Theorem nsgbi 13655

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 13653	. . . 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 5974	. . . . . 6 |- ( x = A -> ( x .+ y ) = ( A .+ y ) )
6	5	eleq1d 2276	. . . . 5 |- ( x = A -> ( ( x .+ y ) e. S <-> ( A .+ y ) e. S ) )
7		oveq2 5975	. . . . . 6 |- ( x = A -> ( y .+ x ) = ( y .+ A ) )
8	7	eleq1d 2276	. . . . 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 5975	. . . . . 6 |- ( y = B -> ( A .+ y ) = ( A .+ B ) )
11	10	eleq1d 2276	. . . . 5 |- ( y = B -> ( ( A .+ y ) e. S <-> ( A .+ B ) e. S ) )
12		oveq1 5974	. . . . . 6 |- ( y = B -> ( y .+ A ) = ( B .+ A ) )
13	12	eleq1d 2276	. . . . 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 2897	. . 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 2178   A.wral 2486   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024  SubGrpcsubg 13618  NrmSGrpcnsg 13619

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-ov 5970  df-inn 9072  df-2 9130  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-subg 13621  df-nsg 13622

This theorem is referenced by:  nsgconj  13657  eqgcpbl  13679