Theorem zrinitorngc 20693

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 2763	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		zrinitorngc.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4	1, 2, 3	rngcbas 20672	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
5	4	eleq2d 2849	. . . . . . . 8 ⊢ (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Rng)))
6		elin 3921	. . . . . . . . 9 ⊢ (𝑟 ∈ (𝑈 ∩ Rng) ↔ (𝑟 ∈ 𝑈 ∧ 𝑟 ∈ Rng))
7	6	simprbi 501	. . . . . . . 8 ⊢ (𝑟 ∈ (𝑈 ∩ Rng) → 𝑟 ∈ Rng)
8	5, 7	biimtrdi 255	. . . . . . 7 ⊢ (𝜑 → (𝑟 ∈ (Base‘𝐶) → 𝑟 ∈ Rng))
9	8	imp 410	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ Rng)
10		zrinitorngc.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing))
11	10	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Ring ∖ NzRing))
12		eqid 2763	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍)
13		eqid 2763	. . . . . . 7 ⊢ (0g‘𝑟) = (0g‘𝑟)
14		eqid 2763	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟))
15	12, 13, 14	zrrnghm 20587	. . . . . 6 ⊢ ((𝑟 ∈ Rng ∧ 𝑍 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHom 𝑟))
16	9, 11, 15	syl2anc 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHom 𝑟))
17		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHom 𝑟)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHom 𝑟))
18	3	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑈 ∈ 𝑉)
19		eqid 2763	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
20		zrinitorngc.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
21		eldifi 4085	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ (Ring ∖ NzRing) → 𝑍 ∈ Ring)
22		ringrng 20336	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ Ring → 𝑍 ∈ Rng)
23	10, 21, 22	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ Rng)
24	20, 23	elind 4153	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ∈ (𝑈 ∩ Rng))
25	24, 4	eleqtrrd 2866	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
26	25	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Base‘𝐶))
27		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ (Base‘𝐶))
28	1, 2, 18, 19, 26, 27	rngchom 20674	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑍(Hom ‘𝐶)𝑟) = (𝑍 RngHom 𝑟))
29	28	eqcomd 2769	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑍 RngHom 𝑟) = (𝑍(Hom ‘𝐶)𝑟))
30	29	eleq2d 2849	. . . . . . 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 480	. . . . . 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 2849	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ ℎ ∈ (𝑍 RngHom 𝑟)))
33		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Base‘𝑟) = (Base‘𝑟)
34	12, 33	rnghmf 20498	. . . . . . . . . . . 12 ⊢ (ℎ ∈ (𝑍 RngHom 𝑟) → ℎ:(Base‘𝑍)⟶(Base‘𝑟))
35	32, 34	biimtrdi 255	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ℎ:(Base‘𝑍)⟶(Base‘𝑟)))
36	35	imp 410	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) → ℎ:(Base‘𝑍)⟶(Base‘𝑟))
37		ffn 6692	. . . . . . . . . . . 12 ⊢ (ℎ:(Base‘𝑍)⟶(Base‘𝑟) → ℎ Fn (Base‘𝑍))
38	37	adantl 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ ℎ:(Base‘𝑍)⟶(Base‘𝑟)) → ℎ Fn (Base‘𝑍))
39		fvex 6881	. . . . . . . . . . . . 13 ⊢ (0g‘𝑟) ∈ V
40	39, 14	fnmpti 6665	. . . . . . . . . . . 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 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) → ℎ ∈ (𝑍 RngHom 𝑟))
43		rnghmghm 20497	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (𝑍 RngHom 𝑟) → ℎ ∈ (𝑍 GrpHom 𝑟))
44		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝑍) = (0g‘𝑍)
45	44, 13	ghmid 19263	. . . . . . . . . . . . . 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 736	. . . . . . . . . . . 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 20579	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0g‘𝑍)})
49	10, 48	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝑍) = {(0g‘𝑍)})
