Theorem eqger 14010

Description: The subgroup coset equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 13-Jan-2015.)

Hypotheses

Ref	Expression
eqger.x	⊢ 𝑋 = (Base‘𝐺)
eqger.r	⊢ ∼ = (𝐺 ~QG 𝑌)

Assertion

Ref	Expression
eqger	⊢ (𝑌 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)

Proof of Theorem eqger

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subgrcl 13965	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2		eqger.r	. . . 4 ⊢ ∼ = (𝐺 ~QG 𝑌)
3	2	releqgg 14006	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ (SubGrp‘𝐺)) → Rel ∼ )
4	1, 3	mpancom 426	. 2 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → Rel ∼ )
5		eqger.x	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐺)
6	5	subgss 13960	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
7		eqid 2238	. . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺)
8		eqid 2238	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
9	5, 7, 8, 2	eqgval 14009	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌)))
10	1, 6, 9	syl2anc 415	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌)))
11	10	biimpa 296	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌))
12	11	simp2d 1041	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝑦 ∈ 𝑋)
13	11	simp1d 1040	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝑥 ∈ 𝑋)
14	1	adantr 276	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝐺 ∈ Grp)
15	5, 7	grpinvcl 13836	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((invg‘𝐺)‘𝑥) ∈ 𝑋)
16	14, 13, 15	syl2anc 415	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → ((invg‘𝐺)‘𝑥) ∈ 𝑋)
17	5, 8, 7	grpinvadd 13866	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ ((invg‘𝐺)‘𝑥) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((invg‘𝐺)‘(((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)) = (((invg‘𝐺)‘𝑦)(+g‘𝐺)((invg‘𝐺)‘((invg‘𝐺)‘𝑥))))
18	14, 16, 12, 17	syl3anc 1278	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → ((invg‘𝐺)‘(((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)) = (((invg‘𝐺)‘𝑦)(+g‘𝐺)((invg‘𝐺)‘((invg‘𝐺)‘𝑥))))
19	5, 7	grpinvinv 13855	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑥)) = 𝑥)
20	14, 13, 19	syl2anc 415	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑥)) = 𝑥)
21	20	oveq2d 6095	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)((invg‘𝐺)‘((invg‘𝐺)‘𝑥))) = (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑥))
22	18, 21	eqtrd 2271	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → ((invg‘𝐺)‘(((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)) = (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑥))
23	11	simp3d 1042	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌)
24	7	subginvcl 13969	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌) → ((invg‘𝐺)‘(((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)) ∈ 𝑌)
25	23, 24	syldan 282	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → ((invg‘𝐺)‘(((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)) ∈ 𝑌)
26	22, 25	eqeltrrd 2316	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑥) ∈ 𝑌)
27	6	adantr 276	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝑌 ⊆ 𝑋)
28	5, 7, 8, 2	eqgval 14009	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑥) ∈ 𝑌)))
29	14, 27, 28	syl2anc 415	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑥) ∈ 𝑌)))
30	12, 13, 26, 29	mpbir3and 1211	. 2 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝑦 ∼ 𝑥)
31	13	adantrr 483	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑥 ∈ 𝑋)
32	5, 7, 8, 2	eqgval 14009	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑦 ∼ 𝑧 ↔ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌)))
33	1, 6, 32	syl2anc 415	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑦 ∼ 𝑧 ↔ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌)))
34	33	biimpa 296	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∼ 𝑧) → (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌))
35	34	adantrl 482	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌))
36	35	simp2d 1041	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑧 ∈ 𝑋)
37	1	adantr 276	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝐺 ∈ Grp)
38	37, 31, 15	syl2anc 415	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((invg‘𝐺)‘𝑥) ∈ 𝑋)
39	12	adantrr 483	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑦 ∈ 𝑋)
40	5, 7	grpinvcl 13836	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → ((invg‘𝐺)‘𝑦) ∈ 𝑋)
41	37, 39, 40	syl2anc 415	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((invg‘𝐺)‘𝑦) ∈ 𝑋)
42	5, 8, 37, 41, 36	grpcld 13802	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑋)
43	5, 8	grpass 13797	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (((invg‘𝐺)‘𝑥) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑋)) → ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) = (((invg‘𝐺)‘𝑥)(+g‘𝐺)(𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))))
