Theorem nmzsubg 13340

Description: The normalizer N_G(S) of a subset 𝑆 of the group is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)

Hypotheses

Ref	Expression
elnmz.1	⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}
nmzsubg.2	⊢ 𝑋 = (Base‘𝐺)
nmzsubg.3	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
nmzsubg	⊢ (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺))

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑆,𝑦   𝑥, + ,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝑁(𝑥,𝑦)

Proof of Theorem nmzsubg

Dummy variables 𝑧 𝑤 𝑢 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnmz.1	. . . 4 ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}
2	1	ssrab3 3269	. . 3 ⊢ 𝑁 ⊆ 𝑋
3	2	a1i 9	. 2 ⊢ (𝐺 ∈ Grp → 𝑁 ⊆ 𝑋)
4		nmzsubg.2	. . . . 5 ⊢ 𝑋 = (Base‘𝐺)
5		eqid 2196	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
6	4, 5	grpidcl 13161	. . . 4 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋)
7		nmzsubg.3	. . . . . . . 8 ⊢ + = (+g‘𝐺)
8	4, 7, 5	grplid 13163	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((0g‘𝐺) + 𝑧) = 𝑧)
9	4, 7, 5	grprid 13164	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → (𝑧 + (0g‘𝐺)) = 𝑧)
10	8, 9	eqtr4d 2232	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((0g‘𝐺) + 𝑧) = (𝑧 + (0g‘𝐺)))
11	10	eleq1d 2265	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → (((0g‘𝐺) + 𝑧) ∈ 𝑆 ↔ (𝑧 + (0g‘𝐺)) ∈ 𝑆))
12	11	ralrimiva 2570	. . . 4 ⊢ (𝐺 ∈ Grp → ∀𝑧 ∈ 𝑋 (((0g‘𝐺) + 𝑧) ∈ 𝑆 ↔ (𝑧 + (0g‘𝐺)) ∈ 𝑆))
13	1	elnmz 13338	. . . 4 ⊢ ((0g‘𝐺) ∈ 𝑁 ↔ ((0g‘𝐺) ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 (((0g‘𝐺) + 𝑧) ∈ 𝑆 ↔ (𝑧 + (0g‘𝐺)) ∈ 𝑆)))
14	6, 12, 13	sylanbrc 417	. . 3 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑁)
15		elex2 2779	. . 3 ⊢ ((0g‘𝐺) ∈ 𝑁 → ∃𝑎 𝑎 ∈ 𝑁)
16	14, 15	syl 14	. 2 ⊢ (𝐺 ∈ Grp → ∃𝑎 𝑎 ∈ 𝑁)
17		id 19	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Grp)
18	2	sseli 3179	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑁 → 𝑧 ∈ 𝑋)
19	2	sseli 3179	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑁 → 𝑤 ∈ 𝑋)
20	4, 7	grpcl 13140	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → (𝑧 + 𝑤) ∈ 𝑋)
21	17, 18, 19, 20	syl3an 1291	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) → (𝑧 + 𝑤) ∈ 𝑋)
22		simpl1 1002	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → 𝐺 ∈ Grp)
23		simpl2 1003	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → 𝑧 ∈ 𝑁)
24	2, 23	sselid 3181	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → 𝑧 ∈ 𝑋)
25		simpl3 1004	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → 𝑤 ∈ 𝑁)
26	2, 25	sselid 3181	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → 𝑤 ∈ 𝑋)
27		simpr 110	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → 𝑢 ∈ 𝑋)
28	4, 7	grpass 13141	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋)) → ((𝑧 + 𝑤) + 𝑢) = (𝑧 + (𝑤 + 𝑢)))
