Theorem cntzsubg 19272

Description: Centralizers in a group are subgroups. (Contributed by Stefan O'Rear, 6-Sep-2015.)

Hypotheses

Ref	Expression
cntzrec.b	⊢ 𝐵 = (Base‘𝑀)
cntzrec.z	⊢ 𝑍 = (Cntz‘𝑀)

Assertion

Ref	Expression
cntzsubg	⊢ ((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubGrp‘𝑀))

Proof of Theorem cntzsubg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpmnd 18874	. . 3 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
2		cntzrec.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
3		cntzrec.z	. . . 4 ⊢ 𝑍 = (Cntz‘𝑀)
4	2, 3	cntzsubm 19271	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀))
5	1, 4	sylan 581	. 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀))
6		simpll 767	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → 𝑀 ∈ Grp)
7	2, 3	cntzssv 19261	. . . . . . . . . . . . 13 ⊢ (𝑍‘𝑆) ⊆ 𝐵
8		simprl 771	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ (𝑍‘𝑆))
9	7, 8	sselid 3932	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝐵)
10		eqid 2737	. . . . . . . . . . . . 13 ⊢ (invg‘𝑀) = (invg‘𝑀)
11	2, 10	grpinvcl 18921	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝑀)‘𝑥) ∈ 𝐵)
12	6, 9, 11	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → ((invg‘𝑀)‘𝑥) ∈ 𝐵)
13		ssel2 3929	. . . . . . . . . . . 12 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝐵)
14	13	ad2ant2l 747	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝐵)
15		eqid 2737	. . . . . . . . . . . . 13 ⊢ (+g‘𝑀) = (+g‘𝑀)
16	2, 15	grpcl 18875	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ ((invg‘𝑀)‘𝑥) ∈ 𝐵) → (𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥)) ∈ 𝐵)
17	6, 9, 12, 16	syl3anc 1374	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥)) ∈ 𝐵)
18	2, 15	grpass 18876	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ Grp ∧ (((invg‘𝑀)‘𝑥) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥)) ∈ 𝐵)) → ((((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦)(+g‘𝑀)(𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥))) = (((invg‘𝑀)‘𝑥)(+g‘𝑀)(𝑦(+g‘𝑀)(𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥)))))
19	6, 12, 14, 17, 18	syl13anc 1375	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → ((((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦)(+g‘𝑀)(𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥))) = (((invg‘𝑀)‘𝑥)(+g‘𝑀)(𝑦(+g‘𝑀)(𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥)))))
20	2, 15	grpass 18876	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ Grp ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ ((invg‘𝑀)‘𝑥) ∈ 𝐵)) → ((𝑦(+g‘𝑀)𝑥)(+g‘𝑀)((invg‘𝑀)‘𝑥)) = (𝑦(+g‘𝑀)(𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥))))
21	6, 14, 9, 12, 20	syl13anc 1375	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → ((𝑦(+g‘𝑀)𝑥)(+g‘𝑀)((invg‘𝑀)‘𝑥)) = (𝑦(+g‘𝑀)(𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥))))
22	21	oveq2d 7376	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (((invg‘𝑀)‘𝑥)(+g‘𝑀)((𝑦(+g‘𝑀)𝑥)(+g‘𝑀)((invg‘𝑀)‘𝑥))) = (((invg‘𝑀)‘𝑥)(+g‘𝑀)(𝑦(+g‘𝑀)(𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥)))))
23	19, 22	eqtr4d 2775	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → ((((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦)(+g‘𝑀)(𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥))) = (((invg‘𝑀)‘𝑥)(+g‘𝑀)((𝑦(+g‘𝑀)𝑥)(+g‘𝑀)((invg‘𝑀)‘𝑥))))
24	15, 3	cntzi 19262	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))
25	24	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))
26	25	oveq1d 7375	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → ((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)((invg‘𝑀)‘𝑥)) = ((𝑦(+g‘𝑀)𝑥)(+g‘𝑀)((invg‘𝑀)‘𝑥)))
27	26	oveq2d 7376	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (((invg‘𝑀)‘𝑥)(+g‘𝑀)((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)((invg‘𝑀)‘𝑥))) = (((invg‘𝑀)‘𝑥)(+g‘𝑀)((𝑦(+g‘𝑀)𝑥)(+g‘𝑀)((invg‘𝑀)‘𝑥))))
