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Theorem cntzsubr 20390

Description: Centralizers in a ring are subrings. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)

**Hypotheses**
Ref	Expression
cntzsubr.b	⊢ 𝐵 = (Base‘𝑅)
cntzsubr.m	⊢ 𝑀 = (mulGrp‘𝑅)
cntzsubr.z	⊢ 𝑍 = (Cntz‘𝑀)

**Assertion**
Ref	Expression
cntzsubr	⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubRing‘𝑅))

Proof of Theorem cntzsubr Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cntzsubr.m	. . . . . 6 ⊢ 𝑀 = (mulGrp‘𝑅)
2		cntzsubr.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
3	1, 2	mgpbas 19987	. . . . 5 ⊢ 𝐵 = (Base‘𝑀)
4		cntzsubr.z	. . . . 5 ⊢ 𝑍 = (Cntz‘𝑀)
5	3, 4	cntzssv 19186	. . . 4 ⊢ (𝑍‘𝑆) ⊆ 𝐵
6	5	a1i 11	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ⊆ 𝐵)
7		simpll 765	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑧 ∈ 𝑆) → 𝑅 ∈ Ring)
8		ssel2 3976	. . . . . . . . 9 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝐵)
9	8	adantll 712	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝐵)
10		eqid 2732	. . . . . . . . 9 ⊢ (._r‘𝑅) = (._r‘𝑅)
11		eqid 2732	. . . . . . . . 9 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
12	2, 10, 11	ringlz 20100	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑧 ∈ 𝐵) → ((0_g‘𝑅)(._r‘𝑅)𝑧) = (0_g‘𝑅))
13	7, 9, 12	syl2anc 584	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑧 ∈ 𝑆) → ((0_g‘𝑅)(._r‘𝑅)𝑧) = (0_g‘𝑅))
14	2, 10, 11	ringrz 20101	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑧 ∈ 𝐵) → (𝑧(._r‘𝑅)(0_g‘𝑅)) = (0_g‘𝑅))
15	7, 9, 14	syl2anc 584	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑧 ∈ 𝑆) → (𝑧(._r‘𝑅)(0_g‘𝑅)) = (0_g‘𝑅))
16	13, 15	eqtr4d 2775	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑧 ∈ 𝑆) → ((0_g‘𝑅)(._r‘𝑅)𝑧) = (𝑧(._r‘𝑅)(0_g‘𝑅)))
17	16	ralrimiva 3146	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → ∀𝑧 ∈ 𝑆 ((0_g‘𝑅)(._r‘𝑅)𝑧) = (𝑧(._r‘𝑅)(0_g‘𝑅)))
18		simpr 485	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → 𝑆 ⊆ 𝐵)
19	2, 11	ring0cl 20077	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (0_g‘𝑅) ∈ 𝐵)
20	19	adantr 481	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (0_g‘𝑅) ∈ 𝐵)
21	1, 10	mgpplusg 19985	. . . . . . 7 ⊢ (._r‘𝑅) = (+_g‘𝑀)
22	3, 21, 4	cntzel 19181	. . . . . 6 ⊢ ((𝑆 ⊆ 𝐵 ∧ (0_g‘𝑅) ∈ 𝐵) → ((0_g‘𝑅) ∈ (𝑍‘𝑆) ↔ ∀𝑧 ∈ 𝑆 ((0_g‘𝑅)(._r‘𝑅)𝑧) = (𝑧(._r‘𝑅)(0_g‘𝑅))))
23	18, 20, 22	syl2anc 584	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → ((0_g‘𝑅) ∈ (𝑍‘𝑆) ↔ ∀𝑧 ∈ 𝑆 ((0_g‘𝑅)(._r‘𝑅)𝑧) = (𝑧(._r‘𝑅)(0_g‘𝑅))))
24	17, 23	mpbird 256	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (0_g‘𝑅) ∈ (𝑍‘𝑆))
