Theorem zrinitorngc 20562

Description: The zero ring is an initial object in the category of non-unital rings. (Contributed by AV, 18-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
zrinitorngc	⊢ (𝜑 → 𝑍 ∈ (InitO‘𝐶))

Proof of Theorem zrinitorngc

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 2729	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		zrinitorngc.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4	1, 2, 3	rngcbas 20541	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
5	4	eleq2d 2814	. . . . . . . 8 ⊢ (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Rng)))
6		elin 3927	. . . . . . . . 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 2729	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍)
13		eqid 2729	. . . . . . 7 ⊢ (0g‘𝑟) = (0g‘𝑟)
14		eqid 2729	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟))
15	12, 13, 14	zrrnghm 20456	. . . . . 6 ⊢ ((𝑟 ∈ Rng ∧ 𝑍 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHom 𝑟))
16	9, 11, 15	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHom 𝑟))
17		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHom 𝑟)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHom 𝑟))
18	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑈 ∈ 𝑉)
19		eqid 2729	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
20		zrinitorngc.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
21		eldifi 4090	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ (Ring ∖ NzRing) → 𝑍 ∈ Ring)
22		ringrng 20205	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ Ring → 𝑍 ∈ Rng)
23	10, 21, 22	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ Rng)
24	20, 23	elind 4159	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ∈ (𝑈 ∩ Rng))
25	24, 4	eleqtrrd 2831	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
26	25	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Base‘𝐶))
27		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ (Base‘𝐶))
28	1, 2, 18, 19, 26, 27	rngchom 20543	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑍(Hom ‘𝐶)𝑟) = (𝑍 RngHom 𝑟))
29	28	eqcomd 2735	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑍 RngHom 𝑟) = (𝑍(Hom ‘𝐶)𝑟))
30	29	eleq2d 2814	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHom 𝑟) ↔ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟)))
31	30	biimpa 476	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHom 𝑟)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟))
32	28	eleq2d 2814	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ ℎ ∈ (𝑍 RngHom 𝑟)))
33		eqid 2729	. . . . . . . . . . . . 13 ⊢ (Base‘𝑟) = (Base‘𝑟)
34	12, 33	rnghmf 20368	. . . . . . . . . . . 12 ⊢ (ℎ ∈ (𝑍 RngHom 𝑟) → ℎ:(Base‘𝑍)⟶(Base‘𝑟))
35	32, 34	biimtrdi 253	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ℎ:(Base‘𝑍)⟶(Base‘𝑟)))
36	35	imp 406	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) → ℎ:(Base‘𝑍)⟶(Base‘𝑟))
37		ffn 6670	. . . . . . . . . . . 12 ⊢ (ℎ:(Base‘𝑍)⟶(Base‘𝑟) → ℎ Fn (Base‘𝑍))
38	37	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ ℎ:(Base‘𝑍)⟶(Base‘𝑟)) → ℎ Fn (Base‘𝑍))
39		fvex 6853	. . . . . . . . . . . . 13 ⊢ (0g‘𝑟) ∈ V
40	39, 14	fnmpti 6643	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) Fn (Base‘𝑍)
41	40	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ ℎ:(Base‘𝑍)⟶(Base‘𝑟)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) Fn (Base‘𝑍))
42	32	biimpa 476	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) → ℎ ∈ (𝑍 RngHom 𝑟))
43		rnghmghm 20367	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (𝑍 RngHom 𝑟) → ℎ ∈ (𝑍 GrpHom 𝑟))
44		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝑍) = (0g‘𝑍)
45	44, 13	ghmid 19136	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (𝑍 GrpHom 𝑟) → (ℎ‘(0g‘𝑍)) = (0g‘𝑟))
46	42, 43, 45	3syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) → (ℎ‘(0g‘𝑍)) = (0g‘𝑟))
47	46	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ ℎ:(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → (ℎ‘(0g‘𝑍)) = (0g‘𝑟))
48	12, 44	0ringbas 20448	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0g‘𝑍)})
49	10, 48	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝑍) = {(0g‘𝑍)})
50	49	eleq2d 2814	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑎 ∈ (Base‘𝑍) ↔ 𝑎 ∈ {(0g‘𝑍)}))
51		elsni 4602	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ {(0g‘𝑍)} → 𝑎 = (0g‘𝑍))
52	51	fveq2d 6844	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ {(0g‘𝑍)} → (ℎ‘𝑎) = (ℎ‘(0g‘𝑍)))
