Theorem nmzbi 13578

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.

Step	Hyp	Ref	Expression
1		elnmz.1	. . . 4 |- N = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) }
2	1	elnmz 13577	. . 3 |- ( A e. N <-> ( A e. X /\ A. z e. X ( ( A .+ z ) e. S <-> ( z .+ A ) e. S ) ) )
3	2	simprbi 275	. 2 |- ( A e. N -> A. z e. X ( ( A .+ z ) e. S <-> ( z .+ A ) e. S ) )
4		oveq2 5954	. . . . 5 |- ( z = B -> ( A .+ z ) = ( A .+ B ) )
5	4	eleq1d 2274	. . . 4 |- ( z = B -> ( ( A .+ z ) e. S <-> ( A .+ B ) e. S ) )
6		oveq1 5953	. . . . 5 |- ( z = B -> ( z .+ A ) = ( B .+ A ) )
7	6	eleq1d 2274	. . . 4 |- ( z = B -> ( ( z .+ A ) e. S <-> ( B .+ A ) e. S ) )
8	5, 7	bibi12d 235	. . 3 |- ( z = B -> ( ( ( A .+ z ) e. S <-> ( z .+ A ) e. S ) <-> ( ( A .+ B ) e. S <-> ( B .+ A ) e. S ) ) )
9	8	rspccva 2876	. 2 |- ( ( A. z e. X ( ( A .+ z ) e. S <-> ( z .+ A ) e. S ) /\ B e. X ) -> ( ( A .+ B ) e. S <-> ( B .+ A ) e. S ) )
10	3, 9	sylan 283	1 |- ( ( A e. N /\ B e. X ) -> ( ( A .+ B ) e. S <-> ( B .+ A ) e. S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   A.wral 2484   {crab 2488  (class class class)co 5946

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-iota 5233  df-fv 5280  df-ov 5949

This theorem is referenced by:  nmzsubg  13579  nmznsg  13582  conjnmz  13648