25	24	ne0d 4334	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ≠ ∅)
26		simpl2 1192	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑥 ∈ (𝑍‘𝑆))
27		simpr 485	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑆)
28	21, 4	cntzi 19187	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ 𝑆) → (𝑥(._r‘𝑅)𝑧) = (𝑧(._r‘𝑅)𝑥))
29	26, 27, 28	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → (𝑥(._r‘𝑅)𝑧) = (𝑧(._r‘𝑅)𝑥))
30		simpl3 1193	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑦 ∈ (𝑍‘𝑆))
31	21, 4	cntzi 19187	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ 𝑆) → (𝑦(._r‘𝑅)𝑧) = (𝑧(._r‘𝑅)𝑦))
32	30, 27, 31	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → (𝑦(._r‘𝑅)𝑧) = (𝑧(._r‘𝑅)𝑦))
33	29, 32	oveq12d 7423	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → ((𝑥(._r‘𝑅)𝑧)(+_g‘𝑅)(𝑦(._r‘𝑅)𝑧)) = ((𝑧(._r‘𝑅)𝑥)(+_g‘𝑅)(𝑧(._r‘𝑅)𝑦)))
34		simpl1l 1224	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑅 ∈ Ring)
35	5, 26	sselid 3979	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑥 ∈ 𝐵)
36	5, 30	sselid 3979	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑦 ∈ 𝐵)
37		simp1r 1198	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → 𝑆 ⊆ 𝐵)
38	37	sselda 3981	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝐵)
39		eqid 2732	. . . . . . . . . . . 12 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
40	2, 39, 10	ringdir 20075	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥(+_g‘𝑅)𝑦)(._r‘𝑅)𝑧) = ((𝑥(._r‘𝑅)𝑧)(+_g‘𝑅)(𝑦(._r‘𝑅)𝑧)))
41	34, 35, 36, 38, 40	syl13anc 1372	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → ((𝑥(+_g‘𝑅)𝑦)(._r‘𝑅)𝑧) = ((𝑥(._r‘𝑅)𝑧)(+_g‘𝑅)(𝑦(._r‘𝑅)𝑧)))
42	2, 39, 10	ringdi 20074	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑧(._r‘𝑅)(𝑥(+_g‘𝑅)𝑦)) = ((𝑧(._r‘𝑅)𝑥)(+_g‘𝑅)(𝑧(._r‘𝑅)𝑦)))
43	34, 38, 35, 36, 42	syl13anc 1372	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → (𝑧(._r‘𝑅)(𝑥(+_g‘𝑅)𝑦)) = ((𝑧(._r‘𝑅)𝑥)(+_g‘𝑅)(𝑧(._r‘𝑅)𝑦)))
44	33, 41, 43	3eqtr4d 2782	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → ((𝑥(+_g‘𝑅)𝑦)(._r‘𝑅)𝑧) = (𝑧(._r‘𝑅)(𝑥(+_g‘𝑅)𝑦)))
45	44	ralrimiva 3146	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → ∀𝑧 ∈ 𝑆 ((𝑥(+_g‘𝑅)𝑦)(._r‘𝑅)𝑧) = (𝑧(._r‘𝑅)(𝑥(+_g‘𝑅)𝑦)))
46		simp1l 1197	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → 𝑅 ∈ Ring)
47		simp2 1137	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → 𝑥 ∈ (𝑍‘𝑆))
48	5, 47	sselid 3979	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → 𝑥 ∈ 𝐵)
49		simp3 1138	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → 𝑦 ∈ (𝑍‘𝑆))