28	23, 27	eqtr4d 2775	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → ((((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦)(+g‘𝑀)(𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥))) = (((invg‘𝑀)‘𝑥)(+g‘𝑀)((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)((invg‘𝑀)‘𝑥))))
29	2, 15	grpcl 18875	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ ((invg‘𝑀)‘𝑥) ∈ 𝐵) → (𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)) ∈ 𝐵)
30	6, 14, 12, 29	syl3anc 1374	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)) ∈ 𝐵)
31	2, 15	grpass 18876	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Grp ∧ (((invg‘𝑀)‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ (𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)) ∈ 𝐵)) → ((((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑥)(+g‘𝑀)(𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥))) = (((invg‘𝑀)‘𝑥)(+g‘𝑀)(𝑥(+g‘𝑀)(𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)))))
32	6, 12, 9, 30, 31	syl13anc 1375	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → ((((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑥)(+g‘𝑀)(𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥))) = (((invg‘𝑀)‘𝑥)(+g‘𝑀)(𝑥(+g‘𝑀)(𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)))))
33	2, 15	grpass 18876	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ ((invg‘𝑀)‘𝑥) ∈ 𝐵)) → ((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)((invg‘𝑀)‘𝑥)) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥))))
34	6, 9, 14, 12, 33	syl13anc 1375	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → ((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)((invg‘𝑀)‘𝑥)) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥))))
35	34	oveq2d 7376	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (((invg‘𝑀)‘𝑥)(+g‘𝑀)((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)((invg‘𝑀)‘𝑥))) = (((invg‘𝑀)‘𝑥)(+g‘𝑀)(𝑥(+g‘𝑀)(𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)))))
36	32, 35	eqtr4d 2775	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → ((((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑥)(+g‘𝑀)(𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥))) = (((invg‘𝑀)‘𝑥)(+g‘𝑀)((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)((invg‘𝑀)‘𝑥))))
37	28, 36	eqtr4d 2775	. . . . . . 7 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → ((((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦)(+g‘𝑀)(𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥))) = ((((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑥)(+g‘𝑀)(𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥))))
38		eqid 2737	. . . . . . . . . . 11 ⊢ (0g‘𝑀) = (0g‘𝑀)
39	2, 15, 38, 10	grprinv 18924	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥)) = (0g‘𝑀))
40	6, 9, 39	syl2anc 585	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥)) = (0g‘𝑀))
41	40	oveq2d 7376	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → ((((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦)(+g‘𝑀)(𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥))) = ((((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦)(+g‘𝑀)(0g‘𝑀)))
42	2, 15	grpcl 18875	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Grp ∧ ((invg‘𝑀)‘𝑥) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦) ∈ 𝐵)
43	6, 12, 14, 42	syl3anc 1374	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦) ∈ 𝐵)
44	2, 15, 38	grprid 18902	. . . . . . . . 9 ⊢ ((𝑀 ∈ Grp ∧ (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦) ∈ 𝐵) → ((((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦)(+g‘𝑀)(0g‘𝑀)) = (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦))
45	6, 43, 44	syl2anc 585	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → ((((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦)(+g‘𝑀)(0g‘𝑀)) = (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦))
