Theorem eqgid 16918

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 16913	. . . 4 |- Rel .~
3		relelec 7430	. . . 4 |- ( Rel .~ -> ( x e. [ .0. ] .~ <-> .0. .~ x ) )
4	2, 3	ax-mp 5	. . 3 |- ( x e. [ .0. ] .~ <-> .0. .~ x )
5		subgrcl 16871	. . . . . . . . . 10 |- ( Y e. (SubGrp ` G ) -> G e. Grp )
6	5	adantr 471	. . . . . . . . 9 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> G e. Grp )
7		eqgid.3	. . . . . . . . . 10 |- .0. = ( 0g ` G )
8		eqid 2462	. . . . . . . . . 10 |- ( invg ` G ) = ( invg ` G )
9	7, 8	grpinvid 16766	. . . . . . . . 9 |- ( G e. Grp -> ( ( invg ` G ) ` .0. ) = .0. )
10	6, 9	syl 17	. . . . . . . 8 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( invg ` G ) ` .0. ) = .0. )
11	10	oveq1d 6330	. . . . . . 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 2462	. . . . . . . . 9 |- ( +g ` G ) = ( +g ` G )
14	12, 13, 7	grplid 16745	. . . . . . . 8 |- ( ( G e. Grp /\ x e. X ) -> ( .0. ( +g ` G ) x ) = x )
15	5, 14	sylan 478	. . . . . . 7 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( .0. ( +g ` G ) x ) = x )
16	11, 15	eqtrd 2496	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) = x )
17	16	eleq1d 2524	. . . . 5 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y <-> x e. Y ) )
18	17	pm5.32da 651	. . . 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 16867	. . . . 5 |- ( Y e. (SubGrp ` G ) -> Y C_ X )
20	12, 7	grpidcl 16743	. . . . . 6 |- ( G e. Grp -> .0. e. X )
21	5, 20	syl 17	. . . . 5 |- ( Y e. (SubGrp ` G ) -> .0. e. X )
22	12, 8, 13, 1	eqgval 16915	. . . . . . 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 995	. . . . . . 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 269	. . . . . 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 925	. . . . 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 1275	. . . 4 |- ( Y e. (SubGrp ` G ) -> ( .0. .~ x <-> ( x e. X /\ ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y ) ) )
27	19	sseld 3443	. . . . 5 |- ( Y e. (SubGrp ` G ) -> ( x e. Y -> x e. X ) )
28	27	pm4.71rd 645	. . . 4 |- ( Y e. (SubGrp ` G ) -> ( x e. Y <-> ( x e. X /\ x e. Y ) ) )
29	18, 26, 28	3bitr4d 293	. . 3 |- ( Y e. (SubGrp ` G ) -> ( .0. .~ x <-> x e. Y ) )
30	4, 29	syl5bb 265	. 2 |- ( Y e. (SubGrp ` G ) -> ( x e. [ .0. ] .~ <-> x e. Y ) )
31	30	eqrdv 2460	1 |- ( Y e. (SubGrp ` G ) -> [ .0. ] .~ = Y )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   C_ wss 3416   class class class wbr 4416   Rel wrel 4858   ` cfv 5601  (class class class)co 6315   [cec 7387   Basecbs 15170   +g cplusg 15239   0gc0g 15387   Grpcgrp 16718   invgcminusg 16719  SubGrpcsubg 16860   ~_QG cqg 16862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-1st 6820  df-2nd 6821  df-ec 7391  df-0g 15389  df-mgm 16537  df-sgrp 16576  df-mnd 16586  df-grp 16722  df-minusg 16723  df-subg 16863  df-eqg 16865

This theorem is referenced by:  cldsubg  21174  qustgphaus  21186