Theorem conjnmz 13811

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		conjsubg.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴))
2		oveq2 6008	. . . . . . . 8 ⊢ (𝑥 = (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) → (𝐴 + 𝑥) = (𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))))
3	2	oveq1d 6015	. . . . . . 7 ⊢ (𝑥 = (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) → ((𝐴 + 𝑥) − 𝐴) = ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴))
4		subgrcl 13711	. . . . . . . . . 10 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5	4	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝐺 ∈ Grp)
6		conjghm.x	. . . . . . . . . 10 ⊢ 𝑋 = (Base‘𝐺)
7		eqid 2229	. . . . . . . . . 10 ⊢ (invg‘𝐺) = (invg‘𝐺)
8		conjnmz.1	. . . . . . . . . . . 12 ⊢ 𝑁 = {𝑦 ∈ 𝑋 ∣ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}
9	8	ssrab3 3310	. . . . . . . . . . 11 ⊢ 𝑁 ⊆ 𝑋
10		simplr 528	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝐴 ∈ 𝑁)
11	9, 10	sselid 3222	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝐴 ∈ 𝑋)
12	6, 7, 5, 11	grpinvcld 13577	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
13	6	subgss 13706	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑋)
14	13	adantr 276	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝑆 ⊆ 𝑋)
15	14	sselda 3224	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝑤 ∈ 𝑋)
16		conjghm.p	. . . . . . . . . 10 ⊢ + = (+g‘𝐺)
17	6, 16	grpass 13537	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐴) + 𝑤) + 𝐴) = (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)))
18	5, 12, 15, 11, 17	syl13anc 1273	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((((invg‘𝐺)‘𝐴) + 𝑤) + 𝐴) = (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)))
19		eqid 2229	. . . . . . . . . . . . 13 ⊢ (0g‘𝐺) = (0g‘𝐺)
20	6, 16, 19, 7, 5, 11	grprinvd 13584	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐴 + ((invg‘𝐺)‘𝐴)) = (0g‘𝐺))
21	20	oveq1d 6015	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + 𝑤) = ((0g‘𝐺) + 𝑤))
22	6, 16	grpass 13537	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ ((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + 𝑤) = (𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)))
23	5, 11, 12, 15, 22	syl13anc 1273	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + 𝑤) = (𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)))
24	6, 16, 19, 5, 15	grplidd 13561	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((0g‘𝐺) + 𝑤) = 𝑤)
25	21, 23, 24	3eqtr3d 2270	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)) = 𝑤)
26		simpr 110	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝑤 ∈ 𝑆)
27	25, 26	eqeltrd 2306	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)) ∈ 𝑆)
28	6, 16, 5, 12, 15	grpcld 13542	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (((invg‘𝐺)‘𝐴) + 𝑤) ∈ 𝑋)
29	8	nmzbi 13741	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑁 ∧ (((invg‘𝐺)‘𝐴) + 𝑤) ∈ 𝑋) → ((𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)) ∈ 𝑆 ↔ ((((invg‘𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆))
30	10, 28, 29	syl2anc 411	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)) ∈ 𝑆 ↔ ((((invg‘𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆))
31	27, 30	mpbid 147	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((((invg‘𝐺)‘𝐴) + 𝑤) + 𝐴) ∈ 𝑆)
32	18, 31	eqeltrrd 2307	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆)
33	6, 16, 5, 15, 11	grpcld 13542	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝑤 + 𝐴) ∈ 𝑋)
34	6, 16, 5, 12, 33	grpcld 13542	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑋)
35	6, 16, 5, 11, 34	grpcld 13542	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) ∈ 𝑋)
36		conjghm.m	. . . . . . . . 9 ⊢ − = (-g‘𝐺)
37	6, 36	grpsubcl 13608	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴) ∈ 𝑋)
38	5, 35, 11, 37	syl3anc 1271	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴) ∈ 𝑋)
39	1, 3, 32, 38	fvmptd3 5727	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐹‘(((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) = ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴))
40	20	oveq1d 6015	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + (𝑤 + 𝐴)) = ((0g‘𝐺) + (𝑤 + 𝐴)))
41	6, 16	grpass 13537	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ ((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ (𝑤 + 𝐴) ∈ 𝑋)) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + (𝑤 + 𝐴)) = (𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))))
42	5, 11, 12, 33, 41	syl13anc 1273	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + (𝑤 + 𝐴)) = (𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))))
