Theorem nzerooringczr 41864

Description: There is no zero object in the category of unital rings (at least in a universe which contains the zero ring and the ring of integers). Example 7.9 (3) in [Adamek] p. 103. (Contributed by AV, 18-Apr-2020.)

Hypotheses

Ref	Expression
nzerooringczr.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
nzerooringczr.c	⊢ 𝐶 = (RingCat‘𝑈)
nzerooringczr.z	⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing))
nzerooringczr.e	⊢ (𝜑 → 𝑍 ∈ 𝑈)
nzerooringczr.i	⊢ (𝜑 → ℤring ∈ 𝑈)

Assertion

Ref	Expression
nzerooringczr	⊢ (𝜑 → (ZeroO‘𝐶) = ∅)

Proof of Theorem nzerooringczr

Dummy variables 𝑓 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-1 6	. 2 ⊢ ((ZeroO‘𝐶) = ∅ → (𝜑 → (ZeroO‘𝐶) = ∅))
2		neq0 3889	. . 3 ⊢ (¬ (ZeroO‘𝐶) = ∅ ↔ ∃ℎ ℎ ∈ (ZeroO‘𝐶))
3		nzerooringczr.u	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		nzerooringczr.c	. . . . . . . . 9 ⊢ 𝐶 = (RingCat‘𝑈)
5	4	ringccat 41816	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
6	3, 5	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		iszeroi 16482	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ ℎ ∈ (ZeroO‘𝐶)) → (ℎ ∈ (Base‘𝐶) ∧ (ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶))))
8	6, 7	sylan 487	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ (ZeroO‘𝐶)) → (ℎ ∈ (Base‘𝐶) ∧ (ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶))))
9		nzerooringczr.z	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing))
10		nzerooringczr.e	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
11	3, 4, 9, 10	zrtermoringc 41862	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶))
12		nzerooringczr.i	. . . . . . . . . 10 ⊢ (𝜑 → ℤring ∈ 𝑈)
13	3, 12, 4	irinitoringc 41861	. . . . . . . . 9 ⊢ (𝜑 → ℤring ∈ (InitO‘𝐶))
14	6	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → 𝐶 ∈ Cat)
15		simplr 788	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → ℎ ∈ (InitO‘𝐶))
16		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → ℤring ∈ (InitO‘𝐶))
17	14, 15, 16	initoeu1w 16485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → ℎ( ≃𝑐 ‘𝐶)ℤring)
18	6	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ℎ ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → 𝐶 ∈ Cat)
19		simpr 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ℎ ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → 𝑍 ∈ (TermO‘𝐶))
20		simplr 788	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ℎ ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → ℎ ∈ (TermO‘𝐶))
21	18, 19, 20	termoeu1w 16492	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ℎ ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → 𝑍( ≃𝑐 ‘𝐶)ℎ)
22		cictr 16288	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐶 ∈ Cat ∧ 𝑍( ≃𝑐 ‘𝐶)ℎ ∧ ℎ( ≃𝑐 ‘𝐶)ℤring) → 𝑍( ≃𝑐 ‘𝐶)ℤring)
23	6, 22	syl3an1 1351	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑍( ≃𝑐 ‘𝐶)ℎ ∧ ℎ( ≃𝑐 ‘𝐶)ℤring) → 𝑍( ≃𝑐 ‘𝐶)ℤring)
24		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
25		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Base‘𝐶) = (Base‘𝐶)
26	9	eldifad 3552	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑍 ∈ Ring)
27	10, 26	elind 3760	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑍 ∈ (𝑈 ∩ Ring))
28	4, 25, 3	ringcbas 41803	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring))
29	27, 28	eleqtrrd 2691	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
30		zringring 19640	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ℤring ∈ Ring
31	30	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → ℤring ∈ Ring)
32	12, 31	elind 3760	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ℤring ∈ (𝑈 ∩ Ring))
