Theorem nsgbi 13095

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 13093	. . . 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 5895	. . . . . 6 |- ( x = A -> ( x .+ y ) = ( A .+ y ) )
6	5	eleq1d 2256	. . . . 5 |- ( x = A -> ( ( x .+ y ) e. S <-> ( A .+ y ) e. S ) )
7		oveq2 5896	. . . . . 6 |- ( x = A -> ( y .+ x ) = ( y .+ A ) )
8	7	eleq1d 2256	. . . . 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 5896	. . . . . 6 |- ( y = B -> ( A .+ y ) = ( A .+ B ) )
11	10	eleq1d 2256	. . . . 5 |- ( y = B -> ( ( A .+ y ) e. S <-> ( A .+ B ) e. S ) )
12		oveq1 5895	. . . . . 6 |- ( y = B -> ( y .+ A ) = ( B .+ A ) )
13	12	eleq1d 2256	. . . . 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 2866	. . 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 1202	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 979   = wceq 1363   e. wcel 2158   A.wral 2465   ` cfv 5228  (class class class)co 5888   Basecbs 12475   +g cplusg 12550  SubGrpcsubg 13058  NrmSGrpcnsg 13059

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-ov 5891  df-inn 8933  df-2 8991  df-ndx 12478  df-slot 12479  df-base 12481  df-plusg 12563  df-subg 13061  df-nsg 13062

This theorem is referenced by:  nsgconj  13097  eqgcpbl  13119