Theorem nsgbi 13540

Description: Defining property of a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)

Hypotheses

Ref	Expression
isnsg.1	|- X = ( Base ` G )
isnsg.2	|- .+ = ( +g ` G )

Assertion

Ref	Expression
nsgbi	|- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. X ) -> ( ( A .+ B ) e. S <-> ( B .+ A ) e. S ) )

Proof of Theorem nsgbi

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isnsg.1	. . . . 5 |- X = ( Base ` G )
2		isnsg.2	. . . . 5 |- .+ = ( +g ` G )
3	1, 2	isnsg 13538	. . . 4 |- ( S e. (NrmSGrp ` G ) <-> ( S e. (SubGrp ` G ) /\ A. x e. X A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) ) )
4	3	simprbi 275	. . 3 |- ( S e. (NrmSGrp ` G ) -> A. x e. X A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) )
5		oveq1 5951	. . . . . 6 |- ( x = A -> ( x .+ y ) = ( A .+ y ) )
6	5	eleq1d 2274	. . . . 5 |- ( x = A -> ( ( x .+ y ) e. S <-> ( A .+ y ) e. S ) )
7		oveq2 5952	. . . . . 6 |- ( x = A -> ( y .+ x ) = ( y .+ A ) )
8	7	eleq1d 2274	. . . . 5 |- ( x = A -> ( ( y .+ x ) e. S <-> ( y .+ A ) e. S ) )
9	6, 8	bibi12d 235	. . . 4 |- ( x = A -> ( ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) <-> ( ( A .+ y ) e. S <-> ( y .+ A ) e. S ) ) )
10		oveq2 5952	. . . . . 6 |- ( y = B -> ( A .+ y ) = ( A .+ B ) )
11	10	eleq1d 2274	. . . . 5 |- ( y = B -> ( ( A .+ y ) e. S <-> ( A .+ B ) e. S ) )
12		oveq1 5951	. . . . . 6 |- ( y = B -> ( y .+ A ) = ( B .+ A ) )
13	12	eleq1d 2274	. . . . 5 |- ( y = B -> ( ( y .+ A ) e. S <-> ( B .+ A ) e. S ) )
14	11, 13	bibi12d 235	. . . 4 |- ( y = B -> ( ( ( A .+ y ) e. S <-> ( y .+ A ) e. S ) <-> ( ( A .+ B ) e. S <-> ( B .+ A ) e. S ) ) )
15	9, 14	rspc2v 2890	. . 3 |- ( ( A e. X /\ B e. X ) -> ( A. x e. X A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) -> ( ( A .+ B ) e. S <-> ( B .+ A ) e. S ) ) )
16	4, 15	syl5com 29	. 2 |- ( S e. (NrmSGrp ` G ) -> ( ( A e. X /\ B e. X ) -> ( ( A .+ B ) e. S <-> ( B .+ A ) e. S ) ) )
17	16	3impib 1204	1 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. X ) -> ( ( A .+ B ) e. S <-> ( B .+ A ) e. S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   A.wral 2484   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909  SubGrpcsubg 13503  NrmSGrpcnsg 13504

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-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  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-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5947  df-inn 9037  df-2 9095  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-subg 13506  df-nsg 13507

This theorem is referenced by:  nsgconj  13542  eqgcpbl  13564