Theorem nmzsubg 13877

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 3314	. . 3 ⊢ 𝑁 ⊆ 𝑋
3	2	a1i 9	. 2 ⊢ (𝐺 ∈ Grp → 𝑁 ⊆ 𝑋)
4		nmzsubg.2	. . . . 5 ⊢ 𝑋 = (Base‘𝐺)
5		eqid 2231	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
6	4, 5	grpidcl 13692	. . . 4 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋)
7		nmzsubg.3	. . . . . . . 8 ⊢ + = (+g‘𝐺)
8	4, 7, 5	grplid 13694	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((0g‘𝐺) + 𝑧) = 𝑧)
9	4, 7, 5	grprid 13695	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → (𝑧 + (0g‘𝐺)) = 𝑧)
10	8, 9	eqtr4d 2267	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((0g‘𝐺) + 𝑧) = (𝑧 + (0g‘𝐺)))
11	10	eleq1d 2300	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → (((0g‘𝐺) + 𝑧) ∈ 𝑆 ↔ (𝑧 + (0g‘𝐺)) ∈ 𝑆))
12	11	ralrimiva 2606	. . . 4 ⊢ (𝐺 ∈ Grp → ∀𝑧 ∈ 𝑋 (((0g‘𝐺) + 𝑧) ∈ 𝑆 ↔ (𝑧 + (0g‘𝐺)) ∈ 𝑆))
13	1	elnmz 13875	. . . 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 2820	. . 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 3224	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑁 → 𝑧 ∈ 𝑋)
19	2	sseli 3224	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑁 → 𝑤 ∈ 𝑋)
20	4, 7	grpcl 13671	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → (𝑧 + 𝑤) ∈ 𝑋)
21	17, 18, 19, 20	syl3an 1316	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) → (𝑧 + 𝑤) ∈ 𝑋)
22		simpl1 1027	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → 𝐺 ∈ Grp)
23		simpl2 1028	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → 𝑧 ∈ 𝑁)
24	2, 23	sselid 3226	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → 𝑧 ∈ 𝑋)
25		simpl3 1029	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → 𝑤 ∈ 𝑁)
26	2, 25	sselid 3226	. . . . . . . . . . 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 13672	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋)) → ((𝑧 + 𝑤) + 𝑢) = (𝑧 + (𝑤 + 𝑢)))
29	22, 24, 26, 27, 28	syl13anc 1276	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((𝑧 + 𝑤) + 𝑢) = (𝑧 + (𝑤 + 𝑢)))
30	29	eleq1d 2300	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (((𝑧 + 𝑤) + 𝑢) ∈ 𝑆 ↔ (𝑧 + (𝑤 + 𝑢)) ∈ 𝑆))
31	4, 7, 22, 26, 27	grpcld 13677	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (𝑤 + 𝑢) ∈ 𝑋)
32	1	nmzbi 13876	. . . . . . . . . . 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 13672	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝑤 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑤 + 𝑢) + 𝑧) = (𝑤 + (𝑢 + 𝑧)))
35	22, 26, 27, 24, 34	syl13anc 1276	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((𝑤 + 𝑢) + 𝑧) = (𝑤 + (𝑢 + 𝑧)))
36	35	eleq1d 2300	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (((𝑤 + 𝑢) + 𝑧) ∈ 𝑆 ↔ (𝑤 + (𝑢 + 𝑧)) ∈ 𝑆))
37	4, 7, 22, 27, 24	grpcld 13677	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (𝑢 + 𝑧) ∈ 𝑋)
38	1	nmzbi 13876	. . . . . . . . . . 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 13672	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑢 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝑢 + 𝑧) + 𝑤) = (𝑢 + (𝑧 + 𝑤)))
42	22, 27, 24, 26, 41	syl13anc 1276	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((𝑢 + 𝑧) + 𝑤) = (𝑢 + (𝑧 + 𝑤)))
43	42	eleq1d 2300	. . . . . . . . 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 2606	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) → ∀𝑢 ∈ 𝑋 (((𝑧 + 𝑤) + 𝑢) ∈ 𝑆 ↔ (𝑢 + (𝑧 + 𝑤)) ∈ 𝑆))
