Theorem nmzbi 17634

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 17633	. . 3 |- ( A e. N <-> ( A e. X /\ A. z e. X ( ( A .+ z ) e. S <-> ( z .+ A ) e. S ) ) )
3	2	simprbi 480	. 2 |- ( A e. N -> A. z e. X ( ( A .+ z ) e. S <-> ( z .+ A ) e. S ) )
4		oveq2 6658	. . . . 5 |- ( z = B -> ( A .+ z ) = ( A .+ B ) )
5	4	eleq1d 2686	. . . 4 |- ( z = B -> ( ( A .+ z ) e. S <-> ( A .+ B ) e. S ) )
6		oveq1 6657	. . . . 5 |- ( z = B -> ( z .+ A ) = ( B .+ A ) )
7	6	eleq1d 2686	. . . 4 |- ( z = B -> ( ( z .+ A ) e. S <-> ( B .+ A ) e. S ) )
8	5, 7	bibi12d 335	. . 3 |- ( z = B -> ( ( ( A .+ z ) e. S <-> ( z .+ A ) e. S ) <-> ( ( A .+ B ) e. S <-> ( B .+ A ) e. S ) ) )
9	8	rspccva 3308	. 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 488	1 |- ( ( A e. N /\ B e. X ) -> ( ( A .+ B ) e. S <-> ( B .+ A ) e. S ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  nmzsubg  17635  nmznsg  17638  conjnmz  17694