Theorem eqger 19202

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 19199	. . 3 ⊢ Rel ∼
3	2	a1i 11	. 2 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → Rel ∼ )
4		subgrcl 19156	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5		eqger.x	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐺)
6	5	subgss 19152	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
7		eqid 2761	. . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺)
8		eqid 2761	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
9	5, 7, 8, 1	eqgval 19201	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌)))
10	4, 6, 9	syl2anc 593	. . . . 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 1155	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝑦 ∈ 𝑋)
13	11	simp1d 1154	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝑥 ∈ 𝑋)
14	4	adantr 484	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝐺 ∈ Grp)
15	5, 7, 14, 13	grpinvcld 19013	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → ((invg‘𝐺)‘𝑥) ∈ 𝑋)
16	5, 8, 7	grpinvadd 19043	. . . . . 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))))
17	14, 15, 12, 16	syl3anc 1389	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → ((invg‘𝐺)‘(((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)) = (((invg‘𝐺)‘𝑦)(+g‘𝐺)((invg‘𝐺)‘((invg‘𝐺)‘𝑥))))
18	5, 7	grpinvinv 19030	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑥)) = 𝑥)
19	14, 13, 18	syl2anc 593	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑥)) = 𝑥)
20	19	oveq2d 7408	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)((invg‘𝐺)‘((invg‘𝐺)‘𝑥))) = (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑥))
21	17, 20	eqtrd 2796	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → ((invg‘𝐺)‘(((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)) = (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑥))
22	11	simp3d 1156	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌)
23	7	subginvcl 19160	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌) → ((invg‘𝐺)‘(((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)) ∈ 𝑌)
24	22, 23	syldan 600	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → ((invg‘𝐺)‘(((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)) ∈ 𝑌)
25	21, 24	eqeltrrd 2862	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑥) ∈ 𝑌)
26	6	adantr 484	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝑌 ⊆ 𝑋)
27	5, 7, 8, 1	eqgval 19201	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑥) ∈ 𝑌)))
28	14, 26, 27	syl2anc 593	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑥) ∈ 𝑌)))
29	12, 13, 25, 28	mpbir3and 1355	. 2 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∼ 𝑦) → 𝑦 ∼ 𝑥)
30	13	adantrr 727	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑥 ∈ 𝑋)
31	5, 7, 8, 1	eqgval 19201	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑦 ∼ 𝑧 ↔ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌)))
32	4, 6, 31	syl2anc 593	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑦 ∼ 𝑧 ↔ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌)))
33	32	biimpa 480	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∼ 𝑧) → (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌))
34	33	adantrl 726	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌))
35	34	simp2d 1155	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑧 ∈ 𝑋)
36	4	adantr 484	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝐺 ∈ Grp)
37	5, 7, 36, 30	grpinvcld 19013	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((invg‘𝐺)‘𝑥) ∈ 𝑋)
38	12	adantrr 727	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑦 ∈ 𝑋)
39	5, 7, 36, 38	grpinvcld 19013	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((invg‘𝐺)‘𝑦) ∈ 𝑋)
40	5, 8, 36, 39, 35	grpcld 18972	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑋)
41	5, 8, 36, 37, 38, 40	grpassd 18970	. . . . 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))))
42		eqid 2761	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
43	5, 8, 42, 7	grprinv 19015	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦)) = (0g‘𝐺))
44	36, 38, 43	syl2anc 593	. . . . . . . 8 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦)) = (0g‘𝐺))
45	44	oveq1d 7407	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦))(+g‘𝐺)𝑧) = ((0g‘𝐺)(+g‘𝐺)𝑧))
46	5, 8, 36, 38, 39, 35	grpassd 18970	. . . . . . 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)))
