)co 6048
This theorem is referenced by:  nmzsubg  14944  nmznsg  14947  conjnmz  15002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-ov 6051
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