Theorem eqgid 15731

Description: The left coset containing the identity is the original subgroup. (Contributed by Mario Carneiro, 20-Sep-2015.)

Hypotheses

Ref	Expression
eqger.x	|- X = ( Base ` G )
eqger.r	|- .~ = ( G ~QG Y )
eqgid.3	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
eqgid	|- ( Y e. (SubGrp ` G ) -> [ .0. ] .~ = Y )

Proof of Theorem eqgid

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqger.r	. . . . 5 |- .~ = ( G ~QG Y )
2	1	releqg 15726	. . . 4 |- Rel .~
3		relelec 7139	. . . 4 |- ( Rel .~ -> ( x e. [ .0. ] .~ <-> .0. .~ x ) )
4	2, 3	ax-mp 5	. . 3 |- ( x e. [ .0. ] .~ <-> .0. .~ x )
5		subgrcl 15684	. . . . . . . . . 10 |- ( Y e. (SubGrp ` G ) -> G e. Grp )
6	5	adantr 465	. . . . . . . . 9 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> G e. Grp )
7		eqgid.3	. . . . . . . . . 10 |- .0. = ( 0g ` G )
8		eqid 2441	. . . . . . . . . 10 |- ( invg ` G ) = ( invg ` G )
9	7, 8	grpinvid 15587	. . . . . . . . 9 |- ( G e. Grp -> ( ( invg ` G ) ` .0. ) = .0. )
10	6, 9	syl 16	. . . . . . . 8 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( invg ` G ) ` .0. ) = .0. )
11	10	oveq1d 6104	. . . . . . 7 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) = ( .0. ( +g ` G ) x ) )
12		eqger.x	. . . . . . . . 9 |- X = ( Base ` G )
13		eqid 2441	. . . . . . . . 9 |- ( +g ` G ) = ( +g ` G )
14	12, 13, 7	grplid 15566	. . . . . . . 8 |- ( ( G e. Grp /\ x e. X ) -> ( .0. ( +g ` G ) x ) = x )
15	5, 14	sylan 471	. . . . . . 7 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( .0. ( +g ` G ) x ) = x )
16	11, 15	eqtrd 2473	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) = x )
17	16	eleq1d 2507	. . . . 5 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y <-> x e. Y ) )
18	17	pm5.32da 641	. . . 4 |- ( Y e. (SubGrp ` G ) -> ( ( x e. X /\ ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y ) <-> ( x e. X /\ x e. Y ) ) )
19	12	subgss 15680	. . . . 5 |- ( Y e. (SubGrp ` G ) -> Y C_ X )
20	12, 7	grpidcl 15564	. . . . . 6 |- ( G e. Grp -> .0. e. X )
21	5, 20	syl 16	. . . . 5 |- ( Y e. (SubGrp ` G ) -> .0. e. X )
22	12, 8, 13, 1	eqgval 15728	. . . . . . 7 |- ( ( G e. Grp /\ Y C_ X ) -> ( .0. .~ x <-> ( .0. e. X /\ x e. X /\ ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y ) ) )
23		3anass 969	. . . . . . 7 |- ( ( .0. e. X /\ x e. X /\ ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y ) <-> ( .0. e. X /\ ( x e. X /\ ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y ) ) )
24	22, 23	syl6bb 261	. . . . . 6 |- ( ( G e. Grp /\ Y C_ X ) -> ( .0. .~ x <-> ( .0. e. X /\ ( x e. X /\ ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y ) ) ) )
25	24	baibd 900	. . . . 5 |- ( ( ( G e. Grp /\ Y C_ X ) /\ .0. e. X ) -> ( .0. .~ x <-> ( x e. X /\ ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y ) ) )
26	5, 19, 21, 25	syl21anc 1217	. . . 4 |- ( Y e. (SubGrp ` G ) -> ( .0. .~ x <-> ( x e. X /\ ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y ) ) )
27	19	sseld 3353	. . . . 5 |- ( Y e. (SubGrp ` G ) -> ( x e. Y -> x e. X ) )
28	27	pm4.71rd 635	. . . 4 |- ( Y e. (SubGrp ` G ) -> ( x e. Y <-> ( x e. X /\ x e. Y ) ) )
29	18, 26, 28	3bitr4d 285	. . 3 |- ( Y e. (SubGrp ` G ) -> ( .0. .~ x <-> x e. Y ) )
30	4, 29	syl5bb 257	. 2 |- ( Y e. (SubGrp ` G ) -> ( x e. [ .0. ] .~ <-> x e. Y ) )
31	30	eqrdv 2439	1 |- ( Y e. (SubGrp ` G ) -> [ .0. ] .~ = Y )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   C_ wss 3326   class class class wbr 4290   Rel wrel 4843   ` cfv 5416  (class class class)co 6089   [cec 7097   Basecbs 14172   +g cplusg 14236   0gc0g 14376   Grpcgrp 15408   invgcminusg 15409  SubGrpcsubg 15673   ~_QG cqg 15675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-ec 7101  df-0g 14378  df-mnd 15413  df-grp 15543  df-minusg 15544  df-subg 15676  df-eqg 15678

This theorem is referenced by:  cldsubg  19679  divstgphaus  19691