Theorem zrtermorngc 20805

Description: The zero ring is a terminal object in the category of non-unital rings. (Contributed by AV, 17-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
zrtermorngc	⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶))

Proof of Theorem zrtermorngc

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 2760	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		zrinitorngc.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4	1, 2, 3	rngcbas 20783	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng))
5	4	eleq2d 2846	. . . . . . . 8 ⊢ (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Rng)))
6		elin 3915	. . . . . . . . 9 ⊢ (𝑟 ∈ (𝑈 ∩ Rng) ↔ (𝑟 ∈ 𝑈 ∧ 𝑟 ∈ Rng))
7	6	simprbi 503	. . . . . . . 8 ⊢ (𝑟 ∈ (𝑈 ∩ Rng) → 𝑟 ∈ Rng)
8	5, 7	biimtrdi 256	. . . . . . 7 ⊢ (𝜑 → (𝑟 ∈ (Base‘𝐶) → 𝑟 ∈ Rng))
9	8	imp 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ Rng)
10		zrinitorngc.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing))
11	10	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Ring ∖ NzRing))
12		eqid 2760	. . . . . . 7 ⊢ (Base‘𝑟) = (Base‘𝑟)
13		eqid 2760	. . . . . . 7 ⊢ (0g‘𝑍) = (0g‘𝑍)
14		eqid 2760	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))
15	12, 13, 14	c0rnghm 20697	. . . . . 6 ⊢ ((𝑟 ∈ Rng ∧ 𝑍 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍))
16	9, 11, 15	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍))
17		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍))
18	3	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑈 ∈ 𝑉)
19		eqid 2760	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
20		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ (Base‘𝐶))
21		zrinitorngc.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
22		eldifi 4078	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ (Ring ∖ NzRing) → 𝑍 ∈ Ring)
23		ringrng 20426	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ Ring → 𝑍 ∈ Rng)
24	10, 22, 23	3syl 19	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ Rng)
25	21, 24	elind 4146	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ∈ (𝑈 ∩ Rng))
26	25, 4	eleqtrrd 2863	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
27	26	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Base‘𝐶))
28	1, 2, 18, 19, 20, 27	rngchom 20785	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑟(Hom ‘𝐶)𝑍) = (𝑟 RngHom 𝑍))
29	28	eqcomd 2766	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑟 RngHom 𝑍) = (𝑟(Hom ‘𝐶)𝑍))
30	29	eleq2d 2846	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍) ↔ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍)))
31	30	biimpa 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍))
32	28	eleq2d 2846	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ ℎ ∈ (𝑟 RngHom 𝑍)))
33		eqid 2760	. . . . . . . . . . 11 ⊢ (Base‘𝑍) = (Base‘𝑍)
34	12, 33	rnghmf 20589	. . . . . . . . . 10 ⊢ (ℎ ∈ (𝑟 RngHom 𝑍) → ℎ:(Base‘𝑟)⟶(Base‘𝑍))
35	32, 34	biimtrdi 256	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ:(Base‘𝑟)⟶(Base‘𝑍)))
36	35	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ:(Base‘𝑟)⟶(Base‘𝑍)))
37		ffn 6702	. . . . . . . . . . 11 ⊢ (ℎ:(Base‘𝑟)⟶(Base‘𝑍) → ℎ Fn (Base‘𝑟))
38	37	adantl 487	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) → ℎ Fn (Base‘𝑟))
39		fvex 6891	. . . . . . . . . . . 12 ⊢ (0g‘𝑍) ∈ V
40	39, 14	fnmpti 6675	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) Fn (Base‘𝑟)
41	40	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) Fn (Base‘𝑟))
42	33, 13	0ringbas 20689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑍 ∈ (Ring ∖ NzRing) → (Base‘𝑍) = {(0g‘𝑍)})
43	10, 42	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝑍) = {(0g‘𝑍)})
44	43	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (Base‘𝑍) = {(0g‘𝑍)})
45	44	feq3d 6687	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ:(Base‘𝑟)⟶(Base‘𝑍) ↔ ℎ:(Base‘𝑟)⟶{(0g‘𝑍)}))
46		fvconst 7160	. . . . . . . . . . . . . . 15 ⊢ ((ℎ:(Base‘𝑟)⟶{(0g‘𝑍)} ∧ 𝑎 ∈ (Base‘𝑟)) → (ℎ‘𝑎) = (0g‘𝑍))
47	46	ex 418	. . . . . . . . . . . . . 14 ⊢ (ℎ:(Base‘𝑟)⟶{(0g‘𝑍)} → (𝑎 ∈ (Base‘𝑟) → (ℎ‘𝑎) = (0g‘𝑍)))
