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Theorem nmzbi 14657
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 14656 . . 3  |-  ( A  e.  N  <->  ( A  e.  X  /\  A. z  e.  X  ( ( A  .+  z )  e.  S  <->  ( z  .+  A )  e.  S
) ) )
32simprbi 450 . 2  |-  ( A  e.  N  ->  A. z  e.  X  ( ( A  .+  z )  e.  S  <->  ( z  .+  A )  e.  S
) )
4 oveq2 5866 . . . . 5  |-  ( z  =  B  ->  ( A  .+  z )  =  ( A  .+  B
) )
54eleq1d 2349 . . . 4  |-  ( z  =  B  ->  (
( A  .+  z
)  e.  S  <->  ( A  .+  B )  e.  S
) )
6 oveq1 5865 . . . . 5  |-  ( z  =  B  ->  (
z  .+  A )  =  ( B  .+  A ) )
76eleq1d 2349 . . . 4  |-  ( z  =  B  ->  (
( z  .+  A
)  e.  S  <->  ( B  .+  A )  e.  S
) )
85, 7bibi12d 312 . . 3  |-  ( z  =  B  ->  (
( ( A  .+  z )  e.  S  <->  ( z  .+  A )  e.  S )  <->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S
) ) )
98rspccva 2883 . 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 457 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547  (class class class)co 5858
This theorem is referenced by:  nmzsubg  14658  nmznsg  14661  conjnmz  14716
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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