Theorem eqgid 17847

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 17842	. . . 4 ⊢ Rel ∼
3		relelec 7937	. . . 4 ⊢ (Rel ∼ → (𝑥 ∈ [ 0 ] ∼ ↔ 0 ∼ 𝑥))
4	2, 3	ax-mp 5	. . 3 ⊢ (𝑥 ∈ [ 0 ] ∼ ↔ 0 ∼ 𝑥)
5		subgrcl 17800	. . . . . . . . . 10 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
6	5	adantr 466	. . . . . . . . 9 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝐺 ∈ Grp)
7		eqgid.3	. . . . . . . . . 10 ⊢ 0 = (0g‘𝐺)
8		eqid 2771	. . . . . . . . . 10 ⊢ (invg‘𝐺) = (invg‘𝐺)
9	7, 8	grpinvid 17677	. . . . . . . . 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 6806	. . . . . . 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 2771	. . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺)
14	12, 13, 7	grplid 17653	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ( 0 (+g‘𝐺)𝑥) = 𝑥)
15	5, 14	sylan 569	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → ( 0 (+g‘𝐺)𝑥) = 𝑥)
16	11, 15	eqtrd 2805	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) = 𝑥)
17	16	eleq1d 2835	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → ((((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌 ↔ 𝑥 ∈ 𝑌))
18	17	pm5.32da 568	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ((𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌) ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑌)))
19	12	subgss 17796	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
20	12, 7	grpidcl 17651	. . . . . 6 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋)
21	5, 20	syl 17	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑋)
22	12, 8, 13, 1	eqgval 17844	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → ( 0 ∼ 𝑥 ↔ ( 0 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌)))
23		3anass 1080	. . . . . . 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	syl6bb 276	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → ( 0 ∼ 𝑥 ↔ ( 0 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌))))
25	24	baibd 529	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) ∧ 0 ∈ 𝑋) → ( 0 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌)))
26	5, 19, 21, 25	syl21anc 1475	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ( 0 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌)))
27	19	sseld 3751	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝑌 → 𝑥 ∈ 𝑋))
28	27	pm4.71rd 552	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝑌 ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑌)))
29	18, 26, 28	3bitr4d 300	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ( 0 ∼ 𝑥 ↔ 𝑥 ∈ 𝑌))
30	4, 29	syl5bb 272	. 2 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ [ 0 ] ∼ ↔ 𝑥 ∈ 𝑌))
31	30	eqrdv 2769	1 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → [ 0 ] ∼ = 𝑌)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   ⊆ wss 3723   class class class wbr 4786  Rel wrel 5254  ‘cfv 6029  (class class class)co 6791  [cec 7892  Basecbs 16057  +_gcplusg 16142  0_gc0g 16301  Grpcgrp 17623  inv_gcminusg 17624  SubGrpcsubg 17789   ~_QG cqg 17791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-1st 7313  df-2nd 7314  df-ec 7896  df-0g 16303  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-minusg 17627  df-subg 17792  df-eqg 17794

This theorem is referenced by:  cldsubg  22127  qustgphaus  22139