29	22, 24, 26, 27, 28	syl13anc 1251	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((𝑧 + 𝑤) + 𝑢) = (𝑧 + (𝑤 + 𝑢)))
30	29	eleq1d 2265	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (((𝑧 + 𝑤) + 𝑢) ∈ 𝑆 ↔ (𝑧 + (𝑤 + 𝑢)) ∈ 𝑆))
31	4, 7, 22, 26, 27	grpcld 13146	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (𝑤 + 𝑢) ∈ 𝑋)
32	1	nmzbi 13339	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑁 ∧ (𝑤 + 𝑢) ∈ 𝑋) → ((𝑧 + (𝑤 + 𝑢)) ∈ 𝑆 ↔ ((𝑤 + 𝑢) + 𝑧) ∈ 𝑆))
33	23, 31, 32	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((𝑧 + (𝑤 + 𝑢)) ∈ 𝑆 ↔ ((𝑤 + 𝑢) + 𝑧) ∈ 𝑆))
34	4, 7	grpass 13141	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝑤 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑤 + 𝑢) + 𝑧) = (𝑤 + (𝑢 + 𝑧)))
35	22, 26, 27, 24, 34	syl13anc 1251	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((𝑤 + 𝑢) + 𝑧) = (𝑤 + (𝑢 + 𝑧)))
36	35	eleq1d 2265	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (((𝑤 + 𝑢) + 𝑧) ∈ 𝑆 ↔ (𝑤 + (𝑢 + 𝑧)) ∈ 𝑆))
37	4, 7, 22, 27, 24	grpcld 13146	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (𝑢 + 𝑧) ∈ 𝑋)
38	1	nmzbi 13339	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝑁 ∧ (𝑢 + 𝑧) ∈ 𝑋) → ((𝑤 + (𝑢 + 𝑧)) ∈ 𝑆 ↔ ((𝑢 + 𝑧) + 𝑤) ∈ 𝑆))
39	25, 37, 38	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((𝑤 + (𝑢 + 𝑧)) ∈ 𝑆 ↔ ((𝑢 + 𝑧) + 𝑤) ∈ 𝑆))
40	33, 36, 39	3bitrd 214	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((𝑧 + (𝑤 + 𝑢)) ∈ 𝑆 ↔ ((𝑢 + 𝑧) + 𝑤) ∈ 𝑆))
41	4, 7	grpass 13141	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑢 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝑢 + 𝑧) + 𝑤) = (𝑢 + (𝑧 + 𝑤)))
42	22, 27, 24, 26, 41	syl13anc 1251	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((𝑢 + 𝑧) + 𝑤) = (𝑢 + (𝑧 + 𝑤)))
43	42	eleq1d 2265	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (((𝑢 + 𝑧) + 𝑤) ∈ 𝑆 ↔ (𝑢 + (𝑧 + 𝑤)) ∈ 𝑆))
44	30, 40, 43	3bitrd 214	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (((𝑧 + 𝑤) + 𝑢) ∈ 𝑆 ↔ (𝑢 + (𝑧 + 𝑤)) ∈ 𝑆))
45	44	ralrimiva 2570	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) → ∀𝑢 ∈ 𝑋 (((𝑧 + 𝑤) + 𝑢) ∈ 𝑆 ↔ (𝑢 + (𝑧 + 𝑤)) ∈ 𝑆))
46	1	elnmz 13338	. . . . . . 7 ⊢ ((𝑧 + 𝑤) ∈ 𝑁 ↔ ((𝑧 + 𝑤) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑋 (((𝑧 + 𝑤) + 𝑢) ∈ 𝑆 ↔ (𝑢 + (𝑧 + 𝑤)) ∈ 𝑆)))
47	21, 45, 46	sylanbrc 417	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) → (𝑧 + 𝑤) ∈ 𝑁)
48	47	3expa 1205	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑤 ∈ 𝑁) → (𝑧 + 𝑤) ∈ 𝑁)
49	48	ralrimiva 2570	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) → ∀𝑤 ∈ 𝑁 (𝑧 + 𝑤) ∈ 𝑁)
50		eqid 2196	. . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺)
51	4, 50	grpinvcl 13180	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((invg‘𝐺)‘𝑧) ∈ 𝑋)
52	18, 51	sylan2 286	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) → ((invg‘𝐺)‘𝑧) ∈ 𝑋)
