Theorem cntzsubg 19311

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 18913	. . 3 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
2		cntzrec.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
3		cntzrec.z	. . . 4 ⊢ 𝑍 = (Cntz‘𝑀)
4	2, 3	cntzsubm 19310	. . 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 19300	. . . . . . . . . . . . 13 ⊢ (𝑍‘𝑆) ⊆ 𝐵
8		simprl 771	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ (𝑍‘𝑆))
9	7, 8	sselid 3920	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝐵)
10		eqid 2737	. . . . . . . . . . . . 13 ⊢ (invg‘𝑀) = (invg‘𝑀)
11	2, 10	grpinvcl 18960	. . . . . . . . . . . 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 3917	. . . . . . . . . . . 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 18914	. . . . . . . . . . . 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 18915	. . . . . . . . . . 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 18915	. . . . . . . . . . . 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 7380	. . . . . . . . . 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 19301	. . . . . . . . . . . 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 7379	. . . . . . . . . 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 7380	. . . . . . . . 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 18914	. . . . . . . . . . 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 18915	. . . . . . . . . 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 18915	. . . . . . . . . . 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 7380	. . . . . . . . 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 18963	. . . . . . . . . 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 7380	. . . . . . . 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 18914	. . . . . . . . . 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 18941	. . . . . . . . 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 18962	. . . . . . . . . 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 7379	. . . . . . . 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 18940	. . . . . . . . 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 3130	. . . 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 3920	. . . . . 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 19295	. . . . 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 3130	. 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → ∀𝑥 ∈ (𝑍‘𝑆)((invg‘𝑀)‘𝑥) ∈ (𝑍‘𝑆))
65	10	issubg3 19117	. . 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 3890  ‘cfv 6496  (class class class)co 7364  Basecbs 17176  +_gcplusg 17217  0_gc0g 17399  Mndcmnd 18699  SubMndcsubmnd 18747  Grpcgrp 18906  inv_gcminusg 18907  SubGrpcsubg 19093  Cntzccntz 19287

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5306  ax-pr 5374  ax-un 7686  ax-cnex 11091  ax-resscn 11092  ax-1cn 11093  ax-icn 11094  ax-addcl 11095  ax-addrcl 11096  ax-mulcl 11097  ax-mulrcl 11098  ax-mulcom 11099  ax-addass 11100  ax-mulass 11101  ax-distr 11102  ax-i2m1 11103  ax-1ne0 11104  ax-1rid 11105  ax-rnegex 11106  ax-rrecex 11107  ax-cnre 11108  ax-pre-lttri 11109  ax-pre-lttrn 11110  ax-pre-ltadd 11111  ax-pre-mulgt0 11112

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 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5523  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5581  df-we 5583  df-xp 5634  df-rel 5635  df-cnv 5636  df-co 5637  df-dm 5638  df-rn 5639  df-res 5640  df-ima 5641  df-pred 6263  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7321  df-ov 7367  df-oprab 7368  df-mpo 7369  df-om 7815  df-2nd 7940  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11178  df-mnf 11179  df-xr 11180  df-ltxr 11181  df-le 11182  df-sub 11376  df-neg 11377  df-nn 12172  df-2 12241  df-sets 17131  df-slot 17149  df-ndx 17161  df-base 17177  df-ress 17198  df-plusg 17230  df-0g 17401  df-mgm 18605  df-sgrp 18684  df-mnd 18700  df-submnd 18749  df-grp 18909  df-minusg 18910  df-subg 19096  df-cntz 19289

This theorem is referenced by:  cntrnsg  19316  lsmcntz  19651  cntrabl  19815  dprdz  20004  dprdcntz2  20012  dmdprdsplit2lem  20019  cntzsdrg  20776