Theorem eqgid 16370

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 16365	. . . 4 |- Rel .~
3		relelec 7270	. . . 4 |- ( Rel .~ -> ( x e. [ .0. ] .~ <-> .0. .~ x ) )
4	2, 3	ax-mp 5	. . 3 |- ( x e. [ .0. ] .~ <-> .0. .~ x )
5		subgrcl 16323	. . . . . . . . . 10 |- ( Y e. (SubGrp ` G ) -> G e. Grp )
6	5	adantr 463	. . . . . . . . 9 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> G e. Grp )
7		eqgid.3	. . . . . . . . . 10 |- .0. = ( 0g ` G )
8		eqid 2382	. . . . . . . . . 10 |- ( invg ` G ) = ( invg ` G )
9	7, 8	grpinvid 16218	. . . . . . . . 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 6211	. . . . . . 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 2382	. . . . . . . . 9 |- ( +g ` G ) = ( +g ` G )
14	12, 13, 7	grplid 16197	. . . . . . . 8 |- ( ( G e. Grp /\ x e. X ) -> ( .0. ( +g ` G ) x ) = x )
15	5, 14	sylan 469	. . . . . . 7 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( .0. ( +g ` G ) x ) = x )
16	11, 15	eqtrd 2423	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) = x )
17	16	eleq1d 2451	. . . . 5 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y <-> x e. Y ) )
18	17	pm5.32da 639	. . . 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 16319	. . . . 5 |- ( Y e. (SubGrp ` G ) -> Y C_ X )
20	12, 7	grpidcl 16195	. . . . . 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 16367	. . . . . . 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 975	. . . . . . 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 907	. . . . 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 1225	. . . 4 |- ( Y e. (SubGrp ` G ) -> ( .0. .~ x <-> ( x e. X /\ ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y ) ) )
27	19	sseld 3416	. . . . 5 |- ( Y e. (SubGrp ` G ) -> ( x e. Y -> x e. X ) )
28	27	pm4.71rd 633	. . . 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 2379	1 |- ( Y e. (SubGrp ` G ) -> [ .0. ] .~ = Y )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 971   = wceq 1399   e. wcel 1826   C_ wss 3389   class class class wbr 4367   Rel wrel 4918   ` cfv 5496  (class class class)co 6196   [cec 7227   Basecbs 14634   +g cplusg 14702   0gc0g 14847   Grpcgrp 16170   invgcminusg 16171  SubGrpcsubg 16312   ~_QG cqg 16314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-ec 7231  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174  df-minusg 16175  df-subg 16315  df-eqg 16317

This theorem is referenced by:  cldsubg  20694  qustgphaus  20706