Theorem conjnmz 19042

Description: A subgroup is unchanged under conjugation by an element of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)

Hypotheses

Ref	Expression
conjghm.x	⊢ 𝑋 = (Base‘𝐺)
conjghm.p	⊢ + = (+g‘𝐺)
conjghm.m	⊢ − = (-g‘𝐺)
conjsubg.f	⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴))
conjnmz.1	⊢ 𝑁 = {𝑦 ∈ 𝑋 ∣ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}

Assertion

Ref	Expression
conjnmz	⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝑆 = ran 𝐹)

Distinct variable groups:   𝑥,𝑦, −   𝑥,𝑧, + ,𝑦   𝑥,𝐴,𝑦,𝑧   𝑦,𝐹,𝑧   𝑥,𝑁   𝑥,𝐺,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Allowed substitution hints:   𝐹(𝑥)   − (𝑧)   𝑁(𝑦,𝑧)

Proof of Theorem conjnmz

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subgrcl 18933	. . . . . . . . . 10 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2	1	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝐺 ∈ Grp)
3		conjnmz.1	. . . . . . . . . . . 12 ⊢ 𝑁 = {𝑦 ∈ 𝑋 ∣ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}
4	3	ssrab3 4040	. . . . . . . . . . 11 ⊢ 𝑁 ⊆ 𝑋
5		simplr 767	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝐴 ∈ 𝑁)
6	4, 5	sselid 3942	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝐴 ∈ 𝑋)
7		conjghm.x	. . . . . . . . . . 11 ⊢ 𝑋 = (Base‘𝐺)
8		eqid 2736	. . . . . . . . . . 11 ⊢ (invg‘𝐺) = (invg‘𝐺)
9	7, 8	grpinvcl 18798	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
10	2, 6, 9	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
11	7	subgss 18929	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑋)
12	11	adantr 481	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝑆 ⊆ 𝑋)
13	12	sselda 3944	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝑤 ∈ 𝑋)
14		conjghm.p	. . . . . . . . . 10 ⊢ + = (+g‘𝐺)
15	7, 14	grpass 18757	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐴) + 𝑤) + 𝐴) = (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)))
16	2, 10, 13, 6, 15	syl13anc 1372	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((((invg‘𝐺)‘𝐴) + 𝑤) + 𝐴) = (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)))
17		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (0g‘𝐺) = (0g‘𝐺)
18	7, 14, 17, 8	grprinv 18801	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐴 + ((invg‘𝐺)‘𝐴)) = (0g‘𝐺))
19	2, 6, 18	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐴 + ((invg‘𝐺)‘𝐴)) = (0g‘𝐺))
20	19	oveq1d 7372	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + 𝑤) = ((0g‘𝐺) + 𝑤))
21	7, 14	grpass 18757	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ ((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + 𝑤) = (𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)))
22	2, 6, 10, 13, 21	syl13anc 1372	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + 𝑤) = (𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)))
23	7, 14, 17	grplid 18780	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋) → ((0g‘𝐺) + 𝑤) = 𝑤)
24	2, 13, 23	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((0g‘𝐺) + 𝑤) = 𝑤)
25	20, 22, 24	3eqtr3d 2784	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)) = 𝑤)
26		simpr 485	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝑤 ∈ 𝑆)
27	25, 26	eqeltrd 2838	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)) ∈ 𝑆)
28	7, 14	grpcl 18756	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ ((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → (((invg‘𝐺)‘𝐴) + 𝑤) ∈ 𝑋)
29	2, 10, 13, 28	syl3anc 1371	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (((invg‘𝐺)‘𝐴) + 𝑤) ∈ 𝑋)
30	3	nmzbi 18966	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑁 ∧ (((invg‘𝐺)‘𝐴) + 𝑤) ∈ 𝑋) → ((𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)) ∈ 𝑆 ↔ ((((invg‘𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆))
31	5, 29, 30	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)) ∈ 𝑆 ↔ ((((invg‘𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆))
32	27, 31	mpbid 231	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((((invg‘𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆)
33	16, 32	eqeltrrd 2839	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆)
34		oveq2 7365	. . . . . . . . 9 ⊢ (𝑥 = (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) → (𝐴 + 𝑥) = (𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))))
35	34	oveq1d 7372	. . . . . . . 8 ⊢ (𝑥 = (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) → ((𝐴 + 𝑥) − 𝐴) = ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴))
36		conjsubg.f	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴))
37		ovex 7390	. . . . . . . 8 ⊢ ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴) ∈ V
