Theorem ricdomn1 33716

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 20869	. . 3 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
2		ricnzr1 33715	. . 3 ⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ NzRing) → 𝑆 ∈ NzRing)
3	1, 2	sylan2 605	. 2 ⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) → 𝑆 ∈ NzRing)
4		ricsym 20663	. . . . . . . 8 ⊢ (𝑅 ≃𝑟 𝑆 → 𝑆 ≃𝑟 𝑅)
5		brric 20657	. . . . . . . 8 ⊢ (𝑆 ≃𝑟 𝑅 ↔ (𝑆 RingIso 𝑅) ≠ ∅)
6	4, 5	sylib 221	. . . . . . 7 ⊢ (𝑅 ≃𝑟 𝑆 → (𝑆 RingIso 𝑅) ≠ ∅)
7	6	ad4antr 745	. . . . . 6 ⊢ (((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) → (𝑆 RingIso 𝑅) ≠ ∅)
8		simpr 490	. . . . . . . . 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 6886	. . . . . . . 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 2762	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
11		eqid 2762	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	10, 11	rimf1o 20641	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) → 𝑓:(Base‘𝑆)–1-1-onto→(Base‘𝑅))
13	12	ad2antlr 740	. . . . . . . . 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 796	. . . . . . . . . 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 486	. . . . . . . . 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 7280	. . . . . . . . 9 ⊢ ((𝑓:(Base‘𝑆)–1-1-onto→(Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑆)) → (◡𝑓‘(𝑓‘𝑥)) = 𝑥)
17	13, 15, 16	syl2anc 596	. . . . . . . 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 20625	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) ↔ (𝑓 ∈ (𝑆 RingHom 𝑅) ∧ ◡𝑓 ∈ (𝑅 RingHom 𝑆)))
19	18	simprbi 503	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) → ◡𝑓 ∈ (𝑅 RingHom 𝑆))
20	19	ad2antlr 740	. . . . . . . . 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 20626	. . . . . . . . 9 ⊢ (◡𝑓 ∈ (𝑅 RingHom 𝑆) → ◡𝑓 ∈ (𝑅 GrpHom 𝑆))
22		eqid 2762	. . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅)
23		eqid 2762	. . . . . . . . . 10 ⊢ (0g‘𝑆) = (0g‘𝑆)
24	22, 23	ghmid 19350	. . . . . . . . 9 ⊢ (◡𝑓 ∈ (𝑅 GrpHom 𝑆) → (◡𝑓‘(0g‘𝑅)) = (0g‘𝑆))
25	20, 21, 24	3syl 19	. . . . . . . 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 2805	. . . . . . 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 490	. . . . . . . . 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 6886	. . . . . . . 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 740	. . . . . . . . 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 788	. . . . . . . . . 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 486	. . . . . . . . 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 7280	. . . . . . . . 9 ⊢ ((𝑓:(Base‘𝑆)–1-1-onto→(Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → (◡𝑓‘(𝑓‘𝑦)) = 𝑦)
33	29, 31, 32	syl2anc 596	. . . . . . . 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 740	. . . . . . . . 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 19	. . . . . . . 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 2805	. . . . . . 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 798	. . . . . . . 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 20643	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) → 𝑓 ∈ (𝑆 RingHom 𝑅))
39	10, 11	rhmf 20627	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑆 RingHom 𝑅) → 𝑓:(Base‘𝑆)⟶(Base‘𝑅))
40	38, 39	syl 18	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) → 𝑓:(Base‘𝑆)⟶(Base‘𝑅))
41	40	adantl 487	. . . . . . . . 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 7081	. . . . . . . 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 7081	. . . . . . . 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 781	. . . . . . . . . 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 6886	. . . . . . . . 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 487	. . . . . . . . . 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 2762	. . . . . . . . . . 11 ⊢ (.r‘𝑆) = (.r‘𝑆)
48		eqid 2762	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
49	10, 47, 48	rhmmul 20632	. . . . . . . . . 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 1398	. . . . . . . . 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 20626	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑆 RingHom 𝑅) → 𝑓 ∈ (𝑆 GrpHom 𝑅))
52	23, 22	ghmid 19350	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑆 GrpHom 𝑅) → (𝑓‘(0g‘𝑆)) = (0g‘𝑅))
53	46, 51, 52	3syl 19	. . . . . . . . 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 2805	. . . . . . . 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 20871	. . . . . . . . 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 482	. . . . . . . 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 1400	. . . . . . 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 980	. . . . . 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 4304	. . . . 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 418	. . . 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 472	. . 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 3207	. 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 20868	. 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 595	1 ⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) → 𝑆 ∈ Domn)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ∅c0 4282   class class class wbr 5107  ^◡ccnv 5658  ⟶wf 6533  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7416  Basecbs 17305  ._rcmulr 17347  0_gc0g 17528   GrpHom cghm 19341   RingHom crh 20611   RingIso crs 20612   ≃_𝑟 cric 20613  NzRingcnzr 20673  Domncdomn 20855

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-sets 17260  df-slot 17278  df-ndx 17290  df-base 17306  df-plusg 17359  df-0g 17530  df-mgm 18734  df-sgrp 18823  df-mnd 18839  df-mhm 18892  df-grp 19061  df-minusg 19062  df-ghm 19342  df-cmn 19910  df-abl 19911  df-mgp 20275  df-rng 20289  df-ur 20322  df-ring 20375  df-rhm 20614  df-rim 20615  df-ric 20655  df-nzr 20674  df-domn 20858

This theorem is used by:  ricdomn  33717