zrinitorngc - Metamath Proof Explorer

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Theorem zrinitorngc 20538

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 2726	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		zrinitorngc.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4	1, 2, 3	rngcbas 20517	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
5	4	eleq2d 2813	. . . . . . . 8 ⊢ (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Rng)))
6		elin 3959	. . . . . . . . 9 ⊢ (𝑟 ∈ (𝑈 ∩ Rng) ↔ (𝑟 ∈ 𝑈 ∧ 𝑟 ∈ Rng))
7	6	simprbi 496	. . . . . . . 8 ⊢ (𝑟 ∈ (𝑈 ∩ Rng) → 𝑟 ∈ Rng)
8	5, 7	biimtrdi 252	. . . . . . 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 2726	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍)
13		eqid 2726	. . . . . . 7 ⊢ (0_g‘𝑟) = (0_g‘𝑟)
14		eqid 2726	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟))
15	12, 13, 14	zrrnghm 20436	. . . . . 6 ⊢ ((𝑟 ∈ Rng ∧ 𝑍 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) ∈ (𝑍 RngHom 𝑟))
16	9, 11, 15	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) ∈ (𝑍 RngHom 𝑟))
17		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) ∈ (𝑍 RngHom 𝑟)) → (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) ∈ (𝑍 RngHom 𝑟))
18	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑈 ∈ 𝑉)
19		eqid 2726	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
20		zrinitorngc.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
21		eldifi 4121	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ (Ring ∖ NzRing) → 𝑍 ∈ Ring)
22		ringrng 20184	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ Ring → 𝑍 ∈ Rng)
23	10, 21, 22	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ Rng)
24	20, 23	elind 4189	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ∈ (𝑈 ∩ Rng))
25	24, 4	eleqtrrd 2830	. . . . . . . . . . 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 20519	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑍(Hom ‘𝐶)𝑟) = (𝑍 RngHom 𝑟))
29	28	eqcomd 2732	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑍 RngHom 𝑟) = (𝑍(Hom ‘𝐶)𝑟))
30	29	eleq2d 2813	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) ∈ (𝑍 RngHom 𝑟) ↔ (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟)))
31	30	biimpa 476	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) ∈ (𝑍 RngHom 𝑟)) → (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟))
32	28	eleq2d 2813	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ ℎ ∈ (𝑍 RngHom 𝑟)))
33		eqid 2726	. . . . . . . . . . . . 13 ⊢ (Base‘𝑟) = (Base‘𝑟)
34	12, 33	rnghmf 20350	. . . . . . . . . . . 12 ⊢ (ℎ ∈ (𝑍 RngHom 𝑟) → ℎ:(Base‘𝑍)⟶(Base‘𝑟))
35	32, 34	biimtrdi 252	. . . . . . . . . . 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 6711	. . . . . . . . . . . 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 6898	. . . . . . . . . . . . 13 ⊢ (0_g‘𝑟) ∈ V
40	39, 14	fnmpti 6687	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘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) ↦ (0_g‘𝑟)) Fn (Base‘𝑍))
42	32	biimpa 476	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) → ℎ ∈ (𝑍 RngHom 𝑟))
43		rnghmghm 20349	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (𝑍 RngHom 𝑟) → ℎ ∈ (𝑍 GrpHom 𝑟))
44		eqid 2726	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝑍) = (0_g‘𝑍)
45	44, 13	ghmid 19147	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (𝑍 GrpHom 𝑟) → (ℎ‘(0_g‘𝑍)) = (0_g‘𝑟))
46	42, 43, 45	3syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) → (ℎ‘(0_g‘𝑍)) = (0_g‘𝑟))
47	46	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ ℎ:(Base‘𝑍)⟶(Base‘𝑟)) ∧ 𝑎 ∈ (Base‘𝑍)) → (ℎ‘(0_g‘𝑍)) = (0_g‘𝑟))
48	12, 44	0ringbas 20428	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0_g‘𝑍)})
49	10, 48	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝑍) = {(0_g‘𝑍)})
50	49	eleq2d 2813	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑎 ∈ (Base‘𝑍) ↔ 𝑎 ∈ {(0_g‘𝑍)}))
51		elsni 4640	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ {(0_g‘𝑍)} → 𝑎 = (0_g‘𝑍))
