Theorem ricdomn1 33373

Description: A ring isomorphism maps a domain to a domain. (Contributed by Thierry Arnoux, 4-May-2026.)

Assertion

Ref	Expression
ricdomn1	⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) → 𝑆 ∈ Domn)

Proof of Theorem ricdomn1

Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		domnnzr 20681	. . 3 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
2		ricnzr1 33372	. . 3 ⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ NzRing) → 𝑆 ∈ NzRing)
3	1, 2	sylan2 595	. 2 ⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) → 𝑆 ∈ NzRing)
4		ricsym 20480	. . . . . . . 8 ⊢ (𝑅 ≃𝑟 𝑆 → 𝑆 ≃𝑟 𝑅)
5		brric 20478	. . . . . . . 8 ⊢ (𝑆 ≃𝑟 𝑅 ↔ (𝑆 RingIso 𝑅) ≠ ∅)
6	4, 5	sylib 219	. . . . . . 7 ⊢ (𝑅 ≃𝑟 𝑆 → (𝑆 RingIso 𝑅) ≠ ∅)
7	6	ad4antr 734	. . . . . 6 ⊢ (((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) → (𝑆 RingIso 𝑅) ≠ ∅)
8		simpr 485	. . . . . . . . 9 ⊢ (((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) ∧ (𝑓‘𝑥) = (0g‘𝑅)) → (𝑓‘𝑥) = (0g‘𝑅))
9	8	fveq2d 6834	. . . . . . . 8 ⊢ (((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) ∧ (𝑓‘𝑥) = (0g‘𝑅)) → (◡𝑓‘(𝑓‘𝑥)) = (◡𝑓‘(0g‘𝑅)))
10		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
11		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	10, 11	rimf1o 20467	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) → 𝑓:(Base‘𝑆)–1-1-onto→(Base‘𝑅))
13	12	ad2antlr 729	. . . . . . . . 9 ⊢ (((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) ∧ (𝑓‘𝑥) = (0g‘𝑅)) → 𝑓:(Base‘𝑆)–1-1-onto→(Base‘𝑅))
14		simp-4r 785	. . . . . . . . . 10 ⊢ ((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) → 𝑥 ∈ (Base‘𝑆))
15	14	adantr 481	. . . . . . . . 9 ⊢ (((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) ∧ (𝑓‘𝑥) = (0g‘𝑅)) → 𝑥 ∈ (Base‘𝑆))
16		f1ocnvfv1 7223	. . . . . . . . 9 ⊢ ((𝑓:(Base‘𝑆)–1-1-onto→(Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑆)) → (◡𝑓‘(𝑓‘𝑥)) = 𝑥)
17	13, 15, 16	syl2anc 586	. . . . . . . 8 ⊢ (((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) ∧ (𝑓‘𝑥) = (0g‘𝑅)) → (◡𝑓‘(𝑓‘𝑥)) = 𝑥)
18		isrim0 20456	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) ↔ (𝑓 ∈ (𝑆 RingHom 𝑅) ∧ ◡𝑓 ∈ (𝑅 RingHom 𝑆)))
19	18	simprbi 498	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) → ◡𝑓 ∈ (𝑅 RingHom 𝑆))
20	19	ad2antlr 729	. . . . . . . . 9 ⊢ (((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) ∧ (𝑓‘𝑥) = (0g‘𝑅)) → ◡𝑓 ∈ (𝑅 RingHom 𝑆))
21		rhmghm 20457	. . . . . . . . 9 ⊢ (◡𝑓 ∈ (𝑅 RingHom 𝑆) → ◡𝑓 ∈ (𝑅 GrpHom 𝑆))
22		eqid 2736	. . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅)
23		eqid 2736	. . . . . . . . . 10 ⊢ (0g‘𝑆) = (0g‘𝑆)
24	22, 23	ghmid 19191	. . . . . . . . 9 ⊢ (◡𝑓 ∈ (𝑅 GrpHom 𝑆) → (◡𝑓‘(0g‘𝑅)) = (0g‘𝑆))
25	20, 21, 24	3syl 18	. . . . . . . 8 ⊢ (((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) ∧ (𝑓‘𝑥) = (0g‘𝑅)) → (◡𝑓‘(0g‘𝑅)) = (0g‘𝑆))
26	9, 17, 25	3eqtr3d 2779	. . . . . . 7 ⊢ (((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) ∧ (𝑓‘𝑥) = (0g‘𝑅)) → 𝑥 = (0g‘𝑆))
27		simpr 485	. . . . . . . . 9 ⊢ (((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) ∧ (𝑓‘𝑦) = (0g‘𝑅)) → (𝑓‘𝑦) = (0g‘𝑅))
28	27	fveq2d 6834	. . . . . . . 8 ⊢ (((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) ∧ (𝑓‘𝑦) = (0g‘𝑅)) → (◡𝑓‘(𝑓‘𝑦)) = (◡𝑓‘(0g‘𝑅)))
29	12	ad2antlr 729	. . . . . . . . 9 ⊢ (((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) ∧ (𝑓‘𝑦) = (0g‘𝑅)) → 𝑓:(Base‘𝑆)–1-1-onto→(Base‘𝑅))
30		simpllr 777	. . . . . . . . . 10 ⊢ ((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) → 𝑦 ∈ (Base‘𝑆))
31	30	adantr 481	. . . . . . . . 9 ⊢ (((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) ∧ (𝑓‘𝑦) = (0g‘𝑅)) → 𝑦 ∈ (Base‘𝑆))
32		f1ocnvfv1 7223	. . . . . . . . 9 ⊢ ((𝑓:(Base‘𝑆)–1-1-onto→(Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → (◡𝑓‘(𝑓‘𝑦)) = 𝑦)
33	29, 31, 32	syl2anc 586	. . . . . . . 8 ⊢ (((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) ∧ (𝑓‘𝑦) = (0g‘𝑅)) → (◡𝑓‘(𝑓‘𝑦)) = 𝑦)