46	1	elnmz 13875	. . . . . . 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 1230	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑤 ∈ 𝑁) → (𝑧 + 𝑤) ∈ 𝑁)
49	48	ralrimiva 2606	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) → ∀𝑤 ∈ 𝑁 (𝑧 + 𝑤) ∈ 𝑁)
50		eqid 2231	. . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺)
51	4, 50	grpinvcl 13711	. . . . . 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 529	. . . . . . . 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 13677	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (𝑢 + ((invg‘𝐺)‘𝑧)) ∈ 𝑋)
58	4, 7, 54, 55, 57	grpcld 13677	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (((invg‘𝐺)‘𝑧) + (𝑢 + ((invg‘𝐺)‘𝑧))) ∈ 𝑋)
59	1	nmzbi 13876	. . . . . . . 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 3226	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → 𝑧 ∈ 𝑋)
62	4, 7, 5, 50	grprinv 13714	. . . . . . . . . . 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 6043	. . . . . . . . 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 13672	. . . . . . . . . 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 1276	. . . . . . . . 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 13694	. . . . . . . . . 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 2272	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (𝑧 + (((invg‘𝐺)‘𝑧) + (𝑢 + ((invg‘𝐺)‘𝑧)))) = (𝑢 + ((invg‘𝐺)‘𝑧)))
70	69	eleq1d 2300	. . . . . . 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 13672	. . . . . . . . . 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 1276	. . . . . . . . 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 13672	. . . . . . . . . . . 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 1276	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((𝑢 + ((invg‘𝐺)‘𝑧)) + 𝑧) = (𝑢 + (((invg‘𝐺)‘𝑧) + 𝑧)))
75	4, 7, 5, 50	grplinv 13713	. . . . . . . . . . . . 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 6044	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (𝑢 + (((invg‘𝐺)‘𝑧) + 𝑧)) = (𝑢 + (0g‘𝐺)))
78	4, 7, 5	grprid 13695	. . . . . . . . . . . 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 2268	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((𝑢 + ((invg‘𝐺)‘𝑧)) + 𝑧) = 𝑢)
81	80	oveq2d 6044	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → (((invg‘𝐺)‘𝑧) + ((𝑢 + ((invg‘𝐺)‘𝑧)) + 𝑧)) = (((invg‘𝐺)‘𝑧) + 𝑢))
82	72, 81	eqtrd 2264	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) ∧ 𝑢 ∈ 𝑋) → ((((invg‘𝐺)‘𝑧) + (𝑢 + ((invg‘𝐺)‘𝑧))) + 𝑧) = (((invg‘𝐺)‘𝑧) + 𝑢))
83	82	eleq1d 2300	. . . . . . 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 2606	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑁) → ∀𝑢 ∈ 𝑋 ((((invg‘𝐺)‘𝑧) + 𝑢) ∈ 𝑆 ↔ (𝑢 + ((invg‘𝐺)‘𝑧)) ∈ 𝑆))
86	1	elnmz 13875	. . . . 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 2606	. 2 ⊢ (𝐺 ∈ Grp → ∀𝑧 ∈ 𝑁 (∀𝑤 ∈ 𝑁 (𝑧 + 𝑤) ∈ 𝑁 ∧ ((invg‘𝐺)‘𝑧) ∈ 𝑁))
90	4, 7, 50	issubg2m 13856	. 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 1207	1 ⊢ (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2202  ∀wral 2511  {crab 2515   ⊆ wss 3201  ‘cfv 5333  (class class class)co 6028  Basecbs 13162  +_gcplusg 13240  0_gc0g 13419  Grpcgrp 13663  inv_gcminusg 13664  SubGrpcsubg 13834

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-subg 13837

This theorem is referenced by:  nmznsg  13880