52	51	fveq2d 6889	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ {(0_g‘𝑍)} → (ℎ‘𝑎) = (ℎ‘(0_g‘𝑍)))
53	50, 52	biimtrdi 252	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑎 ∈ (Base‘𝑍) → (ℎ‘𝑎) = (ℎ‘(0_g‘𝑍))))
54	53	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑎 ∈ (Base‘𝑍) → (ℎ‘𝑎) = (ℎ‘(0_g‘𝑍))))
55	54	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ ℎ:(Base‘𝑍)⟶(Base‘𝑟)) → (𝑎 ∈ (Base‘𝑍) → (ℎ‘𝑎) = (ℎ‘(0_g‘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‘(0_g‘𝑍)))
57		eqidd 2727	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (Base‘𝑍) → (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)))
58		eqidd 2727	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ (Base‘𝑍) ∧ 𝑥 = 𝑎) → (0_g‘𝑟) = (0_g‘𝑟))
59		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (Base‘𝑍) → 𝑎 ∈ (Base‘𝑍))
60	39	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (Base‘𝑍) → (0_g‘𝑟) ∈ V)
61	57, 58, 59, 60	fvmptd 6999	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Base‘𝑍) → ((𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟))‘𝑎) = (0_g‘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) ↦ (0_g‘𝑟))‘𝑎) = (0_g‘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) ↦ (0_g‘𝑟))‘𝑎))
64	38, 41, 63	eqfnfvd 7029	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) ∧ ℎ:(Base‘𝑍)⟶(Base‘𝑟)) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)))
65	36, 64	mpdan 684	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)))
66	65	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟))))
67	66	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) ∈ (𝑍 RngHom 𝑟)) → (ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟))))
68	67	alrimiv 1922	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) ∈ (𝑍 RngHom 𝑟)) → ∀ℎ(ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟))))
69	17, 31, 68	3jca 1125	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) ∈ (𝑍 RngHom 𝑟)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) ∈ (𝑍 RngHom 𝑟) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟) ∧ ∀ℎ(ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)))))
70	16, 69	mpdan 684	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) ∈ (𝑍 RngHom 𝑟) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟) ∧ ∀ℎ(ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)))))
71		eleq1 2815	. . . . 5 ⊢ (ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) → (ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟)))
72	71	eqeu 3697	. . . 4 ⊢ (((𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) ∈ (𝑍 RngHom 𝑟) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟) ∧ ∀ℎ(ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0_g‘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 3140	. 2 ⊢ (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟))
75	1	rngccat 20530	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
76	3, 75	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
77	2, 19, 76, 25	isinito 17958	. 2 ⊢ (𝜑 → (𝑍 ∈ (InitO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)))
78	74, 77	mpbird 257	1 ⊢ (𝜑 → 𝑍 ∈ (InitO‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 ∀wal 1531 = wceq 1533 ∈ wcel 2098 ∃!weu 2556 ∀wral 3055 Vcvv 3468 ∖ cdif 3940 ∩ cin 3942 {csn 4623 ↦ cmpt 5224 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 Hom chom 17217 0_gc0g 17394 Catccat 17617 InitOcinito 17943 GrpHom cghm 19138 Rngcrng 20057 Ringcrg 20138 RngHom crnghm 20336 NzRingcnzr 20414 RngCatcrngc 20512

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-hash 14296 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-hom 17230 df-cco 17231 df-0g 17396 df-cat 17621 df-cid 17622 df-homf 17623 df-ssc 17766 df-resc 17767 df-subc 17768 df-inito 17946 df-estrc 18086 df-mgm 18573 df-mgmhm 18625 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-grp 18866 df-minusg 18867 df-ghm 19139 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-rnghm 20338 df-nzr 20415 df-rngc 20513

This theorem is referenced by: zrzeroorngc 20540

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