Theorem conjnmz 17694

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 17599	. . . . . . . . . 10 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2	1	ad2antrr 762	. . . . . . . . 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		ssrab2 3687	. . . . . . . . . . . 12 ⊢ {𝑦 ∈ 𝑋 ∣ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} ⊆ 𝑋
5	3, 4	eqsstri 3635	. . . . . . . . . . 11 ⊢ 𝑁 ⊆ 𝑋
6		simplr 792	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝐴 ∈ 𝑁)
7	5, 6	sseldi 3601	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝐴 ∈ 𝑋)
8		conjghm.x	. . . . . . . . . . 11 ⊢ 𝑋 = (Base‘𝐺)
9		eqid 2622	. . . . . . . . . . 11 ⊢ (invg‘𝐺) = (invg‘𝐺)
10	8, 9	grpinvcl 17467	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
11	2, 7, 10	syl2anc 693	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
12	8	subgss 17595	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑋)
13	12	adantr 481	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝑆 ⊆ 𝑋)
14	13	sselda 3603	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝑤 ∈ 𝑋)
15		conjghm.p	. . . . . . . . . 10 ⊢ + = (+g‘𝐺)
16	8, 15	grpass 17431	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐴) + 𝑤) + 𝐴) = (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)))
17	2, 11, 14, 7, 16	syl13anc 1328	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((((invg‘𝐺)‘𝐴) + 𝑤) + 𝐴) = (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)))
18		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (0g‘𝐺) = (0g‘𝐺)
19	8, 15, 18, 9	grprinv 17469	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐴 + ((invg‘𝐺)‘𝐴)) = (0g‘𝐺))
20	2, 7, 19	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐴 + ((invg‘𝐺)‘𝐴)) = (0g‘𝐺))
21	20	oveq1d 6665	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + 𝑤) = ((0g‘𝐺) + 𝑤))
22	8, 15	grpass 17431	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ ((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + 𝑤) = (𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)))
23	2, 7, 11, 14, 22	syl13anc 1328	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + 𝑤) = (𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)))
24	8, 15, 18	grplid 17452	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋) → ((0g‘𝐺) + 𝑤) = 𝑤)
25	2, 14, 24	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((0g‘𝐺) + 𝑤) = 𝑤)
26	21, 23, 25	3eqtr3d 2664	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)) = 𝑤)
27		simpr 477	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝑤 ∈ 𝑆)
28	26, 27	eqeltrd 2701	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)) ∈ 𝑆)
29	8, 15	grpcl 17430	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ ((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → (((invg‘𝐺)‘𝐴) + 𝑤) ∈ 𝑋)
30	2, 11, 14, 29	syl3anc 1326	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (((invg‘𝐺)‘𝐴) + 𝑤) ∈ 𝑋)
31	3	nmzbi 17634	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑁 ∧ (((invg‘𝐺)‘𝐴) + 𝑤) ∈ 𝑋) → ((𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)) ∈ 𝑆 ↔ ((((invg‘𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆))
32	6, 30, 31	syl2anc 693	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)) ∈ 𝑆 ↔ ((((invg‘𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆))
33	28, 32	mpbid 222	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((((invg‘𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆)
34	17, 33	eqeltrrd 2702	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆)
35		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) → (𝐴 + 𝑥) = (𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))))
36	35	oveq1d 6665	. . . . . . . 8 ⊢ (𝑥 = (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) → ((𝐴 + 𝑥) − 𝐴) = ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴))
37		conjsubg.f	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴))
38		ovex 6678	. . . . . . . 8 ⊢ ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴) ∈ V
39	36, 37, 38	fvmpt 6282	. . . . . . 7 ⊢ ((((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆 → (𝐹‘(((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) = ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴))
40	34, 39	syl 17	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐹‘(((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) = ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴))