43	6, 16, 19, 5, 33	grplidd 13561	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((0g‘𝐺) + (𝑤 + 𝐴)) = (𝑤 + 𝐴))
44	40, 42, 43	3eqtr3d 2270	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) = (𝑤 + 𝐴))
45	44	oveq1d 6015	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴) = ((𝑤 + 𝐴) − 𝐴))
46	6, 16, 36	grppncan 13619	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑤 + 𝐴) − 𝐴) = 𝑤)
47	5, 15, 11, 46	syl3anc 1271	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝑤 + 𝐴) − 𝐴) = 𝑤)
48	39, 45, 47	3eqtrd 2266	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐹‘(((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) = 𝑤)
49	5	adantr 276	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝐺 ∈ Grp)
50	11	adantr 276	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ 𝑋)
51	14	adantr 276	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝑆 ⊆ 𝑋)
52	51	sselda 3224	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑋)
53	6, 16, 49, 50, 52	grpcld 13542	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ 𝑆) → (𝐴 + 𝑥) ∈ 𝑋)
54	6, 36	grpsubcl 13608	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 + 𝑥) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐴 + 𝑥) − 𝐴) ∈ 𝑋)
55	49, 53, 50, 54	syl3anc 1271	. . . . . . . 8 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + 𝑥) − 𝐴) ∈ 𝑋)
56	55	ralrimiva 2603	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ∀𝑥 ∈ 𝑆 ((𝐴 + 𝑥) − 𝐴) ∈ 𝑋)
57	1	fnmpt 5449	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑆 ((𝐴 + 𝑥) − 𝐴) ∈ 𝑋 → 𝐹 Fn 𝑆)
58	56, 57	syl 14	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝐹 Fn 𝑆)
59		fnfvelrn 5766	. . . . . 6 ⊢ ((𝐹 Fn 𝑆 ∧ (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆) → (𝐹‘(((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) ∈ ran 𝐹)
60	58, 32, 59	syl2anc 411	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐹‘(((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) ∈ ran 𝐹)
61	48, 60	eqeltrrd 2307	. . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝑤 ∈ ran 𝐹)
62	61	ex 115	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → (𝑤 ∈ 𝑆 → 𝑤 ∈ ran 𝐹))
63	62	ssrdv 3230	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝑆 ⊆ ran 𝐹)
64	4	ad2antrr 488	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → 𝐺 ∈ Grp)
65		simplr 528	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ 𝑁)
66	9, 65	sselid 3222	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ 𝑋)
67	14	sselda 3224	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑋)
68	6, 16, 36	grpaddsubass 13618	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴 + 𝑥) − 𝐴) = (𝐴 + (𝑥 − 𝐴)))
69	64, 66, 67, 66, 68	syl13anc 1273	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + 𝑥) − 𝐴) = (𝐴 + (𝑥 − 𝐴)))
70	6, 16, 36	grpnpcan 13620	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑥 − 𝐴) + 𝐴) = 𝑥)
71	64, 67, 66, 70	syl3anc 1271	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → ((𝑥 − 𝐴) + 𝐴) = 𝑥)
72		simpr 110	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
73	71, 72	eqeltrd 2306	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → ((𝑥 − 𝐴) + 𝐴) ∈ 𝑆)
74	6, 36	grpsubcl 13608	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑥 − 𝐴) ∈ 𝑋)
75	64, 67, 66, 74	syl3anc 1271	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → (𝑥 − 𝐴) ∈ 𝑋)
76	8	nmzbi 13741	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑁 ∧ (𝑥 − 𝐴) ∈ 𝑋) → ((𝐴 + (𝑥 − 𝐴)) ∈ 𝑆 ↔ ((𝑥 − 𝐴) + 𝐴) ∈ 𝑆))
77	65, 75, 76	syl2anc 411	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + (𝑥 − 𝐴)) ∈ 𝑆 ↔ ((𝑥 − 𝐴) + 𝐴) ∈ 𝑆))
78	73, 77	mpbird 167	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → (𝐴 + (𝑥 − 𝐴)) ∈ 𝑆)
79	69, 78	eqeltrd 2306	. . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + 𝑥) − 𝐴) ∈ 𝑆)
80	79, 1	fmptd 5788	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝐹:𝑆⟶𝑆)
81	80	frnd 5482	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → ran 𝐹 ⊆ 𝑆)
82	63, 81	eqssd 3241	1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝑆 = ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∀wral 2508  {crab 2512   ⊆ wss 3197   ↦ cmpt 4144  ran crn 4719   Fn wfn 5312  ‘cfv 5317  (class class class)co 6000  Basecbs 13027  +_gcplusg 13105  0_gc0g 13284  Grpcgrp 13528  inv_gcminusg 13529  -_gcsg 13530  SubGrpcsubg 13699

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-inn 9107  df-2 9165  df-ndx 13030  df-slot 13031  df-base 13033  df-plusg 13118  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-grp 13531  df-minusg 13532  df-sbg 13533  df-subg 13702

This theorem is referenced by:  conjnmzb  13812  conjnsg  13813