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.

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 12999	. . 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 5880	. . . . 5 |- ( z = B -> ( A .+ z ) = ( A .+ B ) )
5	4	eleq1d 2246	. . . 4 |- ( z = B -> ( ( A .+ z ) e. S <-> ( A .+ B ) e. S ) )
6		oveq1 5879	. . . . 5 |- ( z = B -> ( z .+ A ) = ( B .+ A ) )
7	6	eleq1d 2246	. . . 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 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 ) )
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 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