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Theorem nmzbi 14867
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 14866 . . 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 5989 . . . . 5  |-  ( z  =  B  ->  ( A  .+  z )  =  ( A  .+  B
) )
54eleq1d 2432 . . . 4  |-  ( z  =  B  ->  (
( A  .+  z
)  e.  S  <->  ( A  .+  B )  e.  S
) )
6 oveq1 5988 . . . . 5  |-  ( z  =  B  ->  (
z  .+  A )  =  ( B  .+  A ) )
76eleq1d 2432 . . . 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 2968 . 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 1647    e. wcel 1715   A.wral 2628   {crab 2632  (class class class)co 5981
This theorem is referenced by:  nmzsubg  14868  nmznsg  14871  conjnmz  14926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-iota 5322  df-fv 5366  df-ov 5984
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