Theorem zrinitorngc 20557

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 2731	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		zrinitorngc.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4	1, 2, 3	rngcbas 20536	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
5	4	eleq2d 2817	. . . . . . . 8 ⊢ (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Rng)))
6		elin 3913	. . . . . . . . 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 2731	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍)
13		eqid 2731	. . . . . . 7 ⊢ (0g‘𝑟) = (0g‘𝑟)
14		eqid 2731	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟))
15	12, 13, 14	zrrnghm 20451	. . . . . 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 2731	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
20		zrinitorngc.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
21		eldifi 4078	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ (Ring ∖ NzRing) → 𝑍 ∈ Ring)
22		ringrng 20203	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ Ring → 𝑍 ∈ Rng)
23	10, 21, 22	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ Rng)
24	20, 23	elind 4147	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ∈ (𝑈 ∩ Rng))
25	24, 4	eleqtrrd 2834	. . . . . . . . . . 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 20538	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑍(Hom ‘𝐶)𝑟) = (𝑍 RngHom 𝑟))
29	28	eqcomd 2737	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑍 RngHom 𝑟) = (𝑍(Hom ‘𝐶)𝑟))
30	29	eleq2d 2817	. . . . . . 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 2817	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ ℎ ∈ (𝑍 RngHom 𝑟)))
33		eqid 2731	. . . . . . . . . . . . 13 ⊢ (Base‘𝑟) = (Base‘𝑟)
34	12, 33	rnghmf 20366	. . . . . . . . . . . 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 6651	. . . . . . . . . . . 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 6835	. . . . . . . . . . . . 13 ⊢ (0g‘𝑟) ∈ V
40	39, 14	fnmpti 6624	. . . . . . . . . . . 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 20365	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (𝑍 RngHom 𝑟) → ℎ ∈ (𝑍 GrpHom 𝑟))
44		eqid 2731	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝑍) = (0g‘𝑍)
45	44, 13	ghmid 19134	. . . . . . . . . . . . . 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 20443	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0g‘𝑍)})
49	10, 48	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝑍) = {(0g‘𝑍)})
50	49	eleq2d 2817	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑎 ∈ (Base‘𝑍) ↔ 𝑎 ∈ {(0g‘𝑍)}))
51		elsni 4590	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ {(0g‘𝑍)} → 𝑎 = (0g‘𝑍))
52	51	fveq2d 6826	. . . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (Base‘𝑍) → (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)))
58		eqidd 2732	. . . . . . . . . . . . . 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 6936	. . . . . . . . . . . . 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 2776	. . . . . . . . . . 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 6967	. . . . . . . . . 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 1928	. . . . . 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 2819	. . . . 5 ⊢ (ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) → (ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟)))
72	71	eqeu 3660	. . . 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 3124	. 2 ⊢ (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟))
75	1	rngccat 20549	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
76	3, 75	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
77	2, 19, 76, 25	isinito 17903	. 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 1539   = wceq 1541   ∈ wcel 2111  ∃!weu 2563  ∀wral 3047  Vcvv 3436   ∖ cdif 3894   ∩ cin 3896  {csn 4573   ↦ cmpt 5170   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  Hom chom 17172  0_gc0g 17343  Catccat 17570  InitOcinito 17888   GrpHom cghm 19124  Rngcrng 20070  Ringcrg 20151   RngHom crnghm 20352  NzRingcnzr 20427  RngCatcrngc 20531

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-oadd 8389  df-er 8622  df-map 8752  df-pm 8753  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-dju 9794  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-xnn0 12455  df-z 12469  df-dec 12589  df-uz 12733  df-fz 13408  df-hash 14238  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-hom 17185  df-cco 17186  df-0g 17345  df-cat 17574  df-cid 17575  df-homf 17576  df-ssc 17717  df-resc 17718  df-subc 17719  df-inito 17891  df-estrc 18029  df-mgm 18548  df-mgmhm 18600  df-sgrp 18627  df-mnd 18643  df-mhm 18691  df-grp 18849  df-minusg 18850  df-ghm 19125  df-cmn 19694  df-abl 19695  df-mgp 20059  df-rng 20071  df-ur 20100  df-ring 20153  df-rnghm 20354  df-nzr 20428  df-rngc 20532

This theorem is referenced by:  zrzeroorngc  20559