Theorem eqgid 18723

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 ∈ (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 18718	. . . 4 ⊢ Rel ∼
3		relelec 8501	. . . 4 ⊢ (Rel ∼ → (𝑥 ∈ [ 0 ] ∼ ↔ 0 ∼ 𝑥))
4	2, 3	ax-mp 5	. . 3 ⊢ (𝑥 ∈ [ 0 ] ∼ ↔ 0 ∼ 𝑥)
5		subgrcl 18675	. . . . . . . . . 10 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
6	5	adantr 480	. . . . . . . . 9 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝐺 ∈ Grp)
7		eqgid.3	. . . . . . . . . 10 ⊢ 0 = (0g‘𝐺)
8		eqid 2738	. . . . . . . . . 10 ⊢ (invg‘𝐺) = (invg‘𝐺)
9	7, 8	grpinvid 18551	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘ 0 ) = 0 )
10	6, 9	syl 17	. . . . . . . 8 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → ((invg‘𝐺)‘ 0 ) = 0 )
11	10	oveq1d 7270	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) = ( 0 (+g‘𝐺)𝑥))
12		eqger.x	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝐺)
13		eqid 2738	. . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺)
14	12, 13, 7	grplid 18524	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ( 0 (+g‘𝐺)𝑥) = 𝑥)
15	5, 14	sylan 579	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → ( 0 (+g‘𝐺)𝑥) = 𝑥)
16	11, 15	eqtrd 2778	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) = 𝑥)
17	16	eleq1d 2823	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → ((((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌 ↔ 𝑥 ∈ 𝑌))
18	17	pm5.32da 578	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ((𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌) ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑌)))
19	12	subgss 18671	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
20	12, 7	grpidcl 18522	. . . . . 6 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋)
21	5, 20	syl 17	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑋)
22	12, 8, 13, 1	eqgval 18720	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → ( 0 ∼ 𝑥 ↔ ( 0 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌)))
23		3anass 1093	. . . . . . 7 ⊢ (( 0 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌) ↔ ( 0 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌)))
24	22, 23	bitrdi 286	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → ( 0 ∼ 𝑥 ↔ ( 0 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌))))
25	24	baibd 539	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) ∧ 0 ∈ 𝑋) → ( 0 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌)))
26	5, 19, 21, 25	syl21anc 834	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ( 0 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌)))
27	19	sseld 3916	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝑌 → 𝑥 ∈ 𝑋))
28	27	pm4.71rd 562	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝑌 ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑌)))
29	18, 26, 28	3bitr4d 310	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ( 0 ∼ 𝑥 ↔ 𝑥 ∈ 𝑌))
30	4, 29	syl5bb 282	. 2 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ [ 0 ] ∼ ↔ 𝑥 ∈ 𝑌))
31	30	eqrdv 2736	1 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → [ 0 ] ∼ = 𝑌)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883   class class class wbr 5070  Rel wrel 5585  ‘cfv 6418  (class class class)co 7255  [cec 8454  Basecbs 16840  +_gcplusg 16888  0_gc0g 17067  Grpcgrp 18492  inv_gcminusg 18493  SubGrpcsubg 18664   ~_QG cqg 18666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-ec 8458  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-minusg 18496  df-subg 18667  df-eqg 18669

This theorem is referenced by:  cldsubg  23170  qustgphaus  23182  eqg0el  31459  qsidomlem1  31530  qsidomlem2  31531