50	49	eleq2d 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑎 ∈ (Base‘𝑍) ↔ 𝑎 ∈ {(0g‘𝑍)}))
51		elsni 4600	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ {(0g‘𝑍)} → 𝑎 = (0g‘𝑍))
52	51	fveq2d 6872	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ {(0g‘𝑍)} → (ℎ‘𝑎) = (ℎ‘(0g‘𝑍)))
53	50, 52	biimtrdi 255	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑎 ∈ (Base‘𝑍) → (ℎ‘𝑎) = (ℎ‘(0g‘𝑍))))
54	53	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑎 ∈ (Base‘𝑍) → (ℎ‘𝑎) = (ℎ‘(0g‘𝑍))))
55	54	ad2antrr 736	. . . . . . . . . . . . 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 410	. . . . . . . . . . . 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 2764	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (Base‘𝑍) → (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)))
58		eqidd 2764	. . . . . . . . . . . . . 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 6984	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Base‘𝑍) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟))‘𝑎) = (0g‘𝑟))
62	61	adantl 485	. . . . . . . . . . . 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 2808	. . . . . . . . . . 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 7015	. . . . . . . . . 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 697	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)))
66	65	ex 416	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟))))
67	66	adantr 484	. . . . . . 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 1948	. . . . . 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 1142	. . . . 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 697	. . . 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 2851	. . . . 5 ⊢ (ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) → (ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟)))
72	71	eqeu 3670	. . . 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 3155	. 2 ⊢ (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟))
75	1	rngccat 20685	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
76	3, 75	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
77	2, 19, 76, 25	isinito 18030	. 2 ⊢ (𝜑 → (𝑍 ∈ (InitO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)))
78	74, 77	mpbird 259	1 ⊢ (𝜑 → 𝑍 ∈ (InitO‘𝐶))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099  ∀wal 1559   = wceq 1561   ∈ wcel 2143  ∃!weu 2596  ∀wral 3077  Vcvv 3455   ∖ cdif 3902   ∩ cin 3904  {csn 4583   ↦ cmpt 5182   Fn wfn 6517  ⟶wf 6518  ‘cfv 6522  (class class class)co 7397  Basecbs 17246  Hom chom 17298  0_gc0g 17469  Catccat 17697  InitOcinito 18015   GrpHom cghm 19254  Rngcrng 20199  Ringcrg 20284   RngHom crnghm 20484  NzRingcnzr 20563  RngCatcrngc 20667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-om 7848  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-1o 8438  df-oadd 8442  df-er 8679  df-map 8811  df-pm 8812  df-ixp 8881  df-en 8929  df-dom 8930  df-sdom 8931  df-fin 8932  df-dju 9860  df-card 9898  df-pnf 11219  df-mnf 11220  df-xr 11221  df-ltxr 11222  df-le 11223  df-sub 11417  df-neg 11418  df-nn 12212  df-2 12281  df-3 12282  df-4 12283  df-5 12284  df-6 12285  df-7 12286  df-8 12287  df-9 12288  df-n0 12483  df-xnn0 12556  df-z 12570  df-dec 12690  df-uz 12841  df-fz 13514  df-hash 14345  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17247  df-ress 17268  df-plusg 17300  df-hom 17311  df-cco 17312  df-0g 17471  df-cat 17701  df-cid 17702  df-homf 17703  df-ssc 17844  df-resc 17845  df-subc 17846  df-inito 18018  df-estrc 18156  df-mgm 18675  df-mgmhm 18727  df-sgrp 18754  df-mnd 18770  df-mhm 18818  df-grp 18979  df-minusg 18980  df-ghm 19255  df-cmn 19823  df-abl 19824  df-mgp 20188  df-rng 20200  df-ur 20233  df-ring 20286  df-rnghm 20486  df-nzr 20564  df-rngc 20668

This theorem is referenced by:  zrzeroorngc  20695