41	20	oveq1d 6665	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + (𝑤 + 𝐴)) = ((0g‘𝐺) + (𝑤 + 𝐴)))
42	8, 15	grpcl 17430	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑤 + 𝐴) ∈ 𝑋)
43	2, 14, 7, 42	syl3anc 1326	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝑤 + 𝐴) ∈ 𝑋)
44	8, 15	grpass 17431	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ ((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ (𝑤 + 𝐴) ∈ 𝑋)) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + (𝑤 + 𝐴)) = (𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))))
45	2, 7, 11, 43, 44	syl13anc 1328	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + (𝑤 + 𝐴)) = (𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))))
46	8, 15, 18	grplid 17452	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑤 + 𝐴) ∈ 𝑋) → ((0g‘𝐺) + (𝑤 + 𝐴)) = (𝑤 + 𝐴))
47	2, 43, 46	syl2anc 693	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((0g‘𝐺) + (𝑤 + 𝐴)) = (𝑤 + 𝐴))
48	41, 45, 47	3eqtr3d 2664	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) = (𝑤 + 𝐴))
49	48	oveq1d 6665	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴) = ((𝑤 + 𝐴) − 𝐴))
50		conjghm.m	. . . . . . . 8 ⊢ − = (-g‘𝐺)
51	8, 15, 50	grppncan 17506	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑤 + 𝐴) − 𝐴) = 𝑤)
52	2, 14, 7, 51	syl3anc 1326	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝑤 + 𝐴) − 𝐴) = 𝑤)
53	40, 49, 52	3eqtrd 2660	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐹‘(((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) = 𝑤)
54		ovex 6678	. . . . . . 7 ⊢ ((𝐴 + 𝑥) − 𝐴) ∈ V
55	54, 37	fnmpti 6022	. . . . . 6 ⊢ 𝐹 Fn 𝑆
56		fnfvelrn 6356	. . . . . 6 ⊢ ((𝐹 Fn 𝑆 ∧ (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆) → (𝐹‘(((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) ∈ ran 𝐹)
57	55, 34, 56	sylancr 695	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐹‘(((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) ∈ ran 𝐹)
58	53, 57	eqeltrrd 2702	. . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝑤 ∈ ran 𝐹)
59	58	ex 450	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → (𝑤 ∈ 𝑆 → 𝑤 ∈ ran 𝐹))
60	59	ssrdv 3609	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝑆 ⊆ ran 𝐹)
61	1	ad2antrr 762	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → 𝐺 ∈ Grp)
62		simplr 792	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ 𝑁)
63	5, 62	sseldi 3601	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ 𝑋)
64	13	sselda 3603	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑋)
65	8, 15, 50	grpaddsubass 17505	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴 + 𝑥) − 𝐴) = (𝐴 + (𝑥 − 𝐴)))
66	61, 63, 64, 63, 65	syl13anc 1328	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + 𝑥) − 𝐴) = (𝐴 + (𝑥 − 𝐴)))
67	8, 15, 50	grpnpcan 17507	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑥 − 𝐴) + 𝐴) = 𝑥)
68	61, 64, 63, 67	syl3anc 1326	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → ((𝑥 − 𝐴) + 𝐴) = 𝑥)
69		simpr 477	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
70	68, 69	eqeltrd 2701	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → ((𝑥 − 𝐴) + 𝐴) ∈ 𝑆)
71	8, 50	grpsubcl 17495	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑥 − 𝐴) ∈ 𝑋)
72	61, 64, 63, 71	syl3anc 1326	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → (𝑥 − 𝐴) ∈ 𝑋)
73	3	nmzbi 17634	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑁 ∧ (𝑥 − 𝐴) ∈ 𝑋) → ((𝐴 + (𝑥 − 𝐴)) ∈ 𝑆 ↔ ((𝑥 − 𝐴) + 𝐴) ∈ 𝑆))
74	62, 72, 73	syl2anc 693	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + (𝑥 − 𝐴)) ∈ 𝑆 ↔ ((𝑥 − 𝐴) + 𝐴) ∈ 𝑆))
75	70, 74	mpbird 247	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → (𝐴 + (𝑥 − 𝐴)) ∈ 𝑆)
76	66, 75	eqeltrd 2701	. . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + 𝑥) − 𝐴) ∈ 𝑆)
77	76, 37	fmptd 6385	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝐹:𝑆⟶𝑆)
78		frn 6053	. . 3 ⊢ (𝐹:𝑆⟶𝑆 → ran 𝐹 ⊆ 𝑆)
79	77, 78	syl 17	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → ran 𝐹 ⊆ 𝑆)
80	60, 79	eqssd 3620	1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝑆 = ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916   ⊆ wss 3574   ↦ cmpt 4729  ran crn 5115   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  0_gc0g 16100  Grpcgrp 17422  inv_gcminusg 17423  -_gcsg 17424  SubGrpcsubg 17588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591

This theorem is referenced by:  conjnmzb  17695  conjnsg  17696  sylow3lem2  18043