Theorem eqger 18548

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		eqger.r	. . . 4 ⊢ ∼ = (𝐺 ~QG 𝑌)
2	1	releqg 18545	. . 3 ⊢ Rel ∼
3	2	a1i 11	. 2 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → Rel ∼ )
4		subgrcl 18502	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5		eqger.x	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐺)
6	5	subgss 18498	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
7		eqid 2736	. . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺)
8		eqid 2736	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
9	5, 7, 8, 1	eqgval 18547	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌)))
10	4, 6, 9	syl2anc 587	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌)))
11	10	biimpa 480	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌))
12	11	simp2d 1145	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝑦 ∈ 𝑋)
13	11	simp1d 1144	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝑥 ∈ 𝑋)
14	4	adantr 484	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝐺 ∈ Grp)
15	5, 7	grpinvcl 18369	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((invg‘𝐺)‘𝑥) ∈ 𝑋)
16	14, 13, 15	syl2anc 587	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → ((invg‘𝐺)‘𝑥) ∈ 𝑋)
17	5, 8, 7	grpinvadd 18395	. . . . . 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 1373	. . . . 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 18384	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑥)) = 𝑥)
20	14, 13, 19	syl2anc 587	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑥)) = 𝑥)
21	20	oveq2d 7207	. . . . 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 2771	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → ((invg‘𝐺)‘(((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)) = (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑥))
23	11	simp3d 1146	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌)
24	7	subginvcl 18506	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌) → ((invg‘𝐺)‘(((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)) ∈ 𝑌)
25	23, 24	syldan 594	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → ((invg‘𝐺)‘(((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)) ∈ 𝑌)
26	22, 25	eqeltrrd 2832	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑥) ∈ 𝑌)
27	6	adantr 484	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝑌 ⊆ 𝑋)
28	5, 7, 8, 1	eqgval 18547	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑥) ∈ 𝑌)))
29	14, 27, 28	syl2anc 587	. . 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 1344	. 2 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝑦 ∼ 𝑥)
31	13	adantrr 717	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑥 ∈ 𝑋)
32	5, 7, 8, 1	eqgval 18547	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑦 ∼ 𝑧 ↔ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌)))
33	4, 6, 32	syl2anc 587	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑦 ∼ 𝑧 ↔ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌)))
34	33	biimpa 480	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∼ 𝑧) → (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌))
35	34	adantrl 716	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌))
36	35	simp2d 1145	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑧 ∈ 𝑋)
37	4	adantr 484	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝐺 ∈ Grp)
38	37, 31, 15	syl2anc 587	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((invg‘𝐺)‘𝑥) ∈ 𝑋)
39	12	adantrr 717	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑦 ∈ 𝑋)
40	5, 7	grpinvcl 18369	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → ((invg‘𝐺)‘𝑦) ∈ 𝑋)
41	37, 39, 40	syl2anc 587	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((invg‘𝐺)‘𝑦) ∈ 𝑋)
42	5, 8	grpcl 18327	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ ((invg‘𝐺)‘𝑦) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑋)
43	37, 41, 36, 42	syl3anc 1373	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑋)
44	5, 8	grpass 18328	. . . . . 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))))
45	37, 38, 39, 43, 44	syl13anc 1374	. . . . 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))))
46		eqid 2736	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
47	5, 8, 46, 7	grprinv 18371	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦)) = (0g‘𝐺))
48	37, 39, 47	syl2anc 587	. . . . . . . 8 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦)) = (0g‘𝐺))
49	48	oveq1d 7206	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦))(+g‘𝐺)𝑧) = ((0g‘𝐺)(+g‘𝐺)𝑧))
50	5, 8	grpass 18328	. . . . . . . 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)))
51	37, 39, 41, 36, 50	syl13anc 1374	. . . . . . 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)))
52	5, 8, 46	grplid 18351	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((0g‘𝐺)(+g‘𝐺)𝑧) = 𝑧)
53	37, 36, 52	syl2anc 587	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((0g‘𝐺)(+g‘𝐺)𝑧) = 𝑧)
54	49, 51, 53	3eqtr3d 2779	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) = 𝑧)
55	54	oveq2d 7207	. . . . 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))
56	45, 55	eqtrd 2771	. . . 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))
57		simpl 486	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑌 ∈ (SubGrp‘𝐺))
58	23	adantrr 717	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌)
59	35	simp3d 1146	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌)
60	8	subgcl 18507	. . . . 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)
61	57, 58, 59, 60	syl3anc 1373	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑌)
62	56, 61	eqeltrrd 2832	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑧) ∈ 𝑌)
63	6	adantr 484	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑌 ⊆ 𝑋)
64	5, 7, 8, 1	eqgval 18547	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∼ 𝑧 ↔ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑧) ∈ 𝑌)))
65	37, 63, 64	syl2anc 587	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (𝑥 ∼ 𝑧 ↔ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑧) ∈ 𝑌)))
66	31, 36, 62, 65	mpbir3and 1344	. 2 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑥 ∼ 𝑧)
67	5, 8, 46, 7	grplinv 18370	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) = (0g‘𝐺))
68	4, 67	sylan 583	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) = (0g‘𝐺))
69	46	subg0cl 18505	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑌)
70	69	adantr 484	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (0g‘𝐺) ∈ 𝑌)
71	68, 70	eqeltrd 2831	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌)
72	71	ex 416	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝑋 → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌))
73	72	pm4.71rd 566	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝑋 ↔ ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌 ∧ 𝑥 ∈ 𝑋)))
74	5, 7, 8, 1	eqgval 18547	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌)))
75	4, 6, 74	syl2anc 587	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌)))
76		df-3an 1091	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌))
77		anidm 568	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) ↔ 𝑥 ∈ 𝑋)
78	77	anbi2ci 628	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌) ↔ ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌 ∧ 𝑥 ∈ 𝑋))
79	76, 78	bitri 278	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌) ↔ ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌 ∧ 𝑥 ∈ 𝑋))
80	75, 79	bitrdi 290	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∼ 𝑥 ↔ ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌 ∧ 𝑥 ∈ 𝑋)))
81	73, 80	bitr4d 285	. 2 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∼ 𝑥))
82	3, 30, 66, 81	iserd 8395	1 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112   ⊆ wss 3853   class class class wbr 5039  Rel wrel 5541  ‘cfv 6358  (class class class)co 7191   Er wer 8366  Basecbs 16666  +_gcplusg 16749  0_gc0g 16898  Grpcgrp 18319  inv_gcminusg 18320  SubGrpcsubg 18491   ~_QG cqg 18493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501  ax-cnex 10750  ax-resscn 10751  ax-1cn 10752  ax-icn 10753  ax-addcl 10754  ax-addrcl 10755  ax-mulcl 10756  ax-mulrcl 10757  ax-mulcom 10758  ax-addass 10759  ax-mulass 10760  ax-distr 10761  ax-i2m1 10762  ax-1ne0 10763  ax-1rid 10764  ax-rnegex 10765  ax-rrecex 10766  ax-cnre 10767  ax-pre-lttri 10768  ax-pre-lttrn 10769  ax-pre-ltadd 10770  ax-pre-mulgt0 10771

