Theorem subg0 13703

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 2229	. . . . . 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 13702	. . . . 5 |- ( S e. (SubGrp ` G ) -> G e. Grp )
7	2, 4, 5, 6	ressplusgd 13148	. . . 4 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` H ) )
8	7	oveqd 6011	. . 3 |- ( S e. (SubGrp ` G ) -> ( ( 0g ` H ) ( +g ` G ) ( 0g ` H ) ) = ( ( 0g ` H ) ( +g ` H ) ( 0g ` H ) ) )
9	1	subggrp 13700	. . . 4 |- ( S e. (SubGrp ` G ) -> H e. Grp )
10		eqid 2229	. . . . . 6 |- ( Base ` H ) = ( Base ` H )
11		eqid 2229	. . . . . 6 |- ( 0g ` H ) = ( 0g ` H )
12	10, 11	grpidcl 13548	. . . . 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 2229	. . . . 5 |- ( +g ` H ) = ( +g ` H )
15	10, 14, 11	grplid 13550	. . . 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 2262	. 2 |- ( S e. (SubGrp ` G ) -> ( ( 0g ` H ) ( +g ` G ) ( 0g ` H ) ) = ( 0g ` H ) )
18		eqid 2229	. . . . 5 |- ( Base ` G ) = ( Base ` G )
19	18	subgss 13697	. . . 4 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
20	1	subgbas 13701	. . . . 5 |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
21	13, 20	eleqtrrd 2309	. . . 4 |- ( S e. (SubGrp ` G ) -> ( 0g ` H ) e. S )
22	19, 21	sseldd 3225	. . 3 |- ( S e. (SubGrp ` G ) -> ( 0g ` H ) e. ( Base ` G ) )
23		subg0.i	. . . 4 |- .0. = ( 0g ` G )
24	18, 3, 23	grpid 13558	. . 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 1395   e. wcel 2200   ` cfv 5314  (class class class)co 5994   Basecbs 13018   ↾_s cress 13019   +g cplusg 13096   0gc0g 13275   Grpcgrp 13519  SubGrpcsubg 13690

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-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-addass 8089  ax-i2m1 8092  ax-0lt1 8093  ax-0id 8095  ax-rnegex 8096  ax-pre-ltirr 8099  ax-pre-ltadd 8103

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 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-pnf 8171  df-mnf 8172  df-ltxr 8174  df-inn 9099  df-2 9157  df-ndx 13021  df-slot 13022  df-base 13024  df-sets 13025  df-iress 13026  df-plusg 13109  df-0g 13277  df-mgm 13375  df-sgrp 13421  df-mnd 13436  df-grp 13522  df-subg 13693

This theorem is referenced by:  subginv  13704  subg0cl  13705  subgmulg  13711  subrng0  14156  subrg0  14177  mpl0fi  14651