Theorem subg0 13828

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 2231	. . . . . 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 13827	. . . . 5 |- ( S e. (SubGrp ` G ) -> G e. Grp )
7	2, 4, 5, 6	ressplusgd 13273	. . . 4 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` H ) )
8	7	oveqd 6045	. . 3 |- ( S e. (SubGrp ` G ) -> ( ( 0g ` H ) ( +g ` G ) ( 0g ` H ) ) = ( ( 0g ` H ) ( +g ` H ) ( 0g ` H ) ) )
9	1	subggrp 13825	. . . 4 |- ( S e. (SubGrp ` G ) -> H e. Grp )
10		eqid 2231	. . . . . 6 |- ( Base ` H ) = ( Base ` H )
11		eqid 2231	. . . . . 6 |- ( 0g ` H ) = ( 0g ` H )
12	10, 11	grpidcl 13673	. . . . 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 2231	. . . . 5 |- ( +g ` H ) = ( +g ` H )
15	10, 14, 11	grplid 13675	. . . 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 2264	. 2 |- ( S e. (SubGrp ` G ) -> ( ( 0g ` H ) ( +g ` G ) ( 0g ` H ) ) = ( 0g ` H ) )
18		eqid 2231	. . . . 5 |- ( Base ` G ) = ( Base ` G )
19	18	subgss 13822	. . . 4 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
20	1	subgbas 13826	. . . . 5 |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
21	13, 20	eleqtrrd 2311	. . . 4 |- ( S e. (SubGrp ` G ) -> ( 0g ` H ) e. S )
22	19, 21	sseldd 3229	. . 3 |- ( S e. (SubGrp ` G ) -> ( 0g ` H ) e. ( Base ` G ) )
23		subg0.i	. . . 4 |- .0. = ( 0g ` G )
24	18, 3, 23	grpid 13683	. . 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 2202   ` cfv 5333  (class class class)co 6028   Basecbs 13143   ↾_s cress 13144   +g cplusg 13221   0gc0g 13400   Grpcgrp 13644  SubGrpcsubg 13815

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-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-ltadd 8191

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-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-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8259  df-mnf 8260  df-ltxr 8262  df-inn 9187  df-2 9245  df-ndx 13146  df-slot 13147  df-base 13149  df-sets 13150  df-iress 13151  df-plusg 13234  df-0g 13402  df-mgm 13500  df-sgrp 13546  df-mnd 13561  df-grp 13647  df-subg 13818

This theorem is referenced by:  subginv  13829  subg0cl  13830  subgmulg  13836  subrng0  14283  subrg0  14304  mpl0fi  14783