Theorem eqger 13430

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 13385	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2		eqger.r	. . . 4 ⊢ ∼ = (𝐺 ~QG 𝑌)
3	2	releqgg 13426	. . 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 13380	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
7		eqid 2196	. . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺)
8		eqid 2196	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
9	5, 7, 8, 2	eqgval 13429	. . . . . 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 1012	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝑦 ∈ 𝑋)
13	11	simp1d 1011	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝑥 ∈ 𝑋)
14	1	adantr 276	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝐺 ∈ Grp)
15	5, 7	grpinvcl 13250	. . . . . . 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 13280	. . . . . 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 1249	. . . . 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 13269	. . . . . . 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 5941	. . . . 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 2229	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → ((invg‘𝐺)‘(((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)) = (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑥))
23	11	simp3d 1013	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌)
24	7	subginvcl 13389	. . . . 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 2274	. . 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 13429	. . . 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 1182	. 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 13429	. . . . . . 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 1012	. . 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 13250	. . . . . . . 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 13216	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑋)
43	5, 8	grpass 13211	. . . . . 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 1251	. . . . 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 2196	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
46	5, 8, 45, 7	grprinv 13253	. . . . . . . . 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 5940	. . . . . . 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 13211	. . . . . . . 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 1251	. . . . . . 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 13233	. . . . . . . 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 2237	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) = 𝑧)
54	53	oveq2d 5941	. . . . 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 2229	. . . 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 1013	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌)
59	8	subgcl 13390	. . . . 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 1249	. . . 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 2274	. . 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 13429	. . . 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 1182	. 2 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑥 ∼ 𝑧)
66	5, 8, 45, 7	grplinv 13252	. . . . . . 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 13388	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑌)
69	68	adantr 276	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (0g‘𝐺) ∈ 𝑌)
70	67, 69	eqeltrd 2273	. . . . 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 13429	. . . . 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 982	. . . . 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 6627	1 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2167   ⊆ wss 3157   class class class wbr 4034  Rel wrel 4669  ‘cfv 5259  (class class class)co 5925   Er wer 6598  Basecbs 12703  +_gcplusg 12780  0_gc0g 12958  Grpcgrp 13202  inv_gcminusg 13203  SubGrpcsubg 13373   ~_QG cqg 13375

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-er 6601  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206  df-subg 13376  df-eqg 13378

This theorem is referenced by:  eqgen  13433  eqg0el  13435  qusgrp  13438  qusadd  13440  qusecsub  13537  2idlcpblrng  14155  qus2idrng  14157  qus1  14158  qusrhm  14160  qusmul2  14161  qusmulrng  14164  zndvds  14281