Theorem zrtermorngc 20576

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

Hypotheses

Ref	Expression
zrinitorngc.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
zrinitorngc.c	⊢ 𝐶 = (RngCat‘𝑈)
zrinitorngc.z	⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing))
zrinitorngc.e	⊢ (𝜑 → 𝑍 ∈ 𝑈)

Assertion

Ref	Expression
zrtermorngc	⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶))

Proof of Theorem zrtermorngc

Dummy variables 𝑎 ℎ 𝑟 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zrinitorngc.c	. . . . . . . . . 10 ⊢ 𝐶 = (RngCat‘𝑈)
2		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		zrinitorngc.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4	1, 2, 3	rngcbas 20554	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
5	4	eleq2d 2822	. . . . . . . 8 ⊢ (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Rng)))
6		elin 3917	. . . . . . . . 9 ⊢ (𝑟 ∈ (𝑈 ∩ Rng) ↔ (𝑟 ∈ 𝑈 ∧ 𝑟 ∈ Rng))
7	6	simprbi 496	. . . . . . . 8 ⊢ (𝑟 ∈ (𝑈 ∩ Rng) → 𝑟 ∈ Rng)
8	5, 7	biimtrdi 253	. . . . . . 7 ⊢ (𝜑 → (𝑟 ∈ (Base‘𝐶) → 𝑟 ∈ Rng))
9	8	imp 406	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ Rng)
10		zrinitorngc.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing))
11	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Ring ∖ NzRing))
12		eqid 2736	. . . . . . 7 ⊢ (Base‘𝑟) = (Base‘𝑟)
13		eqid 2736	. . . . . . 7 ⊢ (0g‘𝑍) = (0g‘𝑍)
14		eqid 2736	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))
15	12, 13, 14	c0rnghm 20468	. . . . . 6 ⊢ ((𝑟 ∈ Rng ∧ 𝑍 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍))
16	9, 11, 15	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍))
17		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍))
18	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑈 ∈ 𝑉)
19		eqid 2736	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
20		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ (Base‘𝐶))
21		zrinitorngc.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
22		eldifi 4083	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ (Ring ∖ NzRing) → 𝑍 ∈ Ring)
23		ringrng 20220	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ Ring → 𝑍 ∈ Rng)
24	10, 22, 23	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ Rng)
25	21, 24	elind 4152	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ∈ (𝑈 ∩ Rng))
26	25, 4	eleqtrrd 2839	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
27	26	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Base‘𝐶))
28	1, 2, 18, 19, 20, 27	rngchom 20556	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑟(Hom ‘𝐶)𝑍) = (𝑟 RngHom 𝑍))
29	28	eqcomd 2742	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑟 RngHom 𝑍) = (𝑟(Hom ‘𝐶)𝑍))
30	29	eleq2d 2822	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍) ↔ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍)))
31	30	biimpa 476	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍))
32	28	eleq2d 2822	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ ℎ ∈ (𝑟 RngHom 𝑍)))
33		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘𝑍) = (Base‘𝑍)
34	12, 33	rnghmf 20384	. . . . . . . . . 10 ⊢ (ℎ ∈ (𝑟 RngHom 𝑍) → ℎ:(Base‘𝑟)⟶(Base‘𝑍))
35	32, 34	biimtrdi 253	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ:(Base‘𝑟)⟶(Base‘𝑍)))
36	35	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ:(Base‘𝑟)⟶(Base‘𝑍)))
37		ffn 6662	. . . . . . . . . . 11 ⊢ (ℎ:(Base‘𝑟)⟶(Base‘𝑍) → ℎ Fn (Base‘𝑟))
38	37	adantl 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) → ℎ Fn (Base‘𝑟))
39		fvex 6847	. . . . . . . . . . . 12 ⊢ (0g‘𝑍) ∈ V
40	39, 14	fnmpti 6635	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) Fn (Base‘𝑟)
41	40	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) Fn (Base‘𝑟))
42	33, 13	0ringbas 20461	. . . . . . . . . . . . . . . . 17 ⊢ (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0g‘𝑍)})
43	10, 42	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝑍) = {(0g‘𝑍)})
44	43	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (Base‘𝑍) = {(0g‘𝑍)})
45	44	feq3d 6647	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ:(Base‘𝑟)⟶(Base‘𝑍) ↔ ℎ:(Base‘𝑟)⟶{(0g‘𝑍)}))