50	5, 49	sselid 3979	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → 𝑦 ∈ 𝐵)
51	2, 39	ringacl 20088	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+_g‘𝑅)𝑦) ∈ 𝐵)
52	46, 48, 50, 51	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → (𝑥(+_g‘𝑅)𝑦) ∈ 𝐵)
53	3, 21, 4	cntzel 19181	. . . . . . . . 9 ⊢ ((𝑆 ⊆ 𝐵 ∧ (𝑥(+_g‘𝑅)𝑦) ∈ 𝐵) → ((𝑥(+_g‘𝑅)𝑦) ∈ (𝑍‘𝑆) ↔ ∀𝑧 ∈ 𝑆 ((𝑥(+_g‘𝑅)𝑦)(._r‘𝑅)𝑧) = (𝑧(._r‘𝑅)(𝑥(+_g‘𝑅)𝑦))))
54	37, 52, 53	syl2anc 584	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → ((𝑥(+_g‘𝑅)𝑦) ∈ (𝑍‘𝑆) ↔ ∀𝑧 ∈ 𝑆 ((𝑥(+_g‘𝑅)𝑦)(._r‘𝑅)𝑧) = (𝑧(._r‘𝑅)(𝑥(+_g‘𝑅)𝑦))))
55	45, 54	mpbird 256	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ (𝑍‘𝑆)) → (𝑥(+_g‘𝑅)𝑦) ∈ (𝑍‘𝑆))
56	55	3expa 1118	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑦 ∈ (𝑍‘𝑆)) → (𝑥(+_g‘𝑅)𝑦) ∈ (𝑍‘𝑆))
57	56	ralrimiva 3146	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → ∀𝑦 ∈ (𝑍‘𝑆)(𝑥(+_g‘𝑅)𝑦) ∈ (𝑍‘𝑆))
58	28	adantll 712	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → (𝑥(._r‘𝑅)𝑧) = (𝑧(._r‘𝑅)𝑥))
59	58	fveq2d 6892	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → ((inv_g‘𝑅)‘(𝑥(._r‘𝑅)𝑧)) = ((inv_g‘𝑅)‘(𝑧(._r‘𝑅)𝑥)))
60		eqid 2732	. . . . . . . . 9 ⊢ (inv_g‘𝑅) = (inv_g‘𝑅)
61		simplll 773	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑅 ∈ Ring)
62		simplr 767	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑥 ∈ (𝑍‘𝑆))
63	5, 62	sselid 3979	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑥 ∈ 𝐵)
64		simplr 767	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → 𝑆 ⊆ 𝐵)
65	64	sselda 3981	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝐵)
66	2, 10, 60, 61, 63, 65	ringmneg1 20109	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → (((inv_g‘𝑅)‘𝑥)(._r‘𝑅)𝑧) = ((inv_g‘𝑅)‘(𝑥(._r‘𝑅)𝑧)))
67	2, 10, 60, 61, 65, 63	ringmneg2 20110	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → (𝑧(._r‘𝑅)((inv_g‘𝑅)‘𝑥)) = ((inv_g‘𝑅)‘(𝑧(._r‘𝑅)𝑥)))
68	59, 66, 67	3eqtr4d 2782	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) ∧ 𝑧 ∈ 𝑆) → (((inv_g‘𝑅)‘𝑥)(._r‘𝑅)𝑧) = (𝑧(._r‘𝑅)((inv_g‘𝑅)‘𝑥)))
69	68	ralrimiva 3146	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → ∀𝑧 ∈ 𝑆 (((inv_g‘𝑅)‘𝑥)(._r‘𝑅)𝑧) = (𝑧(._r‘𝑅)((inv_g‘𝑅)‘𝑥)))
70		ringgrp 20054	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
71	70	ad2antrr 724	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → 𝑅 ∈ Grp)
72		simpr 485	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → 𝑥 ∈ (𝑍‘𝑆))
73	5, 72	sselid 3979	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → 𝑥 ∈ 𝐵)
74	2, 60	grpinvcl 18868	. . . . . . . 8 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((inv_g‘𝑅)‘𝑥) ∈ 𝐵)