47	5, 8, 42, 36, 35	grplidd 18994	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((0g‘𝐺)(+g‘𝐺)𝑧) = 𝑧)
48	45, 46, 47	3eqtr3d 2804	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) = 𝑧)
49	48	oveq2d 7408	. . . . 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))
50	41, 49	eqtrd 2796	. . . 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))
51		simpl 486	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑌 ∈ (SubGrp‘𝐺))
52	22	adantrr 727	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝑌)
53	34	simp3d 1156	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌)
54	8	subgcl 19161	. . . . 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)
55	51, 52, 53, 54	syl3anc 1389	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦)(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑌)
56	50, 55	eqeltrrd 2862	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑧) ∈ 𝑌)
57	6	adantr 484	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑌 ⊆ 𝑋)
58	5, 7, 8, 1	eqgval 19201	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∼ 𝑧 ↔ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑧) ∈ 𝑌)))
59	36, 57, 58	syl2anc 593	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → (𝑥 ∼ 𝑧 ↔ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑧) ∈ 𝑌)))
60	30, 35, 56, 59	mpbir3and 1355	. 2 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑥 ∼ 𝑧)
61	5, 8, 42, 7	grplinv 19014	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) = (0g‘𝐺))
62	4, 61	sylan 589	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) = (0g‘𝐺))
63	42	subg0cl 19159	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑌)
64	63	adantr 484	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (0g‘𝐺) ∈ 𝑌)
65	62, 64	eqeltrd 2861	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌)
66	65	ex 416	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝑋 → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌))
67	66	pm4.71rd 570	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝑋 ↔ ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌 ∧ 𝑥 ∈ 𝑋)))
68	5, 7, 8, 1	eqgval 19201	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌)))
69	4, 6, 68	syl2anc 593	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌)))
70		df-3an 1099	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌))
71		anidm 572	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) ↔ 𝑥 ∈ 𝑋)
72	71	anbi2ci 634	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌) ↔ ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌 ∧ 𝑥 ∈ 𝑋))
73	70, 72	bitri 277	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌) ↔ ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌 ∧ 𝑥 ∈ 𝑋))
74	69, 73	bitrdi 289	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∼ 𝑥 ↔ ((((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑥) ∈ 𝑌 ∧ 𝑥 ∈ 𝑋)))
75	67, 74	bitr4d 284	. 2 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∼ 𝑥))
76	3, 29, 60, 75	iserd 8700	1 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ⊆ wss 3904   class class class wbr 5099  Rel wrel 5650  ‘cfv 6517  (class class class)co 7392   Er wer 8670  Basecbs 17228  +_gcplusg 17269  0_gc0g 17451  Grpcgrp 18958  inv_gcminusg 18959  SubGrpcsubg 19145   ~_QG cqg 19147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17250  df-plusg 17282  df-0g 17453  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-grp 18961  df-minusg 18962  df-subg 19148  df-eqg 19150

This theorem is referenced by:  eqg0el  19207  qusgrp  19210  qusadd  19212  lagsubg2  19218  lagsubg  19219  qus0subgadd  19223  ghmqusnsglem1  19303  ghmqusnsglem2  19304  ghmqusnsg  19305  ghmquskerlem1  19306  ghmquskerlem2  19308  ghmquskerlem3  19309  ghmqusker  19310  orbstafun  19334  orbstaval  19335  orbsta  19336  orbsta2  19337  sylow2blem1  19643  sylow2blem2  19644  sylow2blem3  19645  sylow3lem3  19652  sylow3lem4  19653  qusecsub  19858  2idlcpblrng  21321  qus2idrng  21323  qus1  21324  qusrhm  21326  qusmul2idl  21329  qusmulrng  21332  rhmqusnsg  21335  rngqipring1  21366  zndvds  21581  cldsubg  24151  qustgpopn  24160  qustgphaus  24163  tgptsmscls  24190  qusker  33496  qusvscpbl  33498  quslmod  33505  qusxpid  33510  qustrivr  33512  nsgqusf1olem3  33562  lmhmqusker  33564  rhmquskerlem  33572  qsidomlem1  33600  qsidomlem2  33601  qsnzr  33603  qsdrngilem  33643  qsdrnglem2  33645