Theorem conjnmz 13349

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 5926	. . . . . . . 8 ⊢ (𝑥 = (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) → (𝐴 + 𝑥) = (𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))))
3	2	oveq1d 5933	. . . . . . 7 ⊢ (𝑥 = (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) → ((𝐴 + 𝑥) − 𝐴) = ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴))
4		subgrcl 13249	. . . . . . . . . 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 2193	. . . . . . . . . 10 ⊢ (invg‘𝐺) = (invg‘𝐺)
8		conjnmz.1	. . . . . . . . . . . 12 ⊢ 𝑁 = {𝑦 ∈ 𝑋 ∣ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}
9	8	ssrab3 3265	. . . . . . . . . . 11 ⊢ 𝑁 ⊆ 𝑋
10		simplr 528	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝐴 ∈ 𝑁)
11	9, 10	sselid 3177	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝐴 ∈ 𝑋)
12	6, 7, 5, 11	grpinvcld 13121	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
13	6	subgss 13244	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑋)
14	13	adantr 276	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝑆 ⊆ 𝑋)
15	14	sselda 3179	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → 𝑤 ∈ 𝑋)
16		conjghm.p	. . . . . . . . . 10 ⊢ + = (+g‘𝐺)
17	6, 16	grpass 13081	. . . . . . . . 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 1251	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((((invg‘𝐺)‘𝐴) + 𝑤) + 𝐴) = (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)))
19		eqid 2193	. . . . . . . . . . . . 13 ⊢ (0g‘𝐺) = (0g‘𝐺)
20	6, 16, 19, 7, 5, 11	grprinvd 13128	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐴 + ((invg‘𝐺)‘𝐴)) = (0g‘𝐺))
21	20	oveq1d 5933	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + 𝑤) = ((0g‘𝐺) + 𝑤))
22	6, 16	grpass 13081	. . . . . . . . . . . 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 1251	. . . . . . . . . . 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 13105	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((0g‘𝐺) + 𝑤) = 𝑤)
25	21, 23, 24	3eqtr3d 2234	. . . . . . . . . 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 2270	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐴 + (((invg‘𝐺)‘𝐴) + 𝑤)) ∈ 𝑆)
28	6, 16, 5, 12, 15	grpcld 13086	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (((invg‘𝐺)‘𝐴) + 𝑤) ∈ 𝑋)
29	8	nmzbi 13279	. . . . . . . . . 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 2271	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑆)
33	6, 16, 5, 15, 11	grpcld 13086	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝑤 + 𝐴) ∈ 𝑋)
34	6, 16, 5, 12, 33	grpcld 13086	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴)) ∈ 𝑋)
35	6, 16, 5, 11, 34	grpcld 13086	. . . . . . . 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 13152	. . . . . . . 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 1249	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴) ∈ 𝑋)
39	1, 3, 32, 38	fvmptd3 5651	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐹‘(((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) = ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴))
40	20	oveq1d 5933	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + ((invg‘𝐺)‘𝐴)) + (𝑤 + 𝐴)) = ((0g‘𝐺) + (𝑤 + 𝐴)))
41	6, 16	grpass 13081	. . . . . . . . 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 1251	. . . . . . . 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 13105	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((0g‘𝐺) + (𝑤 + 𝐴)) = (𝑤 + 𝐴))
44	40, 42, 43	3eqtr3d 2234	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → (𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) = (𝑤 + 𝐴))
45	44	oveq1d 5933	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝐴 + (((invg‘𝐺)‘𝐴) + (𝑤 + 𝐴))) − 𝐴) = ((𝑤 + 𝐴) − 𝐴))
46	6, 16, 36	grppncan 13163	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑤 + 𝐴) − 𝐴) = 𝑤)
47	5, 15, 11, 46	syl3anc 1249	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ((𝑤 + 𝐴) − 𝐴) = 𝑤)
48	39, 45, 47	3eqtrd 2230	. . . . 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 3179	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑋)
53	6, 16, 49, 50, 52	grpcld 13086	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ 𝑆) → (𝐴 + 𝑥) ∈ 𝑋)
54	6, 36	grpsubcl 13152	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 + 𝑥) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐴 + 𝑥) − 𝐴) ∈ 𝑋)
55	49, 53, 50, 54	syl3anc 1249	. . . . . . . 8 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + 𝑥) − 𝐴) ∈ 𝑋)
56	55	ralrimiva 2567	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑤 ∈ 𝑆) → ∀𝑥 ∈ 𝑆 ((𝐴 + 𝑥) − 𝐴) ∈ 𝑋)
57	1	fnmpt 5380	. . . . . . 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 5690	. . . . . 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 2271	. . . 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 3185	. 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 3177	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ 𝑋)
67	14	sselda 3179	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑋)
68	6, 16, 36	grpaddsubass 13162	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴 + 𝑥) − 𝐴) = (𝐴 + (𝑥 − 𝐴)))
69	64, 66, 67, 66, 68	syl13anc 1251	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + 𝑥) − 𝐴) = (𝐴 + (𝑥 − 𝐴)))
70	6, 16, 36	grpnpcan 13164	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑥 − 𝐴) + 𝐴) = 𝑥)
71	64, 67, 66, 70	syl3anc 1249	. . . . . . 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 2270	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → ((𝑥 − 𝐴) + 𝐴) ∈ 𝑆)
74	6, 36	grpsubcl 13152	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑥 − 𝐴) ∈ 𝑋)
75	64, 67, 66, 74	syl3anc 1249	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → (𝑥 − 𝐴) ∈ 𝑋)
76	8	nmzbi 13279	. . . . . . 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 2270	. . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + 𝑥) − 𝐴) ∈ 𝑆)
80	79, 1	fmptd 5712	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝐹:𝑆⟶𝑆)
81	80	frnd 5413	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → ran 𝐹 ⊆ 𝑆)
82	63, 81	eqssd 3196	1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝑆 = ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ∀wral 2472  {crab 2476   ⊆ wss 3153   ↦ cmpt 4090  ran crn 4660   Fn wfn 5249  ‘cfv 5254  (class class class)co 5918  Basecbs 12618  +_gcplusg 12695  0_gc0g 12867  Grpcgrp 13072  inv_gcminusg 13073  -_gcsg 13074  SubGrpcsubg 13237

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-sbg 13077  df-subg 13240

This theorem is referenced by:  conjnmzb  13350  conjnsg  13351