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-ov 7194  df-oprab 7195  df-mpo 7196  df-om 7623  df-1st 7739  df-2nd 7740  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-er 8369  df-en 8605  df-dom 8606  df-sdom 8607  df-pnf 10834  df-mnf 10835  df-xr 10836  df-ltxr 10837  df-le 10838  df-sub 11029  df-neg 11030  df-nn 11796  df-2 11858  df-ndx 16669  df-slot 16670  df-base 16672  df-sets 16673  df-ress 16674  df-plusg 16762  df-0g 16900  df-mgm 18068  df-sgrp 18117  df-mnd 18128  df-grp 18322  df-minusg 18323  df-subg 18494  df-eqg 18496

This theorem is referenced by:  qusgrp  18553  qusadd  18555  lagsubg2  18559  lagsubg  18560  orbstafun  18659  orbstaval  18660  orbsta  18661  orbsta2  18662  sylow2blem1  18963  sylow2blem2  18964  sylow2blem3  18965  sylow3lem3  18972  sylow3lem4  18973  2idlcpbl  20226  qus1  20227  qusrhm  20229  quscrng  20232  zndvds  20468  cldsubg  22962  qustgpopn  22971  qustgphaus  22974  tgptsmscls  23001  qusker  31217  qusvscpbl  31219  quslmod  31222  eqg0el  31225  qusxpid  31227  qustrivr  31229  nsgqusf1olem3  31268  qsidomlem1  31296  qsidomlem2  31297