53		simplr 528	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → 𝑧 ∈ 𝑁)
54		simpll 527	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → 𝐺 ∈ Grp)
55	52	adantr 276	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((invg‘𝐺)‘𝑧) ∈ 𝑋)
56		simpr 110	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → 𝑢 ∈ 𝑋)
57	4, 7, 54, 56, 55	grpcld 13146	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (𝑢 + ((invg‘𝐺)‘𝑧)) ∈ 𝑋)
58	4, 7, 54, 55, 57	grpcld 13146	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (((invg‘𝐺)‘𝑧) + (𝑢 + ((invg‘𝐺)‘𝑧))) ∈ 𝑋)
59	1	nmzbi 13339	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑁 ∧ (((invg‘𝐺)‘𝑧) + (𝑢 + ((invg‘𝐺)‘𝑧))) ∈ 𝑋) → ((𝑧 + (((invg‘𝐺)‘𝑧) + (𝑢 + ((invg‘𝐺)‘𝑧)))) ∈ 𝑆 ↔ ((((invg‘𝐺)‘𝑧) + (𝑢 + ((invg‘𝐺)‘𝑧))) + 𝑧) ∈ 𝑆))
60	53, 58, 59	syl2anc 411	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((𝑧 + (((invg‘𝐺)‘𝑧) + (𝑢 + ((invg‘𝐺)‘𝑧)))) ∈ 𝑆 ↔ ((((invg‘𝐺)‘𝑧) + (𝑢 + ((invg‘𝐺)‘𝑧))) + 𝑧) ∈ 𝑆))
61	2, 53	sselid 3181	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → 𝑧 ∈ 𝑋)
62	4, 7, 5, 50	grprinv 13183	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → (𝑧 + ((invg‘𝐺)‘𝑧)) = (0g‘𝐺))
63	54, 61, 62	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (𝑧 + ((invg‘𝐺)‘𝑧)) = (0g‘𝐺))
64	63	oveq1d 5937	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((𝑧 + ((invg‘𝐺)‘𝑧)) + (𝑢 + ((invg‘𝐺)‘𝑧))) = ((0g‘𝐺) + (𝑢 + ((invg‘𝐺)‘𝑧))))
65	4, 7	grpass 13141	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑧 ∈ 𝑋 ∧ ((invg‘𝐺)‘𝑧) ∈ 𝑋 ∧ (𝑢 + ((invg‘𝐺)‘𝑧)) ∈ 𝑋)) → ((𝑧 + ((invg‘𝐺)‘𝑧)) + (𝑢 + ((invg‘𝐺)‘𝑧))) = (𝑧 + (((invg‘𝐺)‘𝑧) + (𝑢 + ((invg‘𝐺)‘𝑧)))))
66	54, 61, 55, 57, 65	syl13anc 1251	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((𝑧 + ((invg‘𝐺)‘𝑧)) + (𝑢 + ((invg‘𝐺)‘𝑧))) = (𝑧 + (((invg‘𝐺)‘𝑧) + (𝑢 + ((invg‘𝐺)‘𝑧)))))
67	4, 7, 5	grplid 13163	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑢 + ((invg‘𝐺)‘𝑧)) ∈ 𝑋) → ((0g‘𝐺) + (𝑢 + ((invg‘𝐺)‘𝑧))) = (𝑢 + ((invg‘𝐺)‘𝑧)))
68	54, 57, 67	syl2anc 411	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((0g‘𝐺) + (𝑢 + ((invg‘𝐺)‘𝑧))) = (𝑢 + ((invg‘𝐺)‘𝑧)))
69	64, 66, 68	3eqtr3d 2237	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (𝑧 + (((invg‘𝐺)‘𝑧) + (𝑢 + ((invg‘𝐺)‘𝑧)))) = (𝑢 + ((invg‘𝐺)‘𝑧)))
70	69	eleq1d 2265	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((𝑧 + (((invg‘𝐺)‘𝑧) + (𝑢 + ((invg‘𝐺)‘𝑧)))) ∈ 𝑆 ↔ (𝑢 + ((invg‘𝐺)‘𝑧)) ∈ 𝑆))
71	4, 7	grpass 13141	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (((invg‘𝐺)‘𝑧) ∈ 𝑋 ∧ (𝑢 + ((invg‘𝐺)‘𝑧)) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((((invg‘𝐺)‘𝑧) + (𝑢 + ((invg‘𝐺)‘𝑧))) + 𝑧) = (((invg‘𝐺)‘𝑧) + ((𝑢 + ((invg‘𝐺)‘𝑧)) + 𝑧)))
72	54, 55, 57, 61, 71	syl13anc 1251	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((((invg‘𝐺)‘𝑧) + (𝑢 + ((invg‘𝐺)‘𝑧))) + 𝑧) = (((invg‘𝐺)‘𝑧) + ((𝑢 + ((invg‘𝐺)‘𝑧)) + 𝑧)))
