Theorem 0nsg 13005

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 12990	. 2 |- ( G e. Grp -> { .0. } e. (SubGrp ` G ) )
3		elsni 3610	. . . . . . . . 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 5888	. . . . . . 7 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` G ) .0. ) )
6		eqid 2177	. . . . . . . . 9 |- ( Base ` G ) = ( Base ` G )
7		eqid 2177	. . . . . . . . 9 |- ( +g ` G ) = ( +g ` G )
8	6, 7, 1	grprid 12839	. . . . . . . 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 2210	. . . . . 6 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) = x )
11	10	oveq1d 5887	. . . . 5 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) = ( x ( -g ` G ) x ) )
12		eqid 2177	. . . . . . 7 |- ( -g ` G ) = ( -g ` G )
13	6, 1, 12	grpsubid 12886	. . . . . 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 2210	. . . 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 12836	. . . . . . . . 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 2254	. . . . . . 7 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> y e. ( Base ` G ) )
21	6, 7, 16, 17, 20	grpcld 12822	. . . . . 6 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) e. ( Base ` G ) )
22	6, 12	grpsubcl 12882	. . . . . 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 1238	. . . . 5 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. ( Base ` G ) )
24		elsng 3607	. . . . 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 2559	. 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 12998	. 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 1353   e. wcel 2148   A.wral 2455   {csn 3592   ` cfv 5215  (class class class)co 5872   Basecbs 12454   +g cplusg 12528   0gc0g 12693   Grpcgrp 12809   -gcsg 12811  SubGrpcsubg 12958  NrmSGrpcnsg 12959

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-ltadd 7924

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-submnd 12784  df-grp 12812  df-minusg 12813  df-sbg 12814  df-subg 12961  df-nsg 12962

This theorem is referenced by:  0idnsgd  13007  1nsgtrivd  13010