Theorem zrinitorngc 45446

Description: The zero ring is an initial object in the category of nonunital 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 2738	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		zrinitorngc.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4	1, 2, 3	rngcbas 45411	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
5	4	eleq2d 2824	. . . . . . . 8 ⊢ (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Rng)))
6		elin 3899	. . . . . . . . 9 ⊢ (𝑟 ∈ (𝑈 ∩ Rng) ↔ (𝑟 ∈ 𝑈 ∧ 𝑟 ∈ Rng))
7	6	simprbi 496	. . . . . . . 8 ⊢ (𝑟 ∈ (𝑈 ∩ Rng) → 𝑟 ∈ Rng)
8	5, 7	syl6bi 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 2738	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍)
13		eqid 2738	. . . . . . 7 ⊢ (0g‘𝑟) = (0g‘𝑟)
14		eqid 2738	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟))
15	12, 13, 14	zrrnghm 45363	. . . . . 6 ⊢ ((𝑟 ∈ Rng ∧ 𝑍 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHomo 𝑟))
16	9, 11, 15	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHomo 𝑟))
17		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHomo 𝑟)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHomo 𝑟))
18	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑈 ∈ 𝑉)
19		eqid 2738	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
20		zrinitorngc.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
21		eldifi 4057	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ (Ring ∖ NzRing) → 𝑍 ∈ Ring)
22		ringrng 45325	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ Ring → 𝑍 ∈ Rng)
23	10, 21, 22	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ Rng)
24	20, 23	elind 4124	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ∈ (𝑈 ∩ Rng))
25	24, 4	eleqtrrd 2842	. . . . . . . . . . 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 45413	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑍(Hom ‘𝐶)𝑟) = (𝑍 RngHomo 𝑟))
29	28	eqcomd 2744	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑍 RngHomo 𝑟) = (𝑍(Hom ‘𝐶)𝑟))
30	29	eleq2d 2824	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHomo 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 RngHomo 𝑟)) → (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟))
32	28	eleq2d 2824	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ ℎ ∈ (𝑍 RngHomo 𝑟)))
33		eqid 2738	. . . . . . . . . . . . 13 ⊢ (Base‘𝑟) = (Base‘𝑟)
34	12, 33	rnghmf 45345	. . . . . . . . . . . 12 ⊢ (ℎ ∈ (𝑍 RngHomo 𝑟) → ℎ:(Base‘𝑍)⟶(Base‘𝑟))
35	32, 34	syl6bi 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 6584	. . . . . . . . . . . 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 6769	. . . . . . . . . . . . 13 ⊢ (0g‘𝑟) ∈ V
40	39, 14	fnmpti 6560	. . . . . . . . . . . 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 RngHomo 𝑟))
43		rnghmghm 45344	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (𝑍 RngHomo 𝑟) → ℎ ∈ (𝑍 GrpHom 𝑟))
44		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝑍) = (0g‘𝑍)
45	44, 13	ghmid 18755	. . . . . . . . . . . . . 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 722	. . . . . . . . . . . 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 45317	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0g‘𝑍)})
49	10, 48	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝑍) = {(0g‘𝑍)})
50	49	eleq2d 2824	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑎 ∈ (Base‘𝑍) ↔ 𝑎 ∈ {(0g‘𝑍)}))
51		elsni 4575	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ {(0g‘𝑍)} → 𝑎 = (0g‘𝑍))
52	51	fveq2d 6760	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ {(0g‘𝑍)} → (ℎ‘𝑎) = (ℎ‘(0g‘𝑍)))
53	50, 52	syl6bi 252	. . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (Base‘𝑍) → (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)))
58		eqidd 2739	. . . . . . . . . . . . . 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 6864	. . . . . . . . . . . . 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 2788	. . . . . . . . . . 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 6894	. . . . . . . . . 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 683	. . . . . . . . 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 RngHomo 𝑟)) → (ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟))))
68	67	alrimiv 1931	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHomo 𝑟)) → ∀ℎ(ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟))))
69	17, 31, 68	3jca 1126	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHomo 𝑟)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHomo 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 683	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHomo 𝑟) ∧ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟) ∧ ∀ℎ(ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)))))
71		eleq1 2826	. . . . 5 ⊢ (ℎ = (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) → (ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ (𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍(Hom ‘𝐶)𝑟)))
72	71	eqeu 3636	. . . 4 ⊢ (((𝑥 ∈ (Base‘𝑍) ↦ (0g‘𝑟)) ∈ (𝑍 RngHomo 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 3107	. 2 ⊢ (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟))
75	1	rngccat 45424	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
76	3, 75	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
77	2, 19, 76, 25	isinito 17627	. 2 ⊢ (𝜑 → (𝑍 ∈ (InitO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)))
78	74, 77	mpbird 256	1 ⊢ (𝜑 → 𝑍 ∈ (InitO‘𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085  ∀wal 1537   = wceq 1539   ∈ wcel 2108  ∃!weu 2568  ∀wral 3063  Vcvv 3422   ∖ cdif 3880   ∩ cin 3882  {csn 4558   ↦ cmpt 5153   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Hom chom 16899  0_gc0g 17067  Catccat 17290  InitOcinito 17612   GrpHom cghm 18746  Ringcrg 19698  NzRingcnzr 20441  Rngcrng 45320   RngHomo crngh 45331  RngCatcrngc 45403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-xnn0 12236  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-hash 13973  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-hom 16912  df-cco 16913  df-0g 17069  df-cat 17294  df-cid 17295  df-homf 17296  df-ssc 17439  df-resc 17440  df-subc 17441  df-inito 17615  df-estrc 17755  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-mhm 18345  df-grp 18495  df-minusg 18496  df-ghm 18747  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-nzr 20442  df-mgmhm 45221  df-rng0 45321  df-rnghomo 45333  df-rngc 45405

This theorem is referenced by:  zrzeroorngc  45448