48	45, 47	biimtrdi 256	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℎ:(Base‘𝑟)⟶(Base‘𝑍) → (𝑎 ∈ (Base‘𝑟) → (ℎ‘𝑎) = (0g‘𝑍))))
49	48	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) → (ℎ:(Base‘𝑟)⟶(Base‘𝑍) → (𝑎 ∈ (Base‘𝑟) → (ℎ‘𝑎) = (0g‘𝑍))))
50	49	imp31 423	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) ∧ 𝑎 ∈ (Base‘𝑟)) → (ℎ‘𝑎) = (0g‘𝑍))
51		eqidd 2761	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Base‘𝑟) → (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)))
52		eqidd 2761	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ (Base‘𝑟) ∧ 𝑥 = 𝑎) → (0g‘𝑍) = (0g‘𝑍))
53		id 23	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Base‘𝑟) → 𝑎 ∈ (Base‘𝑟))
54	39	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Base‘𝑟) → (0g‘𝑍) ∈ V)
55	51, 52, 53, 54	fvmptd 6994	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (Base‘𝑟) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))‘𝑎) = (0g‘𝑍))
56	55	adantl 487	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) ∧ 𝑎 ∈ (Base‘𝑟)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))‘𝑎) = (0g‘𝑍))
57	50, 56	eqtr4d 2798	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) ∧ 𝑎 ∈ (Base‘𝑟)) → (ℎ‘𝑎) = ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))‘𝑎))
58	38, 41, 57	eqfnfvd 7025	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) ∧ ℎ:(Base‘𝑟)⟶(Base‘𝑍)) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)))
59	58	ex 418	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) → (ℎ:(Base‘𝑟)⟶(Base‘𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))))
60	36, 59	syld 48	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))))
61	60	alrimiv 1960	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) → ∀ℎ(ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍))))
62	17, 31, 61	3jca 1146	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍) ∧ ∀ℎ(ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)))))
63	16, 62	mpdan 700	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍) ∧ ∀ℎ(ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)))))
64		eleq1 2848	. . . . 5 ⊢ (ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) → (ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) ↔ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍)))
65	64	eqeu 3664	. . . 4 ⊢ (((𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟 RngHom 𝑍) ∧ (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)) ∈ (𝑟(Hom ‘𝐶)𝑍) ∧ ∀ℎ(ℎ ∈ (𝑟(Hom ‘𝐶)𝑍) → ℎ = (𝑥 ∈ (Base‘𝑟) ↦ (0g‘𝑍)))) → ∃!ℎ ℎ ∈ (𝑟(Hom ‘𝐶)𝑍))
66	63, 65	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ∃!ℎ ℎ ∈ (𝑟(Hom ‘𝐶)𝑍))
67	66	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑟(Hom ‘𝐶)𝑍))
68	1	rngccat 20796	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
69	3, 68	syl 18	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
70	2, 19, 69, 26	istermo 18086	. 2 ⊢ (𝜑 → (𝑍 ∈ (TermO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑟(Hom ‘𝐶)𝑍)))
71	67, 70	mpbird 260	1 ⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∃!weu 2593  ∀wral 3076  Vcvv 3450   ∖ cdif 3896   ∩ cin 3898  {csn 4584   ↦ cmpt 5186   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  Hom chom 17353  0_gc0g 17524  Catccat 17752  TermOctermo 18071  Rngcrng 20287  Ringcrg 20372   RngHom crnghm 20575  NzRingcnzr 20672  RngCatcrngc 20778

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-oadd 8459  df-er 8696  df-map 8828  df-pm 8829  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-dju 9906  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-xnn0 12602  df-z 12616  df-dec 12737  df-uz 12888  df-fz 13562  df-hash 14395  df-struct 17239  df-sets 17256  df-slot 17274  df-ndx 17286  df-base 17302  df-ress 17323  df-plusg 17355  df-hom 17366  df-cco 17367  df-0g 17526  df-cat 17756  df-cid 17757  df-homf 17758  df-ssc 17899  df-resc 17900  df-subc 17901  df-termo 18074  df-estrc 18211  df-mgm 18730  df-mgmhm 18794  df-sgrp 18821  df-mnd 18837  df-mhm 18891  df-grp 19060  df-minusg 19061  df-ghm 19341  df-cmn 19909  df-abl 19910  df-mgp 20274  df-rng 20288  df-ur 20321  df-ring 20374  df-rnghm 20577  df-nzr 20673  df-rngc 20779

This theorem is used by:  zrzeroorngc  20806