Theorem cntzsubg 19467

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 19065	. . 3 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
2		cntzrec.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
3		cntzrec.z	. . . 4 ⊢ 𝑍 = (Cntz‘𝑀)
4	2, 3	cntzsubm 19466	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀))
5	1, 4	sylan 592	. 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀))
6		simpll 779	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → 𝑀 ∈ Grp)
7	2, 3	cntzssv 19456	. . . . . . . . . . . . 13 ⊢ (𝑍‘𝑆) ⊆ 𝐵
8		simprl 783	. . . . . . . . . . . . 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 2762	. . . . . . . . . . . . 13 ⊢ (invg‘𝑀) = (invg‘𝑀)
11	2, 10	grpinvcl 19112	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝑀)‘𝑥) ∈ 𝐵)
12	6, 9, 11	syl2anc 596	. . . . . . . . . . 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 759	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝐵)
15		eqid 2762	. . . . . . . . . . . . 13 ⊢ (+g‘𝑀) = (+g‘𝑀)
16	2, 15	grpcl 19066	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ ((invg‘𝑀)‘𝑥) ∈ 𝐵) → (𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥)) ∈ 𝐵)
17	6, 9, 12, 16	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥)) ∈ 𝐵)
18	2, 15	grpass 19067	. . . . . . . . . . 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 1399	. . . . . . . . . 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 19067	. . . . . . . . . . . 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 1399	. . . . . . . . . . 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 7432	. . . . . . . . . 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 2800	. . . . . . . . 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 19457	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))
25	24	adantl 487	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))
26	25	oveq1d 7431	. . . . . . . . . 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 7432	. . . . . . . . 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 2800	. . . . . . . 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 19066	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ ((invg‘𝑀)‘𝑥) ∈ 𝐵) → (𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)) ∈ 𝐵)
30	6, 14, 12, 29	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)) ∈ 𝐵)
31	2, 15	grpass 19067	. . . . . . . . . 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 1399	. . . . . . . . 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 19067	. . . . . . . . . . 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 1399	. . . . . . . . . 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 7432	. . . . . . . . 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 2800	. . . . . . . 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 2800	. . . . . . 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 2762	. . . . . . . . . . 11 ⊢ (0g‘𝑀) = (0g‘𝑀)
39	2, 15, 38, 10	grprinv 19115	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥)) = (0g‘𝑀))
40	6, 9, 39	syl2anc 596	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥)) = (0g‘𝑀))
41	40	oveq2d 7432	. . . . . . . 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 19066	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Grp ∧ ((invg‘𝑀)‘𝑥) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦) ∈ 𝐵)
43	6, 12, 14, 42	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦) ∈ 𝐵)
44	2, 15, 38	grprid 19093	. . . . . . . . 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 596	. . . . . . . 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 2797	. . . . . . 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 19114	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑥) = (0g‘𝑀))
48	6, 9, 47	syl2anc 596	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑥) = (0g‘𝑀))
49	48	oveq1d 7431	. . . . . . . 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 19092	. . . . . . . . 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 596	. . . . . . . 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 2797	. . . . . . 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 2805	. . . . . 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 473	. . . . 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 3156	. . . 4 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → ∀𝑦 ∈ 𝑆 (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)))
56		simplr 781	. . . . 5 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → 𝑆 ⊆ 𝐵)
57		simpll 779	. . . . . 6 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → 𝑀 ∈ Grp)
58		simpr 490	. . . . . . 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 596	. . . . 5 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → ((invg‘𝑀)‘𝑥) ∈ 𝐵)
61	2, 15, 3	cntzel 19451	. . . . 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 596	. . . 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 260	. . 3 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → ((invg‘𝑀)‘𝑥) ∈ (𝑍‘𝑆))
64	63	ralrimiva 3156	. 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → ∀𝑥 ∈ (𝑍‘𝑆)((invg‘𝑀)‘𝑥) ∈ (𝑍‘𝑆))
65	10	issubg3 19269	. . 3 ⊢ (𝑀 ∈ Grp → ((𝑍‘𝑆) ∈ (SubGrp‘𝑀) ↔ ((𝑍‘𝑆) ∈ (SubMnd‘𝑀) ∧ ∀𝑥 ∈ (𝑍‘𝑆)((invg‘𝑀)‘𝑥) ∈ (𝑍‘𝑆))))
66	65	adantr 486	. 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 726	1 ⊢ ((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubGrp‘𝑀))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ⊆ wss 3902  ‘cfv 6537  (class class class)co 7416  Basecbs 17305  +_gcplusg 17346  0_gc0g 17528  Mndcmnd 18838  SubMndcsubmnd 18891  Grpcgrp 19058  inv_gcminusg 19059  SubGrpcsubg 19244  Cntzccntz 19443

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-sets 17260  df-slot 17278  df-ndx 17290  df-base 17306  df-ress 17327  df-plusg 17359  df-0g 17530  df-mgm 18734  df-sgrp 18823  df-mnd 18839  df-submnd 18893  df-grp 19061  df-minusg 19062  df-subg 19247  df-cntz 19445

This theorem is used by:  cntrnsg  19472  lsmcntz  19807  cntrabl  19971  dprdz  20160  dprdcntz2  20168  dmdprdsplit2lem  20175  cntzsdrg  20969