Theorem nsgbi 13017

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 13015	. . . 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 5881	. . . . . 6 |- ( x = A -> ( x .+ y ) = ( A .+ y ) )
6	5	eleq1d 2246	. . . . 5 |- ( x = A -> ( ( x .+ y ) e. S <-> ( A .+ y ) e. S ) )
7		oveq2 5882	. . . . . 6 |- ( x = A -> ( y .+ x ) = ( y .+ A ) )
8	7	eleq1d 2246	. . . . 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 5882	. . . . . 6 |- ( y = B -> ( A .+ y ) = ( A .+ B ) )
11	10	eleq1d 2246	. . . . 5 |- ( y = B -> ( ( A .+ y ) e. S <-> ( A .+ B ) e. S ) )
12		oveq1 5881	. . . . . 6 |- ( y = B -> ( y .+ A ) = ( B .+ A ) )
13	12	eleq1d 2246	. . . . 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 2854	. . 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 1201	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 978   = wceq 1353   e. wcel 2148   A.wral 2455   ` cfv 5216  (class class class)co 5874   Basecbs 12456   +g cplusg 12530  SubGrpcsubg 12980  NrmSGrpcnsg 12981

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-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  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-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-ov 5877  df-inn 8918  df-2 8976  df-ndx 12459  df-slot 12460  df-base 12462  df-plusg 12543  df-subg 12983  df-nsg 12984

This theorem is referenced by:  nsgconj  13019  eqgcpbl  13040