33	32, 28	eleqtrrd 2691	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ℤring ∈ (Base‘𝐶))
34	24, 25, 6, 29, 33	cic 16282	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑍( ≃𝑐 ‘𝐶)ℤring ↔ ∃𝑓 𝑓 ∈ (𝑍(Iso‘𝐶)ℤring)))
35		n0 3890	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑍(Iso‘𝐶)ℤring) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑍(Iso‘𝐶)ℤring))
36		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
37	25, 36, 24, 6, 29, 33	isohom 16259	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring))
38		ssn0 3928	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring) ∧ (𝑍(Iso‘𝐶)ℤring) ≠ ∅) → (𝑍(Hom ‘𝐶)ℤring) ≠ ∅)
39	4, 25, 3, 36, 29, 33	ringchom 41805	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (𝑍(Hom ‘𝐶)ℤring) = (𝑍 RingHom ℤring))
40	39	neeq1d 2841	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → ((𝑍(Hom ‘𝐶)ℤring) ≠ ∅ ↔ (𝑍 RingHom ℤring) ≠ ∅))
41		zringnzr 19649	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ℤring ∈ NzRing
42		nrhmzr 41663	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ ℤring ∈ NzRing) → (𝑍 RingHom ℤring) = ∅)
43	9, 41, 42	sylancl 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (𝑍 RingHom ℤring) = ∅)
44		eqneqall 2793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑍 RingHom ℤring) = ∅ → ((𝑍 RingHom ℤring) ≠ ∅ → (ZeroO‘𝐶) = ∅))
45	43, 44	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → ((𝑍 RingHom ℤring) ≠ ∅ → (ZeroO‘𝐶) = ∅))
46	40, 45	sylbid 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → ((𝑍(Hom ‘𝐶)ℤring) ≠ ∅ → (ZeroO‘𝐶) = ∅))
47	38, 46	syl5com 31	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring) ∧ (𝑍(Iso‘𝐶)ℤring) ≠ ∅) → (𝜑 → (ZeroO‘𝐶) = ∅))
48	47	expcom 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑍(Iso‘𝐶)ℤring) ≠ ∅ → ((𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring) → (𝜑 → (ZeroO‘𝐶) = ∅)))
49	48	com13 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ((𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring) → ((𝑍(Iso‘𝐶)ℤring) ≠ ∅ → (ZeroO‘𝐶) = ∅)))
50	37, 49	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ((𝑍(Iso‘𝐶)ℤring) ≠ ∅ → (ZeroO‘𝐶) = ∅))
51	35, 50	syl5bir 232	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (∃𝑓 𝑓 ∈ (𝑍(Iso‘𝐶)ℤring) → (ZeroO‘𝐶) = ∅))
52	34, 51	sylbid 229	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑍( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅))
53	52	3ad2ant1 1075	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑍( ≃𝑐 ‘𝐶)ℎ ∧ ℎ( ≃𝑐 ‘𝐶)ℤring) → (𝑍( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅))
54	23, 53	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑍( ≃𝑐 ‘𝐶)ℎ ∧ ℎ( ≃𝑐 ‘𝐶)ℤring) → (ZeroO‘𝐶) = ∅)
55	54	3exp 1256	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑍( ≃𝑐 ‘𝐶)ℎ → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅)))
56	55	a1dd 48	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑍( ≃𝑐 ‘𝐶)ℎ → (ℎ ∈ (Base‘𝐶) → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅))))
57	56	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ℎ ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → (𝑍( ≃𝑐 ‘𝐶)ℎ → (ℎ ∈ (Base‘𝐶) → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅))))
58	21, 57	mpd 15	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ℎ ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → (ℎ ∈ (Base‘𝐶) → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅)))
59	58	exp31 628	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℎ ∈ (TermO‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℎ ∈ (Base‘𝐶) → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅)))))
