Theorem eqger 13782

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 13737	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2		eqger.r	. . . 4 ⊢ ∼ = (𝐺 ~QG 𝑌)
3	2	releqgg 13778	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ (SubGrp‘𝐺)) → Rel ∼ )
4	1, 3	mpancom 422	. 2 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → Rel ∼ )
5		eqger.x	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐺)
6	5	subgss 13732	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
7		eqid 2229	. . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺)
8		eqid 2229	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
9	5, 7, 8, 2	eqgval 13781	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌)))
10	1, 6, 9	syl2anc 411	. . . . 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 1034	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝑦 ∈ 𝑋)
13	11	simp1d 1033	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝑥 ∈ 𝑋)
14	1	adantr 276	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝐺 ∈ Grp)
15	5, 7	grpinvcl 13602	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((invg‘𝐺)‘𝑥) ∈ 𝑋)
16	14, 13, 15	syl2anc 411	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → ((invg‘𝐺)‘𝑥) ∈ 𝑋)
17	5, 8, 7	grpinvadd 13632	. . . . . 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 1271	. . . . 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 13621	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑥)) = 𝑥)
20	14, 13, 19	syl2anc 411	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑥)) = 𝑥)
21	20	oveq2d 6026	. . . . 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 2262	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → ((invg‘𝐺)‘(((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)) = (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑥))
23	11	simp3d 1035	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌)
24	7	subginvcl 13741	. . . . 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 2307	. . 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 13781	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑥) ∈ 𝑌)))
29	14, 27, 28	syl2anc 411	. . 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 1204	. 2 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝑦 ∼ 𝑥)
31	13	adantrr 479	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑥 ∈ 𝑋)
32	5, 7, 8, 2	eqgval 13781	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑦 ∼ 𝑧 ↔ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌)))
33	1, 6, 32	syl2anc 411	. . . . . 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 478	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌))
36	35	simp2d 1034	. . 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 411	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((invg‘𝐺)‘𝑥) ∈ 𝑋)
39	12	adantrr 479	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑦 ∈ 𝑋)
40	5, 7	grpinvcl 13602	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → ((invg‘𝐺)‘𝑦) ∈ 𝑋)
41	37, 39, 40	syl2anc 411	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((invg‘𝐺)‘𝑦) ∈ 𝑋)
42	5, 8, 37, 41, 36	grpcld 13568	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑋)
43	5, 8	grpass 13563	. . . . . 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 1273	. . . . 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 2229	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
46	5, 8, 45, 7	grprinv 13605	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦)) = (0g‘𝐺))
47	37, 39, 46	syl2anc 411	. . . . . . . 8 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦)) = (0g‘𝐺))
48	47	oveq1d 6025	. . . . . . 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 13563	. . . . . . . 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 1273	. . . . . . 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 13585	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((0g‘𝐺)(+g‘𝐺)𝑧) = 𝑧)
52	37, 36, 51	syl2anc 411	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((0g‘𝐺)(+g‘𝐺)𝑧) = 𝑧)
53	48, 50, 52	3eqtr3d 2270	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) = 𝑧)
54	53	oveq2d 6026	. . . . 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 2262	. . . 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 479	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌)
58	35	simp3d 1035	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌)
59	8	subgcl 13742	. . . . 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 1271	. . . 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 2307	. . 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 13781	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∼ 𝑧 ↔ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑧) ∈ 𝑌)))
64	37, 62, 63	syl2anc 411	. . 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 1204	. 2 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑥 ∼ 𝑧)
66	5, 8, 45, 7	grplinv 13604	. . . . . . 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 13740	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑌)
69	68	adantr 276	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (0g‘𝐺) ∈ 𝑌)
70	67, 69	eqeltrd 2306	. . . . 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 394	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝑋 ↔ ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌 ∧ 𝑥 ∈ 𝑋)))
73	5, 7, 8, 2	eqgval 13781	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌)))
74	1, 6, 73	syl2anc 411	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌)))
75		df-3an 1004	. . . . 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 396	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) ↔ 𝑥 ∈ 𝑋)
77	76	anbi2ci 459	. . . . 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 6719	1 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200   ⊆ wss 3197   class class class wbr 4083  Rel wrel 4725  ‘cfv 5321  (class class class)co 6010   Er wer 6690  Basecbs 13053  +_gcplusg 13131  0_gc0g 13310  Grpcgrp 13554  inv_gcminusg 13555  SubGrpcsubg 13725   ~_QG cqg 13727

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-pre-ltirr 8127  ax-pre-ltadd 8131

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-er 6693  df-pnf 8199  df-mnf 8200  df-ltxr 8202  df-inn 9127  df-2 9185  df-ndx 13056  df-slot 13057  df-base 13059  df-sets 13060  df-iress 13061  df-plusg 13144  df-0g 13312  df-mgm 13410  df-sgrp 13456  df-mnd 13471  df-grp 13557  df-minusg 13558  df-subg 13728  df-eqg 13730

This theorem is referenced by:  eqgen  13785  eqg0el  13787  qusgrp  13790  qusadd  13792  qusecsub  13889  2idlcpblrng  14508  qus2idrng  14510  qus1  14511  qusrhm  14513  qusmul2  14514  qusmulrng  14517  zndvds  14634