Theorem 0nsg 13881

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 13866	. 2 |- ( G e. Grp -> { .0. } e. (SubGrp ` G ) )
3		elsni 3691	. . . . . . . . 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 6044	. . . . . . 7 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` G ) .0. ) )
6		eqid 2231	. . . . . . . . 9 |- ( Base ` G ) = ( Base ` G )
7		eqid 2231	. . . . . . . . 9 |- ( +g ` G ) = ( +g ` G )
8	6, 7, 1	grprid 13695	. . . . . . . 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 2264	. . . . . 6 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) = x )
11	10	oveq1d 6043	. . . . 5 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) = ( x ( -g ` G ) x ) )
12		eqid 2231	. . . . . . 7 |- ( -g ` G ) = ( -g ` G )
13	6, 1, 12	grpsubid 13747	. . . . . 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 2264	. . . 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 531	. . . . . . 7 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> x e. ( Base ` G ) )
18	6, 1	grpidcl 13692	. . . . . . . . 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 2308	. . . . . . 7 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> y e. ( Base ` G ) )
21	6, 7, 16, 17, 20	grpcld 13677	. . . . . 6 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) e. ( Base ` G ) )
22	6, 12	grpsubcl 13743	. . . . . 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 1274	. . . . 5 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. ( Base ` G ) )
24		elsng 3688	. . . . 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 2615	. 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 13874	. 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 1398   e. wcel 2202   A.wral 2511   {csn 3673   ` cfv 5333  (class class class)co 6028   Basecbs 13162   +g cplusg 13240   0gc0g 13419   Grpcgrp 13663   -gcsg 13665  SubGrpcsubg 13834  NrmSGrpcnsg 13835

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-submnd 13623  df-grp 13666  df-minusg 13667  df-sbg 13668  df-subg 13837  df-nsg 13838

This theorem is referenced by:  0idnsgd  13883  1nsgtrivd  13886  ghmker  13937