Theorem ricdomn1 33618

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 20816	. . 3 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
2		ricnzr1 33617	. . 3 ⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ NzRing) → 𝑆 ∈ NzRing)
3	1, 2	sylan2 604	. 2 ⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) → 𝑆 ∈ NzRing)
4		ricsym 20610	. . . . . . . 8 ⊢ (𝑅 ≃𝑟 𝑆 → 𝑆 ≃𝑟 𝑅)
5		brric 20604	. . . . . . . 8 ⊢ (𝑆 ≃𝑟 𝑅 ↔ (𝑆 RingIso 𝑅) ≠ ∅)
6	4, 5	sylib 221	. . . . . . 7 ⊢ (𝑅 ≃𝑟 𝑆 → (𝑆 RingIso 𝑅) ≠ ∅)
7	6	ad4antr 744	. . . . . 6 ⊢ (((((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) ∧ (𝑥(.r‘𝑆)𝑦) = (0g‘𝑆)) → (𝑆 RingIso 𝑅) ≠ ∅)
8		simpr 489	. . . . . . . . 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 6885	. . . . . . . 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 20588	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) → 𝑓:(Base‘𝑆)–1-1-onto→(Base‘𝑅))
13	12	ad2antlr 739	. . . . . . . . 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 795	. . . . . . . . . 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 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)) → 𝑥 ∈ (Base‘𝑆))
16		f1ocnvfv1 7274	. . . . . . . . 9 ⊢ ((𝑓:(Base‘𝑆)–1-1-onto→(Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑆)) → (◡𝑓‘(𝑓‘𝑥)) = 𝑥)
17	13, 15, 16	syl2anc 595	. . . . . . . 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 20572	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) ↔ (𝑓 ∈ (𝑆 RingHom 𝑅) ∧ ◡𝑓 ∈ (𝑅 RingHom 𝑆)))
19	18	simprbi 502	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) → ◡𝑓 ∈ (𝑅 RingHom 𝑆))
20	19	ad2antlr 739	. . . . . . . . 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 20573	. . . . . . . . 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 19298	. . . . . . . . 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 489	. . . . . . . . 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 6885	. . . . . . . 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 739	. . . . . . . . 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 787	. . . . . . . . . 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 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)) → 𝑦 ∈ (Base‘𝑆))
32		f1ocnvfv1 7274	. . . . . . . . 9 ⊢ ((𝑓:(Base‘𝑆)–1-1-onto→(Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → (◡𝑓‘(𝑓‘𝑦)) = 𝑦)
33	29, 31, 32	syl2anc 595	. . . . . . . 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 739	. . . . . . . . 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 797	. . . . . . . 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 20590	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑆 RingIso 𝑅) → 𝑓 ∈ (𝑆 RingHom 𝑅))
39	10, 11	rhmf 20574	. . . . . . . . . . 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 486	. . . . . . . . 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 7080	. . . . . . . 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 7080	. . . . . . . 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 780	. . . . . . . . . 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 6885	. . . . . . . . 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 486	. . . . . . . . . 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 20579	. . . . . . . . . 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 1397	. . . . . . . . 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 20573	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑆 RingHom 𝑅) → 𝑓 ∈ (𝑆 GrpHom 𝑅))
52	23, 22	ghmid 19298	. . . . . . . . . 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 20818	. . . . . . . . 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 481	. . . . . . . 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 1399	. . . . . . 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 979	. . . . . 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 4307	. . . . 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 417	. . . 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 471	. . 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 20815	. 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 594	1 ⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) → 𝑆 ∈ Domn)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∨ wo 860   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078  ∅c0 4285   class class class wbr 5108  ^◡ccnv 5659  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  ._rcmulr 17317  0_gc0g 17498   GrpHom cghm 19289   RingHom crh 20558   RingIso crs 20559   ≃_𝑟 cric 20560  NzRingcnzr 20620  Domncdomn 20802

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  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 proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  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 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-er 8692  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-sets 17230  df-slot 17248  df-ndx 17260  df-base 17276  df-plusg 17329  df-0g 17500  df-mgm 18704  df-sgrp 18783  df-mnd 18799  df-mhm 18847  df-grp 19009  df-minusg 19010  df-ghm 19290  df-cmn 19858  df-abl 19859  df-mgp 20223  df-rng 20237  df-ur 20270  df-ring 20323  df-rhm 20561  df-rim 20562  df-ric 20602  df-nzr 20621  df-domn 20805

This theorem is used by:  ricdomn  33619