Theorem 0nsg 13665

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 13650	. 2 |- ( G e. Grp -> { .0. } e. (SubGrp ` G ) )
3		elsni 3661	. . . . . . . . 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 5983	. . . . . . 7 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` G ) .0. ) )
6		eqid 2207	. . . . . . . . 9 |- ( Base ` G ) = ( Base ` G )
7		eqid 2207	. . . . . . . . 9 |- ( +g ` G ) = ( +g ` G )
8	6, 7, 1	grprid 13479	. . . . . . . 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 2240	. . . . . 6 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) = x )
11	10	oveq1d 5982	. . . . 5 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) = ( x ( -g ` G ) x ) )
12		eqid 2207	. . . . . . 7 |- ( -g ` G ) = ( -g ` G )
13	6, 1, 12	grpsubid 13531	. . . . . 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 2240	. . . 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 13476	. . . . . . . . 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 2284	. . . . . . 7 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> y e. ( Base ` G ) )
21	6, 7, 16, 17, 20	grpcld 13461	. . . . . 6 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) e. ( Base ` G ) )
22	6, 12	grpsubcl 13527	. . . . . 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 1250	. . . . 5 |- ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. ( Base ` G ) )
24		elsng 3658	. . . . 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 2590	. 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 13658	. 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 1373   e. wcel 2178   A.wral 2486   {csn 3643   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   0gc0g 13203   Grpcgrp 13447   -gcsg 13449  SubGrpcsubg 13618  NrmSGrpcnsg 13619

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955  df-plusg 13037  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-submnd 13407  df-grp 13450  df-minusg 13451  df-sbg 13452  df-subg 13621  df-nsg 13622

This theorem is referenced by:  0idnsgd  13667  1nsgtrivd  13670  ghmker  13721