34	19	ad2antlr 729	. . . . . . . . 9 ⊢ (((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) ∧ (𝑓‘𝑦) = (0g‘𝑅)) → ◡𝑓 ∈ (𝑅 RingHom 𝑆))
35	34, 21, 24	3syl 18	. . . . . . . 8 ⊢ (((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) ∧ (𝑓‘𝑦) = (0g‘𝑅)) → (◡𝑓‘(0g‘𝑅)) = (0g‘𝑆))
36	28, 33, 35	3eqtr3d 2779	. . . . . . 7 ⊢ (((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) ∧ (𝑓‘𝑦) = (0g‘𝑅)) → 𝑦 = (0g‘𝑆))
37		simp-5r 787	. . . . . . . 8 ⊢ ((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) → 𝑅 ∈ Domn)
38		rimrhm 20468	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) → 𝑓 ∈ (𝑆 RingHom 𝑅))
39	10, 11	rhmf 20458	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑆 RingHom 𝑅) → 𝑓:(Base‘𝑆)⟶(Base‘𝑅))
40	38, 39	syl 17	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) → 𝑓:(Base‘𝑆)⟶(Base‘𝑅))
41	40	adantl 482	. . . . . . . . 9 ⊢ ((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) → 𝑓:(Base‘𝑆)⟶(Base‘𝑅))
42	41, 14	ffvelcdmd 7029	. . . . . . . 8 ⊢ ((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) → (𝑓‘𝑥) ∈ (Base‘𝑅))
43	41, 30	ffvelcdmd 7029	. . . . . . . 8 ⊢ ((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) → (𝑓‘𝑦) ∈ (Base‘𝑅))
44		simplr 770	. . . . . . . . . 10 ⊢ ((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) → (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆))
45	44	fveq2d 6834	. . . . . . . . 9 ⊢ ((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) → (𝑓‘(𝑥(.r‘𝑆)𝑦)) = (𝑓‘(0g‘𝑆)))
46	38	adantl 482	. . . . . . . . . 10 ⊢ ((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) → 𝑓 ∈ (𝑆 RingHom 𝑅))
47		eqid 2736	. . . . . . . . . . 11 ⊢ (.r‘𝑆) = (.r‘𝑆)
48		eqid 2736	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
49	10, 47, 48	rhmmul 20460	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝑆 RingHom 𝑅) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑓‘(𝑥(.r‘𝑆)𝑦)) = ((𝑓‘𝑥)(.r‘𝑅)(𝑓‘𝑦)))
50	46, 14, 30, 49	syl3anc 1375	. . . . . . . . 9 ⊢ ((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) → (𝑓‘(𝑥(.r‘𝑆)𝑦)) = ((𝑓‘𝑥)(.r‘𝑅)(𝑓‘𝑦)))
51		rhmghm 20457	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑆 RingHom 𝑅) → 𝑓 ∈ (𝑆 GrpHom 𝑅))
52	23, 22	ghmid 19191	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑆 GrpHom 𝑅) → (𝑓‘(0g‘𝑆)) = (0g‘𝑅))
53	46, 51, 52	3syl 18	. . . . . . . . 9 ⊢ ((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) → (𝑓‘(0g‘𝑆)) = (0g‘𝑅))
54	45, 50, 53	3eqtr3d 2779	. . . . . . . 8 ⊢ ((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) → ((𝑓‘𝑥)(.r‘𝑅)(𝑓‘𝑦)) = (0g‘𝑅))
55	11, 48, 22	domneq0 20683	. . . . . . . . 9 ⊢ ((𝑅 ∈ Domn ∧ (𝑓‘𝑥) ∈ (Base‘𝑅) ∧ (𝑓‘𝑦) ∈ (Base‘𝑅)) → (((𝑓‘𝑥)(.r‘𝑅)(𝑓‘𝑦)) = (0g‘𝑅) ↔ ((𝑓‘𝑥) = (0g‘𝑅) ∨ (𝑓‘𝑦) = (0g‘𝑅))))
56	55	biimpa 477	. . . . . . . 8 ⊢ (((𝑅 ∈ Domn ∧ (𝑓‘𝑥) ∈ (Base‘𝑅) ∧ (𝑓‘𝑦) ∈ (Base‘𝑅)) ∧ ((𝑓‘𝑥)(.r‘𝑅)(𝑓‘𝑦)) = (0g‘𝑅)) → ((𝑓‘𝑥) = (0g‘𝑅) ∨ (𝑓‘𝑦) = (0g‘𝑅)))
57	37, 42, 43, 54, 56	syl31anc 1377	. . . . . . 7 ⊢ ((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) → ((𝑓‘𝑥) = (0g‘𝑅) ∨ (𝑓‘𝑦) = (0g‘𝑅)))
58	26, 36, 57	orim12da 32548	. . . . . 6 ⊢ ((((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) ∧ 𝑓 ∈ (𝑆 RingIso 𝑅)) → (𝑥 = (0g‘𝑆) ∨ 𝑦 = (0g‘𝑆)))
59	7, 58	n0limd 32562	. . . . 5 ⊢ (((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) → (𝑥 = (0g‘𝑆) ∨ 𝑦 = (0g‘𝑆)))
60	59	ex 413	. . . 4 ⊢ ((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝑥(.r‘𝑆)𝑦) = (0g‘𝑆) → (𝑥 = (0g‘𝑆) ∨ 𝑦 = (0g‘𝑆))))
61	60	anasss 467	. . 3 ⊢ (((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝑥(.r‘𝑆)𝑦) = (0g‘𝑆) → (𝑥 = (0g‘𝑆) ∨ 𝑦 = (0g‘𝑆))))
62	61	ralrimivva 3179	. 2 ⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)((𝑥(.r‘𝑆)𝑦) = (0g‘𝑆) → (𝑥 = (0g‘𝑆) ∨ 𝑦 = (0g‘𝑆))))
63	10, 47, 23	isdomn 20680	. 2 ⊢ (𝑆 ∈ Domn ↔ (𝑆 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)((𝑥(.r‘𝑆)𝑦) = (0g‘𝑆) → (𝑥 = (0g‘𝑆) ∨ 𝑦 = (0g‘𝑆)))))
64	3, 62, 63	sylanbrc 585	1 ⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) → 𝑆 ∈ Domn)