46		fvconst 7108	. . . . . . . . . . . . . . 15 ⊢ ((ℎ:(Base‘𝑟)⟶{(0g‘𝑍)} ∧ 𝑎 ∈ (Base‘𝑟)) → (ℎ‘𝑎) = (0g‘𝑍))
47	46	ex 412	. . . . . . . . . . . . . 14 ⊢ (ℎ:(Base‘𝑟)⟶{(0g‘𝑍)} → (𝑎 ∈ (Base‘𝑟) → (ℎ‘𝑎) = (0g‘𝑍)))
48	45, 47	biimtrdi 253	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ:(Base‘𝑟)⟶(Base‘𝑍) → (𝑎 ∈ (Base‘𝑟) → (ℎ‘𝑎) = (0g‘𝑍))))
49	48	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) → (ℎ:(Base‘𝑟)⟶(Base‘𝑍) → (𝑎 ∈ (Base‘𝑟) → (ℎ‘𝑎) = (0g‘𝑍))))
50	49	imp31 417	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) ∧ 𝑎 ∈ (Base‘𝑟)) → (ℎ‘𝑎) = (0g‘𝑍))
51		eqidd 2737	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Base‘𝑟) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)))
52		eqidd 2737	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ (Base‘𝑟) ∧ 𝑥 = 𝑎) → (0g‘𝑍) = (0g‘𝑍))
53		id 22	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Base‘𝑟) → 𝑎 ∈ (Base‘𝑟))
54	39	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Base‘𝑟) → (0g‘𝑍) ∈ V)
55	51, 52, 53, 54	fvmptd 6948	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (Base‘𝑟) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))‘𝑎) = (0g‘𝑍))
56	55	adantl 481	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) ∧ 𝑎 ∈ (Base‘𝑟)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))‘𝑎) = (0g‘𝑍))
57	50, 56	eqtr4d 2774	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) ∧ 𝑎 ∈ (Base‘𝑟)) → (ℎ‘𝑎) = ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))‘𝑎))
58	38, 41, 57	eqfnfvd 6979	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)))
59	58	ex 412	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) → (ℎ:(Base‘𝑟)⟶(Base‘𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))))
60	36, 59	syld 47	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))))
61	60	alrimiv 1928	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) → ∀ℎ(ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))))
62	17, 31, 61	3jca 1128	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍) ∧ ∀ℎ(ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)))))
63	16, 62	mpdan 687	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍) ∧ ∀ℎ(ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)))))
64		eleq1 2824	. . . . 5 ⊢ (ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍)))
65	64	eqeu 3664	. . . 4 ⊢ (((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍) ∧ ∀ℎ(ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)))) → ∃!ℎ ℎ ∈ (𝑟(Hom ‘𝐶)𝑍))
66	63, 65	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ∃!ℎ ℎ ∈ (𝑟(Hom ‘𝐶)𝑍))
67	66	ralrimiva 3128	. 2 ⊢ (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑟(Hom ‘𝐶)𝑍))
68	1	rngccat 20567	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
69	3, 68	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
70	2, 19, 69, 26	istermo 17921	. 2 ⊢ (𝜑 → (𝑍 ∈ (TermO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑟(Hom ‘𝐶)𝑍)))
71	67, 70	mpbird 257	1 ⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   = wceq 1541   ∈ wcel 2113  ∃!weu 2568  ∀wral 3051  Vcvv 3440   ∖ cdif 3898   ∩ cin 3900  {csn 4580   ↦ cmpt 5179   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  Hom chom 17188  0_gc0g 17359  Catccat 17587  TermOctermo 17906  Rngcrng 20087  Ringcrg 20168   RngHom crnghm 20370  NzRingcnzr 20445  RngCatcrngc 20549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-er 8635  df-map 8765  df-pm 8766  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9813  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-xnn0 12475  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-hash 14254  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-hom 17201  df-cco 17202  df-0g 17361  df-cat 17591  df-cid 17592  df-homf 17593  df-ssc 17734  df-resc 17735  df-subc 17736  df-termo 17909  df-estrc 18046  df-mgm 18565  df-mgmhm 18617  df-sgrp 18644  df-mnd 18660  df-mhm 18708  df-grp 18866  df-minusg 18867  df-ghm 19142  df-cmn 19711  df-abl 19712  df-mgp 20076  df-rng 20088  df-ur 20117  df-ring 20170  df-rnghm 20372  df-nzr 20446  df-rngc 20550

This theorem is referenced by:  zrzeroorngc  20577