zrtermoringc - Mathbox for Alexander van der Vekens

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Theorem zrtermoringc 46921

Description: The zero ring is a terminal object in the category of unital rings. (Contributed by AV, 17-Apr-2020.)

**Hypotheses**
Ref	Expression
zrtermoringc.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
zrtermoringc.c	⊢ 𝐶 = (RingCat‘𝑈)
zrtermoringc.z	⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing))
zrtermoringc.e	⊢ (𝜑 → 𝑍 ∈ 𝑈)

**Assertion**
Ref	Expression
zrtermoringc	⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶))

Proof of Theorem zrtermoringc Dummy variables 𝑎 ℎ 𝑟 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zrtermoringc.c	. . . . . . . . . 10 ⊢ 𝐶 = (RingCat‘𝑈)
2		eqid 2732	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		zrtermoringc.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4	1, 2, 3	ringcbas 46862	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring))
5	4	eleq2d 2819	. . . . . . . 8 ⊢ (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Ring)))
6		elin 3963	. . . . . . . . 9 ⊢ (𝑟 ∈ (𝑈 ∩ Ring) ↔ (𝑟 ∈ 𝑈 ∧ 𝑟 ∈ Ring))
7	6	simprbi 497	. . . . . . . 8 ⊢ (𝑟 ∈ (𝑈 ∩ Ring) → 𝑟 ∈ Ring)
8	5, 7	syl6bi 252	. . . . . . 7 ⊢ (𝜑 → (𝑟 ∈ (Base‘𝐶) → 𝑟 ∈ Ring))
9	8	imp 407	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ Ring)
10		zrtermoringc.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing))
11	10	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Ring ∖ NzRing))
12		eqid 2732	. . . . . . 7 ⊢ (Base‘𝑟) = (Base‘𝑟)
13		eqid 2732	. . . . . . 7 ⊢ (0_g‘𝑍) = (0_g‘𝑍)
14		eqid 2732	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍))
15	12, 13, 14	c0rhm 46696	. . . . . 6 ⊢ ((𝑟 ∈ Ring ∧ 𝑍 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍))
16	9, 11, 15	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍))
17		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍))
18	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑈 ∈ 𝑉)
19		eqid 2732	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
20		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ (Base‘𝐶))
21		zrtermoringc.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
22	10	eldifad 3959	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ Ring)
23	21, 22	elind 4193	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ∈ (𝑈 ∩ Ring))
24	23, 4	eleqtrrd 2836	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
25	24	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Base‘𝐶))
26	1, 2, 18, 19, 20, 25	ringchom 46864	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑟(Hom ‘𝐶)𝑍) = (𝑟 RingHom 𝑍))
27	26	eqcomd 2738	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑟 RingHom 𝑍) = (𝑟(Hom ‘𝐶)𝑍))
28	27	eleq2d 2819	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍) ↔ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍)))
29	28	biimpa 477	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍))
30	26	eleq2d 2819	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ ℎ ∈ (𝑟 RingHom 𝑍)))
31		eqid 2732	. . . . . . . . . . 11 ⊢ (Base‘𝑍) = (Base‘𝑍)
32	12, 31	rhmf 20255	. . . . . . . . . 10 ⊢ (ℎ ∈ (𝑟 RingHom 𝑍) → ℎ:(Base‘𝑟)⟶(Base‘𝑍))
33	30, 32	syl6bi 252	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ:(Base‘𝑟)⟶(Base‘𝑍)))
34	33	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ:(Base‘𝑟)⟶(Base‘𝑍)))
35		ffn 6714	. . . . . . . . . . 11 ⊢ (ℎ:(Base‘𝑟)⟶(Base‘𝑍) → ℎ Fn (Base‘𝑟))
36	35	adantl 482	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) → ℎ Fn (Base‘𝑟))
37		fvex 6901	. . . . . . . . . . . 12 ⊢ (0_g‘𝑍) ∈ V
38	37, 14	fnmpti 6690	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) Fn (Base‘𝑟)
39	38	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) Fn (Base‘𝑟))
40	31, 13	0ringbas 46631	. . . . . . . . . . . . . . . . 17 ⊢ (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0_g‘𝑍)})
41	10, 40	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝑍) = {(0_g‘𝑍)})
42	41	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (Base‘𝑍) = {(0_g‘𝑍)})
