Theorem ricdomn1 33783

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 20919	. . 3 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
2		ricnzr1 33782	. . 3 ⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ NzRing) → 𝑆 ∈ NzRing)
3	1, 2	sylan2 605	. 2 ⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) → 𝑆 ∈ NzRing)
4		ricsym 20712	. . . . . . . 8 ⊢ (𝑅 ≃𝑟 𝑆 → 𝑆 ≃𝑟 𝑅)
5		brric 20706	. . . . . . . 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 6877	. . . . . . . 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 2760	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
11		eqid 2760	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	10, 11	rimf1o 20690	. . . . . . . . . 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 7272	. . . . . . . . 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 20674	. . . . . . . . . . 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 20675	. . . . . . . . 9 ⊢ (◡𝑓 ∈ (𝑅 RingHom 𝑆) → ◡𝑓 ∈ (𝑅 GrpHom 𝑆))
22		eqid 2760	. . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅)
23		eqid 2760	. . . . . . . . . 10 ⊢ (0g‘𝑆) = (0g‘𝑆)
24	22, 23	ghmid 19397	. . . . . . . . 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 2803	. . . . . . 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 6877	. . . . . . . 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 7272	. . . . . . . . 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 2803	. . . . . . 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 20692	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) → 𝑓 ∈ (𝑆 RingHom 𝑅))
39	10, 11	rhmf 20676	. . . . . . . . . . 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 7073	. . . . . . . 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 7073	. . . . . . . 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 6877	. . . . . . . . 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 2760	. . . . . . . . . . 11 ⊢ (.r‘𝑆) = (.r‘𝑆)
48		eqid 2760	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
49	10, 47, 48	rhmmul 20681	. . . . . . . . . 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 20675	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑆 RingHom 𝑅) → 𝑓 ∈ (𝑆 GrpHom 𝑅))
52	23, 22	ghmid 19397	. . . . . . . . . 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 2803	. . . . . . . 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 20921	. . . . . . . . 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 4300	. . . . 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 3205	. 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 20918	. 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 2955  ∀wral 3076  ∅c0 4278   class class class wbr 5102  ^◡ccnv 5646  ⟶wf 6523  –1-1-onto→wf1o 6526  ‘cfv 6527  (class class class)co 7408  Basecbs 17348  ._rcmulr 17390  0_gc0g 17571   GrpHom cghm 19388   RingHom crh 20660   RingIso crs 20661   ≃_𝑟 cric 20662  NzRingcnzr 20723  Domncdomn 20905

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 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-er 8695  df-map 8827  df-en 8952  df-dom 8953  df-sdom 8954  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-nn 12305  df-2 12374  df-sets 17303  df-slot 17321  df-ndx 17333  df-base 17349  df-plusg 17402  df-0g 17573  df-mgm 18777  df-sgrp 18869  df-mnd 18885  df-mhm 18939  df-grp 19108  df-minusg 19109  df-ghm 19389  df-cmn 19957  df-abl 19958  df-mgp 20322  df-rng 20336  df-ur 20369  df-ring 20422  df-rhm 20663  df-rim 20664  df-ric 20704  df-nzr 20724  df-domn 20908

This theorem is used by:  ricdomn  33784