75	71, 73, 74	syl2anc 584	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → ((inv_g‘𝑅)‘𝑥) ∈ 𝐵)
76	3, 21, 4	cntzel 19181	. . . . . . 7 ⊢ ((𝑆 ⊆ 𝐵 ∧ ((inv_g‘𝑅)‘𝑥) ∈ 𝐵) → (((inv_g‘𝑅)‘𝑥) ∈ (𝑍‘𝑆) ↔ ∀𝑧 ∈ 𝑆 (((inv_g‘𝑅)‘𝑥)(._r‘𝑅)𝑧) = (𝑧(._r‘𝑅)((inv_g‘𝑅)‘𝑥))))
77	64, 75, 76	syl2anc 584	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → (((inv_g‘𝑅)‘𝑥) ∈ (𝑍‘𝑆) ↔ ∀𝑧 ∈ 𝑆 (((inv_g‘𝑅)‘𝑥)(._r‘𝑅)𝑧) = (𝑧(._r‘𝑅)((inv_g‘𝑅)‘𝑥))))
78	69, 77	mpbird 256	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → ((inv_g‘𝑅)‘𝑥) ∈ (𝑍‘𝑆))
79	57, 78	jca 512	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ (𝑍‘𝑆)) → (∀𝑦 ∈ (𝑍‘𝑆)(𝑥(+_g‘𝑅)𝑦) ∈ (𝑍‘𝑆) ∧ ((inv_g‘𝑅)‘𝑥) ∈ (𝑍‘𝑆)))
80	79	ralrimiva 3146	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → ∀𝑥 ∈ (𝑍‘𝑆)(∀𝑦 ∈ (𝑍‘𝑆)(𝑥(+_g‘𝑅)𝑦) ∈ (𝑍‘𝑆) ∧ ((inv_g‘𝑅)‘𝑥) ∈ (𝑍‘𝑆)))
81	70	adantr 481	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → 𝑅 ∈ Grp)
82	2, 39, 60	issubg2 19015	. . . 4 ⊢ (𝑅 ∈ Grp → ((𝑍‘𝑆) ∈ (SubGrp‘𝑅) ↔ ((𝑍‘𝑆) ⊆ 𝐵 ∧ (𝑍‘𝑆) ≠ ∅ ∧ ∀𝑥 ∈ (𝑍‘𝑆)(∀𝑦 ∈ (𝑍‘𝑆)(𝑥(+_g‘𝑅)𝑦) ∈ (𝑍‘𝑆) ∧ ((inv_g‘𝑅)‘𝑥) ∈ (𝑍‘𝑆)))))
83	81, 82	syl 17	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → ((𝑍‘𝑆) ∈ (SubGrp‘𝑅) ↔ ((𝑍‘𝑆) ⊆ 𝐵 ∧ (𝑍‘𝑆) ≠ ∅ ∧ ∀𝑥 ∈ (𝑍‘𝑆)(∀𝑦 ∈ (𝑍‘𝑆)(𝑥(+_g‘𝑅)𝑦) ∈ (𝑍‘𝑆) ∧ ((inv_g‘𝑅)‘𝑥) ∈ (𝑍‘𝑆)))))
84	6, 25, 80, 83	mpbir3and 1342	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubGrp‘𝑅))
85	1	ringmgp 20055	. . 3 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd)
86	3, 4	cntzsubm 19196	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀))
87	85, 86	sylan 580	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀))
88	1	issubrg3 20384	. . 3 ⊢ (𝑅 ∈ Ring → ((𝑍‘𝑆) ∈ (SubRing‘𝑅) ↔ ((𝑍‘𝑆) ∈ (SubGrp‘𝑅) ∧ (𝑍‘𝑆) ∈ (SubMnd‘𝑀))))
89	88	adantr 481	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → ((𝑍‘𝑆) ∈ (SubRing‘𝑅) ↔ ((𝑍‘𝑆) ∈ (SubGrp‘𝑅) ∧ (𝑍‘𝑆) ∈ (SubMnd‘𝑀))))
90	84, 87, 89	mpbir2and 711	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubRing‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ⊆ wss 3947 ∅c0 4321 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 +_gcplusg 17193 ._rcmulr 17194 0_gc0g 17381 Mndcmnd 18621 SubMndcsubmnd 18666 Grpcgrp 18815 inv_gcminusg 18816 SubGrpcsubg 18994 Cntzccntz 19173 mulGrpcmgp 19981 Ringcrg 20049 SubRingcsubrg 20351

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-subg 18997 df-cntz 19175 df-mgp 19982 df-ur 19999 df-ring 20051 df-subrg 20353

This theorem is referenced by: cntzsdrg 20410 cntrcrng 32201

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