Theorem cntzsubg 19553

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 19151	. . 3 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
2		cntzrec.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
3		cntzrec.z	. . . 4 ⊢ 𝑍 = (Cntz‘𝑀)
4	2, 3	cntzsubm 19552	. . 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 19542	. . . . . . . . . . . . 13 ⊢ (𝑍‘𝑆) ⊆ 𝐵
8		simprl 783	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ (𝑍‘𝑆))
9	7, 8	sselid 3906	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝐵)
10		eqid 2760	. . . . . . . . . . . . 13 ⊢ (invg‘𝑀) = (invg‘𝑀)
11	2, 10	grpinvcl 19198	. . . . . . . . . . . 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 3903	. . . . . . . . . . . 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 2760	. . . . . . . . . . . . 13 ⊢ (+g‘𝑀) = (+g‘𝑀)
16	2, 15	grpcl 19152	. . . . . . . . . . . 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 19153	. . . . . . . . . . 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 19153	. . . . . . . . . . . 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 7436	. . . . . . . . . 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 2798	. . . . . . . . 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 19543	. . . . . . . . . . . 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 7435	. . . . . . . . . 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 7436	. . . . . . . . 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 2798	. . . . . . . 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 19152	. . . . . . . . . . 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 19153	. . . . . . . . . 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 19153	. . . . . . . . . . 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 7436	. . . . . . . . 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 2798	. . . . . . . 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 2798	. . . . . . 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 2760	. . . . . . . . . . 11 ⊢ (0g‘𝑀) = (0g‘𝑀)
39	2, 15, 38, 10	grprinv 19201	. . . . . . . . . 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 7436	. . . . . . . 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 19152	. . . . . . . . . 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 19179	. . . . . . . . 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 2795	. . . . . . 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 19200	. . . . . . . . . 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 7435	. . . . . . . 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 19178	. . . . . . . . 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 2795	. . . . . . 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 2803	. . . . . 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 3154	. . . 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 3906	. . . . . 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 19537	. . . . 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 3154	. 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → ∀𝑥 ∈ (𝑍‘𝑆)((invg‘𝑀)‘𝑥) ∈ (𝑍‘𝑆))
65	10	issubg3 19355	. . 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 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  +_gcplusg 17428  0_gc0g 17610  Mndcmnd 18923  SubMndcsubmnd 18977  Grpcgrp 19144  inv_gcminusg 19145  SubGrpcsubg 19330  Cntzccntz 19529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-0g 17612  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-submnd 18979  df-grp 19147  df-minusg 19148  df-subg 19333  df-cntz 19531

This theorem is used by:  cntrnsg  19558  lsmcntz  19893  cntrabl  20057  dprdz  20246  dprdcntz2  20254  dmdprdsplit2lem  20261  cntzsdrg  21059