Theorem subg0 13889

Description: A subgroup of a group must have the same identity as the group. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
subg0.h	|- H = ( G ↾s S )
subg0.i	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
subg0	|- ( S e. (SubGrp ` G ) -> .0. = ( 0g ` H ) )

Proof of Theorem subg0

Step	Hyp	Ref	Expression
1		subg0.h	. . . . . 6 |- H = ( G ↾s S )
2	1	a1i 9	. . . . 5 |- ( S e. (SubGrp ` G ) -> H = ( G ↾s S ) )
3		eqid 2232	. . . . . 6 |- ( +g ` G ) = ( +g ` G )
4	3	a1i 9	. . . . 5 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` G ) )
5		id 19	. . . . 5 |- ( S e. (SubGrp ` G ) -> S e. (SubGrp ` G ) )
6		subgrcl 13888	. . . . 5 |- ( S e. (SubGrp ` G ) -> G e. Grp )
7	2, 4, 5, 6	ressplusgd 13334	. . . 4 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` H ) )
8	7	oveqd 6066	. . 3 |- ( S e. (SubGrp ` G ) -> ( ( 0g ` H ) ( +g ` G ) ( 0g ` H ) ) = ( ( 0g ` H ) ( +g ` H ) ( 0g ` H ) ) )
9	1	subggrp 13886	. . . 4 |- ( S e. (SubGrp ` G ) -> H e. Grp )
10		eqid 2232	. . . . . 6 |- ( Base ` H ) = ( Base ` H )
11		eqid 2232	. . . . . 6 |- ( 0g ` H ) = ( 0g ` H )
12	10, 11	grpidcl 13734	. . . . 5 |- ( H e. Grp -> ( 0g ` H ) e. ( Base ` H ) )
13	9, 12	syl 14	. . . 4 |- ( S e. (SubGrp ` G ) -> ( 0g ` H ) e. ( Base ` H ) )
14		eqid 2232	. . . . 5 |- ( +g ` H ) = ( +g ` H )
15	10, 14, 11	grplid 13736	. . . 4 |- ( ( H e. Grp /\ ( 0g ` H ) e. ( Base ` H ) ) -> ( ( 0g ` H ) ( +g ` H ) ( 0g ` H ) ) = ( 0g ` H ) )
16	9, 13, 15	syl2anc 411	. . 3 |- ( S e. (SubGrp ` G ) -> ( ( 0g ` H ) ( +g ` H ) ( 0g ` H ) ) = ( 0g ` H ) )
17	8, 16	eqtrd 2265	. 2 |- ( S e. (SubGrp ` G ) -> ( ( 0g ` H ) ( +g ` G ) ( 0g ` H ) ) = ( 0g ` H ) )
18		eqid 2232	. . . . 5 |- ( Base ` G ) = ( Base ` G )
19	18	subgss 13883	. . . 4 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
20	1	subgbas 13887	. . . . 5 |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
21	13, 20	eleqtrrd 2312	. . . 4 |- ( S e. (SubGrp ` G ) -> ( 0g ` H ) e. S )
22	19, 21	sseldd 3238	. . 3 |- ( S e. (SubGrp ` G ) -> ( 0g ` H ) e. ( Base ` G ) )
23		subg0.i	. . . 4 |- .0. = ( 0g ` G )
24	18, 3, 23	grpid 13744	. . 3 |- ( ( G e. Grp /\ ( 0g ` H ) e. ( Base ` G ) ) -> ( ( ( 0g ` H ) ( +g ` G ) ( 0g ` H ) ) = ( 0g ` H ) <-> .0. = ( 0g ` H ) ) )
25	6, 22, 24	syl2anc 411	. 2 |- ( S e. (SubGrp ` G ) -> ( ( ( 0g ` H ) ( +g ` G ) ( 0g ` H ) ) = ( 0g ` H ) <-> .0. = ( 0g ` H ) ) )
26	17, 25	mpbid 147	1 |- ( S e. (SubGrp ` G ) -> .0. = ( 0g ` H ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2203   ` cfv 5351  (class class class)co 6049   Basecbs 13204   ↾_s cress 13205   +g cplusg 13282   0gc0g 13461   Grpcgrp 13705  SubGrpcsubg 13876

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-pre-ltirr 8238  ax-pre-ltadd 8242

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-pnf 8309  df-mnf 8310  df-ltxr 8312  df-inn 9237  df-2 9295  df-ndx 13207  df-slot 13208  df-base 13210  df-sets 13211  df-iress 13212  df-plusg 13295  df-0g 13463  df-mgm 13561  df-sgrp 13607  df-mnd 13622  df-grp 13708  df-subg 13879

This theorem is referenced by:  subginv  13890  subg0cl  13891  subgmulg  13897  subrng0  14344  subrg0  14365  mpl0fi  14849