53	50, 52	biimtrdi 253	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑎 ∈ (Base‘𝑍) → (ℎ‘𝑎) = (ℎ‘(0g‘𝑍))))
54	53	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑎 ∈ (Base‘𝑍) → (ℎ‘𝑎) = (ℎ‘(0g‘𝑍))))
55	54	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ ℎ:(Base‘𝑍)⟶(Base‘𝑟)) → (𝑎 ∈ (Base‘𝑍) → (ℎ‘𝑎) = (ℎ‘(0g‘𝑍))))
56	55	imp 406	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ ℎ:(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → (ℎ‘𝑎) = (ℎ‘(0g‘𝑍)))
57		eqidd 2730	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (Base‘𝑍) → (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)))
58		eqidd 2730	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ (Base‘𝑍) ∧ 𝑥 = 𝑎) → (0g‘𝑟) = (0g‘𝑟))
59		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (Base‘𝑍) → 𝑎 ∈ (Base‘𝑍))
60	39	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (Base‘𝑍) → (0g‘𝑟) ∈ V)
61	57, 58, 59, 60	fvmptd 6957	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Base‘𝑍) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟))‘𝑎) = (0g‘𝑟))
62	61	adantl 481	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ ℎ:(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟))‘𝑎) = (0g‘𝑟))
63	47, 56, 62	3eqtr4d 2774	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ ℎ:(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → (ℎ‘𝑎) = ((𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟))‘𝑎))
64	38, 41, 63	eqfnfvd 6988	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ ℎ:(Base‘𝑍)⟶(Base‘𝑟)) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)))
65	36, 64	mpdan 687	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)))
66	65	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟))))
67	66	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHom 𝑟)) → (ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟))))
68	67	alrimiv 1927	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHom 𝑟)) → ∀ℎ(ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟))))
69	17, 31, 68	3jca 1128	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHom 𝑟)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHom 𝑟) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟) ∧ ∀ℎ(ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)))))
70	16, 69	mpdan 687	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHom 𝑟) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟) ∧ ∀ℎ(ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)))))
71		eleq1 2816	. . . . 5 ⊢ (ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) → (ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟)))
72	71	eqeu 3674	. . . 4 ⊢ (((𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHom 𝑟) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟) ∧ ∀ℎ(ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)))) → ∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟))
73	70, 72	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟))
74	73	ralrimiva 3125	. 2 ⊢ (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟))
75	1	rngccat 20554	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
76	3, 75	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
77	2, 19, 76, 25	isinito 17938	. 2 ⊢ (𝜑 → (𝑍 ∈ (InitO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)))
78	74, 77	mpbird 257	1 ⊢ (𝜑 → 𝑍 ∈ (InitO‘𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∃!weu 2561  ∀wral 3044  Vcvv 3444   ∖ cdif 3908   ∩ cin 3910  {csn 4585   ↦ cmpt 5183   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  Hom chom 17207  0_gc0g 17378  Catccat 17605  InitOcinito 17923   GrpHom cghm 19126  Rngcrng 20072  Ringcrg 20153   RngHom crnghm 20354  NzRingcnzr 20432  RngCatcrngc 20536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-oadd 8415  df-er 8648  df-map 8778  df-pm 8779  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-dju 9830  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-xnn0 12492  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-hash 14272  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-hom 17220  df-cco 17221  df-0g 17380  df-cat 17609  df-cid 17610  df-homf 17611  df-ssc 17752  df-resc 17753  df-subc 17754  df-inito 17926  df-estrc 18064  df-mgm 18549  df-mgmhm 18601  df-sgrp 18628  df-mnd 18644  df-mhm 18692  df-grp 18850  df-minusg 18851  df-ghm 19127  df-cmn 19696  df-abl 19697  df-mgp 20061  df-rng 20073  df-ur 20102  df-ring 20155  df-rnghm 20356  df-nzr 20433  df-rngc 20537

This theorem is referenced by:  zrzeroorngc  20564