44	37, 38, 39, 42, 43	syl13anc 1280	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) = (((invg‘𝐺)‘𝑥)(+g‘𝐺)(𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))))
45		eqid 2238	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
46	5, 8, 45, 7	grprinv 13839	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦)) = (0g‘𝐺))
47	37, 39, 46	syl2anc 415	. . . . . . . 8 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦)) = (0g‘𝐺))
48	47	oveq1d 6094	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦))(+g‘𝐺)𝑧) = ((0g‘𝐺)(+g‘𝐺)𝑧))
49	5, 8	grpass 13797	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑦 ∈ 𝑋 ∧ ((invg‘𝐺)‘𝑦) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦))(+g‘𝐺)𝑧) = (𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)))
50	37, 39, 41, 36, 49	syl13anc 1280	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦))(+g‘𝐺)𝑧) = (𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)))
51	5, 8, 45	grplid 13819	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((0g‘𝐺)(+g‘𝐺)𝑧) = 𝑧)
52	37, 36, 51	syl2anc 415	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((0g‘𝐺)(+g‘𝐺)𝑧) = 𝑧)
53	48, 50, 52	3eqtr3d 2279	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) = 𝑧)
54	53	oveq2d 6095	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)(𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) = (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑧))
55	44, 54	eqtrd 2271	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) = (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑧))
56		simpl 109	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑌 ∈ (SubGrp‘𝐺))
57	23	adantrr 483	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌)
58	35	simp3d 1042	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌)
59	8	subgcl 13970	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑌)
60	56, 57, 58, 59	syl3anc 1278	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑌)
61	55, 60	eqeltrrd 2316	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑧) ∈ 𝑌)
62	6	adantr 276	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑌 ⊆ 𝑋)
63	5, 7, 8, 2	eqgval 14009	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∼ 𝑧 ↔ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑧) ∈ 𝑌)))
64	37, 62, 63	syl2anc 415	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (𝑥 ∼ 𝑧 ↔ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑧) ∈ 𝑌)))
65	31, 36, 61, 64	mpbir3and 1211	. 2 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑥 ∼ 𝑧)
66	5, 8, 45, 7	grplinv 13838	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) = (0g‘𝐺))
67	1, 66	sylan 283	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) = (0g‘𝐺))
68	45	subg0cl 13968	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑌)
69	68	adantr 276	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (0g‘𝐺) ∈ 𝑌)
70	67, 69	eqeltrd 2315	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌)
71	70	ex 115	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝑋 → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌))
72	71	pm4.71rd 398	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝑋 ↔ ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌 ∧ 𝑥 ∈ 𝑋)))
73	5, 7, 8, 2	eqgval 14009	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌)))
74	1, 6, 73	syl2anc 415	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌)))
75		df-3an 1011	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌))
76		anidm 400	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) ↔ 𝑥 ∈ 𝑋)
77	76	anbi2ci 463	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌) ↔ ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌 ∧ 𝑥 ∈ 𝑋))
78	75, 77	bitri 184	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌) ↔ ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌 ∧ 𝑥 ∈ 𝑋))
79	74, 78	bitrdi 196	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∼ 𝑥 ↔ ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌 ∧ 𝑥 ∈ 𝑋)))
80	72, 79	bitr4d 191	. 2 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∼ 𝑥))
81	4, 30, 65, 80	iserd 6827	1 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209   ⊆ wss 3220   class class class wbr 4128  Rel wrel 4777  ‘cfv 5375  (class class class)co 6079   Er wer 6798  Basecbs 13335  +_gcplusg 13414  0_gc0g 13593  Grpcgrp 13788  inv_gcminusg 13789  SubGrpcsubg 13953   ~_QG cqg 13955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-er 6801  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-iress 13343  df-plusg 13427  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-minusg 13792  df-subg 13956  df-eqg 13958

This theorem is referenced by:  eqgen  14013  eqg0el  14015  qusgrp  14018  qusadd  14020  qusecsub  14118  2idlcpblrng  14843  qus2idrng  14845  qus1  14846  qusrhm  14848  qusmul2  14849  qusmulrng  14852  zndvds  14967