73	4, 7	grpass 13141	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝑢 ∈ 𝑋 ∧ ((invg‘𝐺)‘𝑧) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑢 + ((invg‘𝐺)‘𝑧)) + 𝑧) = (𝑢 + (((invg‘𝐺)‘𝑧) + 𝑧)))
74	54, 56, 55, 61, 73	syl13anc 1251	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((𝑢 + ((invg‘𝐺)‘𝑧)) + 𝑧) = (𝑢 + (((invg‘𝐺)‘𝑧) + 𝑧)))
75	4, 7, 5, 50	grplinv 13182	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → (((invg‘𝐺)‘𝑧) + 𝑧) = (0g‘𝐺))
76	54, 61, 75	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (((invg‘𝐺)‘𝑧) + 𝑧) = (0g‘𝐺))
77	76	oveq2d 5938	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (𝑢 + (((invg‘𝐺)‘𝑧) + 𝑧)) = (𝑢 + (0g‘𝐺)))
78	4, 7, 5	grprid 13164	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋) → (𝑢 + (0g‘𝐺)) = 𝑢)
79	54, 56, 78	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (𝑢 + (0g‘𝐺)) = 𝑢)
80	74, 77, 79	3eqtrd 2233	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((𝑢 + ((invg‘𝐺)‘𝑧)) + 𝑧) = 𝑢)
81	80	oveq2d 5938	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (((invg‘𝐺)‘𝑧) + ((𝑢 + ((invg‘𝐺)‘𝑧)) + 𝑧)) = (((invg‘𝐺)‘𝑧) + 𝑢))
82	72, 81	eqtrd 2229	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((((invg‘𝐺)‘𝑧) + (𝑢 + ((invg‘𝐺)‘𝑧))) + 𝑧) = (((invg‘𝐺)‘𝑧) + 𝑢))
83	82	eleq1d 2265	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (((((invg‘𝐺)‘𝑧) + (𝑢 + ((invg‘𝐺)‘𝑧))) + 𝑧) ∈ 𝑆 ↔ (((invg‘𝐺)‘𝑧) + 𝑢) ∈ 𝑆))
84	60, 70, 83	3bitr3rd 219	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((((invg‘𝐺)‘𝑧) + 𝑢) ∈ 𝑆 ↔ (𝑢 + ((invg‘𝐺)‘𝑧)) ∈ 𝑆))
85	84	ralrimiva 2570	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) → ∀𝑢 ∈ 𝑋 ((((invg‘𝐺)‘𝑧) + 𝑢) ∈ 𝑆 ↔ (𝑢 + ((invg‘𝐺)‘𝑧)) ∈ 𝑆))
86	1	elnmz 13338	. . . . 5 ⊢ (((invg‘𝐺)‘𝑧) ∈ 𝑁 ↔ (((invg‘𝐺)‘𝑧) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑋 ((((invg‘𝐺)‘𝑧) + 𝑢) ∈ 𝑆 ↔ (𝑢 + ((invg‘𝐺)‘𝑧)) ∈ 𝑆)))
87	52, 85, 86	sylanbrc 417	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) → ((invg‘𝐺)‘𝑧) ∈ 𝑁)
88	49, 87	jca 306	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) → (∀𝑤 ∈ 𝑁 (𝑧 + 𝑤) ∈ 𝑁 ∧ ((invg‘𝐺)‘𝑧) ∈ 𝑁))
89	88	ralrimiva 2570	. 2 ⊢ (𝐺 ∈ Grp → ∀𝑧 ∈ 𝑁 (∀𝑤 ∈ 𝑁 (𝑧 + 𝑤) ∈ 𝑁 ∧ ((invg‘𝐺)‘𝑧) ∈ 𝑁))
90	4, 7, 50	issubg2m 13319	. 2 ⊢ (𝐺 ∈ Grp → (𝑁 ∈ (SubGrp‘𝐺) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑎 𝑎 ∈ 𝑁 ∧ ∀𝑧 ∈ 𝑁 (∀𝑤 ∈ 𝑁 (𝑧 + 𝑤) ∈ 𝑁 ∧ ((invg‘𝐺)‘𝑧) ∈ 𝑁))))
91	3, 16, 89, 90	mpbir3and 1182	1 ⊢ (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  {crab 2479   ⊆ wss 3157  ‘cfv 5258  (class class class)co 5922  Basecbs 12678  +_gcplusg 12755  0_gc0g 12927  Grpcgrp 13132  inv_gcminusg 13133  SubGrpcsubg 13297

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 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

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 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-subg 13300

This theorem is referenced by:  nmznsg  13343