38	35, 36, 37	fvmpt 6948	. . . . . . 7 ⊢ ((((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆 → (𝐹‘(((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) = ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴))
39	33, 38	syl 17	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐹‘(((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) = ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴))
40	19	oveq1d 7372	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + (𝑤 + 𝐴)) = ((0g‘𝐺) + (𝑤 + 𝐴)))
41	7, 14	grpcl 18756	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑤 + 𝐴) ∈ 𝑋)
42	2, 13, 6, 41	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝑤 + 𝐴) ∈ 𝑋)
43	7, 14	grpass 18757	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ ((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ (𝑤 + 𝐴) ∈ 𝑋)) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + (𝑤 + 𝐴)) = (𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))))
44	2, 6, 10, 42, 43	syl13anc 1372	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + (𝑤 + 𝐴)) = (𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))))
45	7, 14, 17	grplid 18780	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑤 + 𝐴) ∈ 𝑋) → ((0g‘𝐺) + (𝑤 + 𝐴)) = (𝑤 + 𝐴))
46	2, 42, 45	syl2anc 584	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((0g‘𝐺) + (𝑤 + 𝐴)) = (𝑤 + 𝐴))
47	40, 44, 46	3eqtr3d 2784	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) = (𝑤 + 𝐴))
48	47	oveq1d 7372	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴) = ((𝑤 + 𝐴) − 𝐴))
49		conjghm.m	. . . . . . . 8 ⊢ − = (-g‘𝐺)
50	7, 14, 49	grppncan 18838	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑤 + 𝐴) − 𝐴) = 𝑤)
51	2, 13, 6, 50	syl3anc 1371	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝑤 + 𝐴) − 𝐴) = 𝑤)
52	39, 48, 51	3eqtrd 2780	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐹‘(((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) = 𝑤)
53		ovex 7390	. . . . . . 7 ⊢ ((𝐴 + 𝑥) − 𝐴) ∈ V
54	53, 36	fnmpti 6644	. . . . . 6 ⊢ 𝐹 Fn 𝑆
55		fnfvelrn 7031	. . . . . 6 ⊢ ((𝐹 Fn 𝑆 ∧ (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆) → (𝐹‘(((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) ∈ ran 𝐹)
56	54, 33, 55	sylancr 587	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐹‘(((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) ∈ ran 𝐹)
57	52, 56	eqeltrrd 2839	. . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝑤 ∈ ran 𝐹)
58	57	ex 413	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → (𝑤 ∈ 𝑆 → 𝑤 ∈ ran 𝐹))
59	58	ssrdv 3950	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝑆 ⊆ ran 𝐹)
60	1	ad2antrr 724	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → 𝐺 ∈ Grp)
61		simplr 767	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ 𝑁)
62	4, 61	sselid 3942	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ 𝑋)
63	12	sselda 3944	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑋)
64	7, 14, 49	grpaddsubass 18837	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴 + 𝑥) − 𝐴) = (𝐴 + (𝑥 − 𝐴)))
65	60, 62, 63, 62, 64	syl13anc 1372	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + 𝑥) − 𝐴) = (𝐴 + (𝑥 − 𝐴)))
66	7, 14, 49	grpnpcan 18839	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑥 − 𝐴) + 𝐴) = 𝑥)
67	60, 63, 62, 66	syl3anc 1371	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → ((𝑥 − 𝐴) + 𝐴) = 𝑥)
68		simpr 485	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
69	67, 68	eqeltrd 2838	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → ((𝑥 − 𝐴) + 𝐴) ∈ 𝑆)
70	7, 49	grpsubcl 18827	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑥 − 𝐴) ∈ 𝑋)
71	60, 63, 62, 70	syl3anc 1371	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → (𝑥 − 𝐴) ∈ 𝑋)
72	3	nmzbi 18966	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑁 ∧ (𝑥 − 𝐴) ∈ 𝑋) → ((𝐴 + (𝑥 − 𝐴)) ∈ 𝑆 ↔ ((𝑥 − 𝐴) + 𝐴) ∈ 𝑆))
73	61, 71, 72	syl2anc 584	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + (𝑥 − 𝐴)) ∈ 𝑆 ↔ ((𝑥 − 𝐴) + 𝐴) ∈ 𝑆))
74	69, 73	mpbird 256	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → (𝐴 + (𝑥 − 𝐴)) ∈ 𝑆)
75	65, 74	eqeltrd 2838	. . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + 𝑥) − 𝐴) ∈ 𝑆)
76	75, 36	fmptd 7062	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝐹:𝑆⟶𝑆)
77	76	frnd 6676	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → ran 𝐹 ⊆ 𝑆)
78	59, 77	eqssd 3961	1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝑆 = ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  {crab 3407   ⊆ wss 3910   ↦ cmpt 5188  ran crn 5634   Fn wfn 6491  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  +_gcplusg 17133  0_gc0g 17321  Grpcgrp 18748  inv_gcminusg 18749  -_gcsg 18750  SubGrpcsubg 18922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-sbg 18753  df-subg 18925

This theorem is referenced by:  conjnmzb  19043  conjnsg  19044  sylow3lem2  19410