60	59	com34 89	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℎ ∈ (TermO‘𝐶) → (ℎ ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅)))))
61	60	com25 97	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ℎ ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℎ ∈ (TermO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
62	61	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ℎ ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℎ ∈ (TermO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
63	17, 62	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → (ℎ ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℎ ∈ (TermO‘𝐶) → (ZeroO‘𝐶) = ∅))))
64	63	ex 449	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) → (ℤring ∈ (InitO‘𝐶) → (ℎ ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℎ ∈ (TermO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
65	64	com25 97	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) → (ℎ ∈ (TermO‘𝐶) → (ℎ ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℤring ∈ (InitO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
66	65	expimpd 627	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶)) → (ℎ ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℤring ∈ (InitO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
67	66	com23 84	. . . . . . . . . . 11 ⊢ (𝜑 → (ℎ ∈ (Base‘𝐶) → ((ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶)) → (𝑍 ∈ (TermO‘𝐶) → (ℤring ∈ (InitO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
68	67	impd 446	. . . . . . . . . 10 ⊢ (𝜑 → ((ℎ ∈ (Base‘𝐶) ∧ (ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶))) → (𝑍 ∈ (TermO‘𝐶) → (ℤring ∈ (InitO‘𝐶) → (ZeroO‘𝐶) = ∅))))
69	68	com24 93	. . . . . . . . 9 ⊢ (𝜑 → (ℤring ∈ (InitO‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → ((ℎ ∈ (Base‘𝐶) ∧ (ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶))) → (ZeroO‘𝐶) = ∅))))
70	13, 69	mpd 15	. . . . . . . 8 ⊢ (𝜑 → (𝑍 ∈ (TermO‘𝐶) → ((ℎ ∈ (Base‘𝐶) ∧ (ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶))) → (ZeroO‘𝐶) = ∅)))
71	11, 70	mpd 15	. . . . . . 7 ⊢ (𝜑 → ((ℎ ∈ (Base‘𝐶) ∧ (ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶))) → (ZeroO‘𝐶) = ∅))
72	71	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ (ZeroO‘𝐶)) → ((ℎ ∈ (Base‘𝐶) ∧ (ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶))) → (ZeroO‘𝐶) = ∅))
73	8, 72	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ (ZeroO‘𝐶)) → (ZeroO‘𝐶) = ∅)
74	73	expcom 450	. . . 4 ⊢ (ℎ ∈ (ZeroO‘𝐶) → (𝜑 → (ZeroO‘𝐶) = ∅))
75	74	exlimiv 1845	. . 3 ⊢ (∃ℎ ℎ ∈ (ZeroO‘𝐶) → (𝜑 → (ZeroO‘𝐶) = ∅))
76	2, 75	sylbi 206	. 2 ⊢ (¬ (ZeroO‘𝐶) = ∅ → (𝜑 → (ZeroO‘𝐶) = ∅))
77	1, 76	pm2.61i 175	1 ⊢ (𝜑 → (ZeroO‘𝐶) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  Catccat 16148  Isociso 16229   ≃_𝑐 ccic 16278  InitOcinito 16461  TermOctermo 16462  ZeroOczeroo 16463  Ringcrg 18370   RingHom crh 18535  NzRingcnzr 19078  ℤ_ringzring 19637  RingCatcringc 41795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-addf 9894  ax-mulf 9895

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-xnn0 11241  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-0g 15925  df-cat 16152  df-cid 16153  df-homf 16154  df-sect 16230  df-inv 16231  df-iso 16232  df-cic 16279  df-ssc 16293  df-resc 16294  df-subc 16295  df-inito 16464  df-termo 16465  df-zeroo 16466  df-estrc 16586  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-grp 17248  df-minusg 17249  df-mulg 17364  df-subg 17414  df-ghm 17481  df-cmn 18018  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-rnghom 18538  df-subrg 18601  df-nzr 19079  df-cnfld 19568  df-zring 19638  df-ringc 41797

This theorem is referenced by: (None)