46	41, 45	eqtrd 2772	. . . . . . 7 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → ((((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦)(+g‘𝑀)(𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥))) = (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦))
47	2, 15, 38, 10	grplinv 18923	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑥) = (0g‘𝑀))
48	6, 9, 47	syl2anc 585	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑥) = (0g‘𝑀))
49	48	oveq1d 7375	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → ((((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑥)(+g‘𝑀)(𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥))) = ((0g‘𝑀)(+g‘𝑀)(𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥))))
50	2, 15, 38	grplid 18901	. . . . . . . . 9 ⊢ ((𝑀 ∈ Grp ∧ (𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)) ∈ 𝐵) → ((0g‘𝑀)(+g‘𝑀)(𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥))) = (𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)))
51	6, 30, 50	syl2anc 585	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → ((0g‘𝑀)(+g‘𝑀)(𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥))) = (𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)))
52	49, 51	eqtrd 2772	. . . . . . 7 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → ((((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑥)(+g‘𝑀)(𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥))) = (𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)))
53	37, 46, 52	3eqtr3d 2780	. . . . . 6 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)))
54	53	anassrs 467	. . . . 5 ⊢ ((((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑦 ∈ 𝑆) → (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)))
55	54	ralrimiva 3129	. . . 4 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → ∀𝑦 ∈ 𝑆 (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)))
56		simplr 769	. . . . 5 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → 𝑆 ⊆ 𝐵)
57		simpll 767	. . . . . 6 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → 𝑀 ∈ Grp)
58		simpr 484	. . . . . . 7 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → 𝑥 ∈ (𝑍‘𝑆))
59	7, 58	sselid 3932	. . . . . 6 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → 𝑥 ∈ 𝐵)
60	57, 59, 11	syl2anc 585	. . . . 5 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → ((invg‘𝑀)‘𝑥) ∈ 𝐵)
61	2, 15, 3	cntzel 19256	. . . . 5 ⊢ ((𝑆 ⊆ 𝐵 ∧ ((invg‘𝑀)‘𝑥) ∈ 𝐵) → (((invg‘𝑀)‘𝑥) ∈ (𝑍‘𝑆) ↔ ∀𝑦 ∈ 𝑆 (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥))))
62	56, 60, 61	syl2anc 585	. . . 4 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → (((invg‘𝑀)‘𝑥) ∈ (𝑍‘𝑆) ↔ ∀𝑦 ∈ 𝑆 (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥))))
63	55, 62	mpbird 257	. . 3 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → ((invg‘𝑀)‘𝑥) ∈ (𝑍‘𝑆))
64	63	ralrimiva 3129	. 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → ∀𝑥 ∈ (𝑍‘𝑆)((invg‘𝑀)‘𝑥) ∈ (𝑍‘𝑆))
65	10	issubg3 19078	. . 3 ⊢ (𝑀 ∈ Grp → ((𝑍‘𝑆) ∈ (SubGrp‘𝑀) ↔ ((𝑍‘𝑆) ∈ (SubMnd‘𝑀) ∧ ∀𝑥 ∈ (𝑍‘𝑆)((invg‘𝑀)‘𝑥) ∈ (𝑍‘𝑆))))
66	65	adantr 480	. 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → ((𝑍‘𝑆) ∈ (SubGrp‘𝑀) ↔ ((𝑍‘𝑆) ∈ (SubMnd‘𝑀) ∧ ∀𝑥 ∈ (𝑍‘𝑆)((invg‘𝑀)‘𝑥) ∈ (𝑍‘𝑆))))
67	5, 64, 66	mpbir2and 714	1 ⊢ ((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubGrp‘𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3902  ‘cfv 6493  (class class class)co 7360  Basecbs 17140  +_gcplusg 17181  0_gc0g 17363  Mndcmnd 18663  SubMndcsubmnd 18711  Grpcgrp 18867  inv_gcminusg 18868  SubGrpcsubg 19054  Cntzccntz 19248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-sets 17095  df-slot 17113  df-ndx 17125  df-base 17141  df-ress 17162  df-plusg 17194  df-0g 17365  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-submnd 18713  df-grp 18870  df-minusg 18871  df-subg 19057  df-cntz 19250

This theorem is referenced by:  cntrnsg  19277  lsmcntz  19612  cntrabl  19776  dprdz  19965  dprdcntz2  19973  dmdprdsplit2lem  19980  cntzsdrg  20739