Theorem cntzsubg 17592

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 17252	. . 3 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
2		cntzrec.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
3		cntzrec.z	. . . 4 ⊢ 𝑍 = (Cntz‘𝑀)
4	2, 3	cntzsubm 17591	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀))
5	1, 4	sylan 487	. 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀))
6		simpll 786	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → 𝑀 ∈ Grp)
7	2, 3	cntzssv 17584	. . . . . . . . . . . . 13 ⊢ (𝑍‘𝑆) ⊆ 𝐵
8		simprl 790	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ (𝑍‘𝑆))
9	7, 8	sseldi 3566	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝐵)
10		eqid 2610	. . . . . . . . . . . . 13 ⊢ (invg‘𝑀) = (invg‘𝑀)
11	2, 10	grpinvcl 17290	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝑀)‘𝑥) ∈ 𝐵)
12	6, 9, 11	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → ((invg‘𝑀)‘𝑥) ∈ 𝐵)
13		ssel2 3563	. . . . . . . . . . . 12 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝐵)
14	13	ad2ant2l 778	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝐵)
15		eqid 2610	. . . . . . . . . . . . 13 ⊢ (+g‘𝑀) = (+g‘𝑀)
16	2, 15	grpcl 17253	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ ((invg‘𝑀)‘𝑥) ∈ 𝐵) → (𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥)) ∈ 𝐵)
17	6, 9, 12, 16	syl3anc 1318	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥)) ∈ 𝐵)
18	2, 15	grpass 17254	. . . . . . . . . . 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 1320	. . . . . . . . . 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 17254	. . . . . . . . . . . 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 1320	. . . . . . . . . . 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 6565	. . . . . . . . . 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 2647	. . . . . . . . 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 17585	. . . . . . . . . . . 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 6564	. . . . . . . . . 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 6565	. . . . . . . . 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 2647	. . . . . . . 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 17253	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ ((invg‘𝑀)‘𝑥) ∈ 𝐵) → (𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)) ∈ 𝐵)
30	6, 14, 12, 29	syl3anc 1318	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)) ∈ 𝐵)
31	2, 15	grpass 17254	. . . . . . . . . 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 1320	. . . . . . . . 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 17254	. . . . . . . . . . 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 1320	. . . . . . . . . 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 6565	. . . . . . . . 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 2647	. . . . . . . 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 2647	. . . . . . 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 2610	. . . . . . . . . . 11 ⊢ (0g‘𝑀) = (0g‘𝑀)
39	2, 15, 38, 10	grprinv 17292	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥)) = (0g‘𝑀))
40	6, 9, 39	syl2anc 691	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (𝑥(+g‘𝑀)((invg‘𝑀)‘𝑥)) = (0g‘𝑀))
41	40	oveq2d 6565	. . . . . . . 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 17253	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Grp ∧ ((invg‘𝑀)‘𝑥) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦) ∈ 𝐵)
43	6, 12, 14, 42	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦) ∈ 𝐵)
44	2, 15, 38	grprid 17276	. . . . . . . . 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 691	. . . . . . . 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 2644	. . . . . . 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 17291	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑥) = (0g‘𝑀))
48	6, 9, 47	syl2anc 691	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆)) → (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑥) = (0g‘𝑀))
49	48	oveq1d 6564	. . . . . . . 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 17275	. . . . . . . . 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 691	. . . . . . . 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 2644	. . . . . . 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 2652	. . . . . 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 678	. . . . 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 2949	. . . 4 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → ∀𝑦 ∈ 𝑆 (((invg‘𝑀)‘𝑥)(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)((invg‘𝑀)‘𝑥)))
56		simplr 788	. . . . 5 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → 𝑆 ⊆ 𝐵)
57		simpll 786	. . . . . 6 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → 𝑀 ∈ Grp)
58		simpr 476	. . . . . . 7 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → 𝑥 ∈ (𝑍‘𝑆))
59	7, 58	sseldi 3566	. . . . . 6 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → 𝑥 ∈ 𝐵)
60	57, 59, 11	syl2anc 691	. . . . 5 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → ((invg‘𝑀)‘𝑥) ∈ 𝐵)
61	2, 15, 3	cntzel 17579	. . . . 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 691	. . . 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 246	. . 3 ⊢ (((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → ((invg‘𝑀)‘𝑥) ∈ (𝑍‘𝑆))
64	63	ralrimiva 2949	. 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → ∀𝑥 ∈ (𝑍‘𝑆)((invg‘𝑀)‘𝑥) ∈ (𝑍‘𝑆))
65	10	issubg3 17435	. . 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 959	1 ⊢ ((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubGrp‘𝑀))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +_gcplusg 15768  0_gc0g 15923  Mndcmnd 17117  SubMndcsubmnd 17157  Grpcgrp 17245  inv_gcminusg 17246  SubGrpcsubg 17411  Cntzccntz 17571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-subg 17414  df-cntz 17573

This theorem is referenced by:  cntrnsg  17597  lsmcntz  17915  dprdz  18252  dprdcntz2  18260  dmdprdsplit2lem  18267  cntzsdrg  36791