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Theorem nmzbi 13000
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 12999 . . 3  |-  ( A  e.  N  <->  ( A  e.  X  /\  A. z  e.  X  ( ( A  .+  z )  e.  S  <->  ( z  .+  A )  e.  S
) ) )
32simprbi 275 . 2  |-  ( A  e.  N  ->  A. z  e.  X  ( ( A  .+  z )  e.  S  <->  ( z  .+  A )  e.  S
) )
4 oveq2 5880 . . . . 5  |-  ( z  =  B  ->  ( A  .+  z )  =  ( A  .+  B
) )
54eleq1d 2246 . . . 4  |-  ( z  =  B  ->  (
( A  .+  z
)  e.  S  <->  ( A  .+  B )  e.  S
) )
6 oveq1 5879 . . . . 5  |-  ( z  =  B  ->  (
z  .+  A )  =  ( B  .+  A ) )
76eleq1d 2246 . . . 4  |-  ( z  =  B  ->  (
( z  .+  A
)  e.  S  <->  ( B  .+  A )  e.  S
) )
85, 7bibi12d 235 . . 3  |-  ( z  =  B  ->  (
( ( A  .+  z )  e.  S  <->  ( z  .+  A )  e.  S )  <->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S
) ) )
98rspccva 2840 . 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 283 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    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455   {crab 2459  (class class class)co 5872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-iota 5177  df-fv 5223  df-ov 5875
This theorem is referenced by:  nmzsubg  13001  nmznsg  13004
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