43	42	feq3d 6701	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ:(Base‘𝑟)⟶(Base‘𝑍) ↔ ℎ:(Base‘𝑟)⟶{(0_g‘𝑍)}))
44		fvconst 7158	. . . . . . . . . . . . . . 15 ⊢ ((ℎ:(Base‘𝑟)⟶{(0_g‘𝑍)} ∧ 𝑎 ∈ (Base‘𝑟)) → (ℎ‘𝑎) = (0_g‘𝑍))
45	44	ex 413	. . . . . . . . . . . . . 14 ⊢ (ℎ:(Base‘𝑟)⟶{(0_g‘𝑍)} → (𝑎 ∈ (Base‘𝑟) → (ℎ‘𝑎) = (0_g‘𝑍)))
46	43, 45	syl6bi 252	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ:(Base‘𝑟)⟶(Base‘𝑍) → (𝑎 ∈ (Base‘𝑟) → (ℎ‘𝑎) = (0_g‘𝑍))))
47	46	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) → (ℎ:(Base‘𝑟)⟶(Base‘𝑍) → (𝑎 ∈ (Base‘𝑟) → (ℎ‘𝑎) = (0_g‘𝑍))))
48	47	imp31 418	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) ∧ 𝑎 ∈ (Base‘𝑟)) → (ℎ‘𝑎) = (0_g‘𝑍))
49		eqidd 2733	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Base‘𝑟) → (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)))
50		eqidd 2733	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ (Base‘𝑟) ∧ 𝑥 = 𝑎) → (0_g‘𝑍) = (0_g‘𝑍))
51		id 22	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Base‘𝑟) → 𝑎 ∈ (Base‘𝑟))
52	37	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Base‘𝑟) → (0_g‘𝑍) ∈ V)
53	49, 50, 51, 52	fvmptd 7002	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (Base‘𝑟) → ((𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍))‘𝑎) = (0_g‘𝑍))
54	53	adantl 482	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) ∧ 𝑎 ∈ (Base‘𝑟)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍))‘𝑎) = (0_g‘𝑍))
55	48, 54	eqtr4d 2775	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) ∧ 𝑎 ∈ (Base‘𝑟)) → (ℎ‘𝑎) = ((𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍))‘𝑎))
56	36, 39, 55	eqfnfvd 7032	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)))
57	56	ex 413	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) → (ℎ:(Base‘𝑟)⟶(Base‘𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍))))
58	34, 57	syld 47	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍))))
59	58	alrimiv 1930	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) → ∀ℎ(ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍))))
60	17, 29, 59	3jca 1128	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍) ∧ ∀ℎ(ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)))))
61	16, 60	mpdan 685	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍) ∧ ∀ℎ(ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)))))
62		eleq1 2821	. . . . 5 ⊢ (ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍)))
63	62	eqeu 3701	. . . 4 ⊢ (((𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟 RingHom 𝑍) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍) ∧ ∀ℎ(ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0_g‘𝑍)))) → ∃!ℎ ℎ ∈ (𝑟(Hom ‘𝐶)𝑍))
64	61, 63	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ∃!ℎ ℎ ∈ (𝑟(Hom ‘𝐶)𝑍))
65	64	ralrimiva 3146	. 2 ⊢ (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑟(Hom ‘𝐶)𝑍))
66	1	ringccat 46875	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
67	3, 66	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
68	2, 19, 67, 24	istermo 17943	. 2 ⊢ (𝜑 → (𝑍 ∈ (TermO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑟(Hom ‘𝐶)𝑍)))
69	65, 68	mpbird 256	1 ⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ∃!weu 2562 ∀wral 3061 Vcvv 3474 ∖ cdif 3944 ∩ cin 3946 {csn 4627 ↦ cmpt 5230 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 Hom chom 17204 0_gc0g 17381 Catccat 17604 TermOctermo 17928 Ringcrg 20049 RingHom crh 20240 NzRingcnzr 20283 RingCatcringc 46854

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 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 theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-hom 17217 df-cco 17218 df-0g 17383 df-cat 17608 df-cid 17609 df-homf 17610 df-ssc 17753 df-resc 17754 df-subc 17755 df-termo 17931 df-estrc 18070 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-grp 18818 df-minusg 18819 df-ghm 19084 df-mgp 19982 df-ur 19999 df-ring 20051 df-rnghom 20243 df-nzr 20284 df-ringc 46856

This theorem is referenced by: nzerooringczr 46923

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