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 849   ∧ w3a 1088   = wceq 1543   ∈ wcel 2115   ≠ wne 2931  ∀wral 3050  ∅c0 4264   class class class wbr 5075  ^◡ccnv 5620  ⟶wf 6484  –1-1-onto→wf1o 6487  ‘cfv 6488  (class class class)co 7359  Basecbs 17173  ._rcmulr 17215  0_gc0g 17396   GrpHom cghm 19181   RingHom crh 20443   RingIso crs 20444   ≃_𝑟 cric 20445  NzRingcnzr 20487  Domncdomn 20667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-10 2148  ax-11 2164  ax-12 2185  ax-ext 2708  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7681  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 850  df-3or 1089  df-3an 1090  df-tru 1546  df-fal 1556  df-ex 1783  df-nf 1787  df-sb 2070  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3061  df-rmo 3341  df-reu 3342  df-rab 3389  df-v 3430  df-sbc 3727  df-csb 3835  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-pss 3906  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7316  df-ov 7362  df-oprab 7363  df-mpo 7364  df-om 7810  df-1st 7934  df-2nd 7935  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-map 8768  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-plusg 17227  df-0g 17398  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-mhm 18745  df-grp 18906  df-minusg 18907  df-ghm 19182  df-cmn 19751  df-abl 19752  df-mgp 20116  df-rng 20128  df-ur 20157  df-ring 20210  df-rhm 20446  df-rim 20447  df-ric 20449  df-nzr 20488  df-domn 20670

This theorem is referenced by:  ricdomn  33374