Theorem ricdomn1 33419

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 20724	. . 3 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
2		ricnzr1 33418	. . 3 ⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ NzRing) → 𝑆 ∈ NzRing)
3	1, 2	sylan2 601	. 2 ⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) → 𝑆 ∈ NzRing)
4		ricsym 20523	. . . . . . . 8 ⊢ (𝑅 ≃𝑟 𝑆 → 𝑆 ≃𝑟 𝑅)
5		brric 20521	. . . . . . . 8 ⊢ (𝑆 ≃𝑟 𝑅 ↔ (𝑆 RingIso 𝑅) ≠ ∅)
6	4, 5	sylib 220	. . . . . . 7 ⊢ (𝑅 ≃𝑟 𝑆 → (𝑆 RingIso 𝑅) ≠ ∅)
7	6	ad4antr 740	. . . . . 6 ⊢ (((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) → (𝑆 RingIso 𝑅) ≠ ∅)
8		simpr 487	. . . . . . . . 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 6856	. . . . . . . 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 2752	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
11		eqid 2752	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	10, 11	rimf1o 20510	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) → 𝑓:(Base‘𝑆)–1-1-onto→(Base‘𝑅))
13	12	ad2antlr 735	. . . . . . . . 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 791	. . . . . . . . . 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 483	. . . . . . . . 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 7245	. . . . . . . . 9 ⊢ ((𝑓:(Base‘𝑆)–1-1-onto→(Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑆)) → (◡𝑓‘(𝑓‘𝑥)) = 𝑥)
17	13, 15, 16	syl2anc 592	. . . . . . . 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 20499	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) ↔ (𝑓 ∈ (𝑆 RingHom 𝑅) ∧ ◡𝑓 ∈ (𝑅 RingHom 𝑆)))
19	18	simprbi 500	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) → ◡𝑓 ∈ (𝑅 RingHom 𝑆))
20	19	ad2antlr 735	. . . . . . . . 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 20500	. . . . . . . . 9 ⊢ (◡𝑓 ∈ (𝑅 RingHom 𝑆) → ◡𝑓 ∈ (𝑅 GrpHom 𝑆))
22		eqid 2752	. . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅)
23		eqid 2752	. . . . . . . . . 10 ⊢ (0g‘𝑆) = (0g‘𝑆)
24	22, 23	ghmid 19234	. . . . . . . . 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 2795	. . . . . . 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 487	. . . . . . . . 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 6856	. . . . . . . 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 735	. . . . . . . . 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 783	. . . . . . . . . 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 483	. . . . . . . . 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 7245	. . . . . . . . 9 ⊢ ((𝑓:(Base‘𝑆)–1-1-onto→(Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → (◡𝑓‘(𝑓‘𝑦)) = 𝑦)
33	29, 31, 32	syl2anc 592	. . . . . . . 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 735	. . . . . . . . 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 2795	. . . . . . 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 793	. . . . . . . 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 20511	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) → 𝑓 ∈ (𝑆 RingHom 𝑅))
39	10, 11	rhmf 20501	. . . . . . . . . . 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 484	. . . . . . . . 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 7051	. . . . . . . 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 7051	. . . . . . . 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 776	. . . . . . . . . 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 6856	. . . . . . . . 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 484	. . . . . . . . . 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 2752	. . . . . . . . . . 11 ⊢ (.r‘𝑆) = (.r‘𝑆)
48		eqid 2752	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
49	10, 47, 48	rhmmul 20503	. . . . . . . . . 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 1382	. . . . . . . . 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 20500	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑆 RingHom 𝑅) → 𝑓 ∈ (𝑆 GrpHom 𝑅))
52	23, 22	ghmid 19234	. . . . . . . . . 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 2795	. . . . . . . 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 20726	. . . . . . . . 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 479	. . . . . . . 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 1384	. . . . . . 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 32594	. . . . . 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 32608	. . . . 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 415	. . . 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 469	. . 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 3195	. 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 20723	. 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 591	1 ⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) → 𝑆 ∈ Domn)

Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 856   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132   ≠ wne 2947  ∀wral 3066  ∅c0 4276   class class class wbr 5090  ^◡ccnv 5635  ⟶wf 6502  –1-1-onto→wf1o 6505  ‘cfv 6506  (class class class)co 7381  Basecbs 17217  ._rcmulr 17259  0_gc0g 17440   GrpHom cghm 19225   RingHom crh 20486   RingIso crs 20487   ≃_𝑟 cric 20488  NzRingcnzr 20530  Domncdomn 20710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-nel 3052  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-1st 7955  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-er 8662  df-map 8794  df-en 8913  df-dom 8914  df-sdom 8915  df-pnf 11204  df-mnf 11205  df-xr 11206  df-ltxr 11207  df-le 11208  df-sub 11402  df-neg 11403  df-nn 12197  df-2 12266  df-sets 17172  df-slot 17190  df-ndx 17202  df-base 17218  df-plusg 17271  df-0g 17442  df-mgm 18646  df-sgrp 18725  df-mnd 18741  df-mhm 18789  df-grp 18950  df-minusg 18951  df-ghm 19226  df-cmn 19794  df-abl 19795  df-mgp 20159  df-rng 20171  df-ur 20200  df-ring 20253  df-rhm 20489  df-rim 20490  df-ric 20492  df-nzr 20531  df-domn 20713

This theorem is referenced by:  ricdomn  33420