Theorem 0nsg 13751

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 13736	. 2 |- ( G e. Grp -> { .0. } e. (SubGrp ` G ) )
3		elsni 3684	. . . . . . . . 9 |- ( y e. { .0. } -> y = .0. )
4	3	ad2antll 491	. . . . . . . 8 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> y = .0. )
5	4	oveq2d 6017	. . . . . . 7 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` G ) .0. ) )
6		eqid 2229	. . . . . . . . 9 |- ( Base ` G ) = ( Base ` G )
7		eqid 2229	. . . . . . . . 9 |- ( +g ` G ) = ( +g ` G )
8	6, 7, 1	grprid 13565	. . . . . . . 8 |- ( ( G e. Grp /\ x e. ( Base ` G ) ) -> ( x ( +g ` G ) .0. ) = x )
9	8	adantrr 479	. . . . . . 7 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) .0. ) = x )
10	5, 9	eqtrd 2262	. . . . . 6 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) = x )
11	10	oveq1d 6016	. . . . 5 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) = ( x ( -g ` G ) x ) )
12		eqid 2229	. . . . . . 7 |- ( -g ` G ) = ( -g ` G )
13	6, 1, 12	grpsubid 13617	. . . . . 6 |- ( ( G e. Grp /\ x e. ( Base ` G ) ) -> ( x ( -g ` G ) x ) = .0. )
14	13	adantrr 479	. . . . 5 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( -g ` G ) x ) = .0. )
15	11, 14	eqtrd 2262	. . . 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 529	. . . . . . 7 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> x e. ( Base ` G ) )
18	6, 1	grpidcl 13562	. . . . . . . . 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 2306	. . . . . . 7 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> y e. ( Base ` G ) )
21	6, 7, 16, 17, 20	grpcld 13547	. . . . . 6 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) e. ( Base ` G ) )
22	6, 12	grpsubcl 13613	. . . . . 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 1271	. . . . 5 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. ( Base ` G ) )
24		elsng 3681	. . . . 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 2612	. 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 13744	. 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 417	1 |- ( G e. Grp -> { .0. } e. (NrmSGrp ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   {csn 3666   ` cfv 5318  (class class class)co 6001   Basecbs 13032   +g cplusg 13110   0gc0g 13289   Grpcgrp 13533   -gcsg 13535  SubGrpcsubg 13704  NrmSGrpcnsg 13705

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-pre-ltirr 8111  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-inn 9111  df-2 9169  df-ndx 13035  df-slot 13036  df-base 13038  df-sets 13039  df-iress 13040  df-plusg 13123  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-submnd 13493  df-grp 13536  df-minusg 13537  df-sbg 13538  df-subg 13707  df-nsg 13708

This theorem is referenced by:  0idnsgd  13753  1nsgtrivd  13756  ghmker  13807