Theorem 0nsg 14017

Description: The zero subgroup is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)

Hypothesis

Ref	Expression
0nsg.z	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
0nsg	|- ( G e. Grp -> { .0. } e. (NrmSGrp ` G ) )

Proof of Theorem 0nsg

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

Step	Hyp	Ref	Expression
1		0nsg.z	. . 3 |- .0. = ( 0g ` G )
2	1	0subg 14002	. 2 |- ( G e. Grp -> { .0. } e. (SubGrp ` G ) )
3		elsni 3727	. . . . . . . . 9 |- ( y e. { .0. } -> y = .0. )
4	3	ad2antll 495	. . . . . . . 8 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> y = .0. )
5	4	oveq2d 6101	. . . . . . 7 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` G ) .0. ) )
6		eqid 2238	. . . . . . . . 9 |- ( Base ` G ) = ( Base ` G )
7		eqid 2238	. . . . . . . . 9 |- ( +g ` G ) = ( +g ` G )
8	6, 7, 1	grprid 13837	. . . . . . . 8 |- ( ( G e. Grp /\ x e. ( Base ` G ) ) -> ( x ( +g ` G ) .0. ) = x )
9	8	adantrr 483	. . . . . . 7 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) .0. ) = x )
10	5, 9	eqtrd 2271	. . . . . 6 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) = x )
11	10	oveq1d 6100	. . . . 5 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) = ( x ( -g ` G ) x ) )
12		eqid 2238	. . . . . . 7 |- ( -g ` G ) = ( -g ` G )
13	6, 1, 12	grpsubid 13889	. . . . . 6 |- ( ( G e. Grp /\ x e. ( Base ` G ) ) -> ( x ( -g ` G ) x ) = .0. )
14	13	adantrr 483	. . . . 5 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( -g ` G ) x ) = .0. )
15	11, 14	eqtrd 2271	. . . 4 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) = .0. )
16		simpl 109	. . . . . 6 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> G e. Grp )
17		simprl 535	. . . . . . 7 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> x e. ( Base ` G ) )
18	6, 1	grpidcl 13834	. . . . . . . . 9 |- ( G e. Grp -> .0. e. ( Base ` G ) )
19	18	adantr 276	. . . . . . . 8 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> .0. e. ( Base ` G ) )
20	4, 19	eqeltrd 2315	. . . . . . 7 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> y e. ( Base ` G ) )
21	6, 7, 16, 17, 20	grpcld 13819	. . . . . 6 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) e. ( Base ` G ) )
22	6, 12	grpsubcl 13885	. . . . . 6 |- ( ( G e. Grp /\ ( x ( +g ` G ) y ) e. ( Base ` G ) /\ x e. ( Base ` G ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. ( Base ` G ) )
23	16, 21, 17, 22	syl3anc 1278	. . . . 5 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. ( Base ` G ) )
24		elsng 3724	. . . . 5 |- ( ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. ( Base ` G ) -> ( ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. { .0. } <-> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) = .0. ) )
25	23, 24	syl 14	. . . 4 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. { .0. } <-> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) = .0. ) )
26	15, 25	mpbird 167	. . 3 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. { .0. } )
27	26	ralrimivva 2632	. 2 |- ( G e. Grp -> A. x e. ( Base ` G ) A. y e. { .0. } ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. { .0. } )
28	6, 7, 12	isnsg3 14010	. 2 |- ( { .0. } e. (NrmSGrp ` G ) <-> ( { .0. } e. (SubGrp ` G ) /\ A. x e. ( Base ` G ) A. y e. { .0. } ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. { .0. } ) )
29	2, 27, 28	sylanbrc 421	1 |- ( G e. Grp -> { .0. } e. (NrmSGrp ` G ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   {csn 3709   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   0gc0g 13610   Grpcgrp 13805   -gcsg 13807  SubGrpcsubg 13970  NrmSGrpcnsg 13971

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-submnd 13767  df-grp 13808  df-minusg 13809  df-sbg 13810  df-subg 13973  df-nsg 13974

This theorem is used by:  0idnsgd  14019  1nsgtrivd  14022  ghmker  14073