Theorem nzerooringczr 21779

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 4299	. . 3 ⊢ (¬ (ZeroO‘𝐶) = ∅ ↔ ∃ℎ ℎ ∈ (ZeroO‘𝐶))
3		nzerooringczr.u	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		nzerooringczr.c	. . . . . . . . 9 ⊢ 𝐶 = (RingCat‘𝑈)
5	4	ringccat 20908	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
6	3, 5	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		iszeroi 18177	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ ℎ ∈ (ZeroO‘𝐶)) → (ℎ ∈ (Base‘𝐶) ∧ (ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶))))
8	6, 7	sylan 592	. . . . . 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 20920	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶))
12		nzerooringczr.i	. . . . . . . . . 10 ⊢ (𝜑 → ℤring ∈ 𝑈)
13	3, 12, 4	irinitoringc 21778	. . . . . . . . 9 ⊢ (𝜑 → ℤring ∈ (InitO‘𝐶))
14	6	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → 𝐶 ∈ Cat)
15		simplr 781	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → ℎ ∈ (InitO‘𝐶))
16		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → ℤring ∈ (InitO‘𝐶))
17	14, 15, 16	initoeu1w 18180	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → ℎ( ≃𝑐 ‘𝐶)ℤring)
18	6	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ℎ ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → 𝐶 ∈ Cat)
19		simpr 490	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ℎ ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → 𝑍 ∈ (TermO‘𝐶))
20		simplr 781	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ℎ ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → ℎ ∈ (TermO‘𝐶))
21	18, 19, 20	termoeu1w 18187	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ℎ ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → 𝑍( ≃𝑐 ‘𝐶)ℎ)
22		cictr 17973	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐶 ∈ Cat ∧ 𝑍( ≃𝑐 ‘𝐶)ℎ ∧ ℎ( ≃𝑐 ‘𝐶)ℤring) → 𝑍( ≃𝑐 ‘𝐶)ℤring)
23	6, 22	syl3an1 1181	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑍( ≃𝑐 ‘𝐶)ℎ ∧ ℎ( ≃𝑐 ‘𝐶)ℤring) → 𝑍( ≃𝑐 ‘𝐶)ℤring)
24		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
25		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Base‘𝐶) = (Base‘𝐶)
26	9	eldifad 3911	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑍 ∈ Ring)
27	10, 26	elind 4146	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑍 ∈ (𝑈 ∩ Ring))
28	4, 25, 3	ringcbas 20895	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring))
29	27, 28	eleqtrrd 2864	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
30		zringring 21748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ℤring ∈ Ring
31	30	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → ℤring ∈ Ring)
32	12, 31	elind 4146	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ℤring ∈ (𝑈 ∩ Ring))
33	32, 28	eleqtrrd 2864	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ℤring ∈ (Base‘𝐶))
34	24, 25, 6, 29, 33	cic 17967	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑍( ≃𝑐 ‘𝐶)ℤring ↔ ∃𝑓 𝑓 ∈ (𝑍(Iso‘𝐶)ℤring)))
35		n0 4300	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑍(Iso‘𝐶)ℤring) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑍(Iso‘𝐶)ℤring))
36		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
37	25, 36, 24, 6, 29, 33	isohom 17944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring))
38		ssn0 4355	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20897	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (𝑍(Hom ‘𝐶)ℤring) = (𝑍 RingHom ℤring))
40	39	neeq1d 3015	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → ((𝑍(Hom ‘𝐶)ℤring) ≠ ∅ ↔ (𝑍 RingHom ℤring) ≠ ∅))
41		zringnzr 21759	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ℤring ∈ NzRing
42		nrhmzr 20782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ ℤring ∈ NzRing) → (𝑍 RingHom ℤring) = ∅)
43	9, 41, 42	sylancl 598	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (𝑍 RingHom ℤring) = ∅)
44		eqneqall 2967	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑍 RingHom ℤring) = ∅ → ((𝑍 RingHom ℤring) ≠ ∅ → (ZeroO‘𝐶) = ∅))
45	43, 44	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → ((𝑍 RingHom ℤring) ≠ ∅ → (ZeroO‘𝐶) = ∅))
46	40, 45	sylbid 243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → ((𝑍(Hom ‘𝐶)ℤring) ≠ ∅ → (ZeroO‘𝐶) = ∅))
47	38, 46	syl5com 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring) ∧ (𝑍(Iso‘𝐶)ℤring) ≠ ∅) → (𝜑 → (ZeroO‘𝐶) = ∅))
48	47	expcom 419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑍(Iso‘𝐶)ℤring) ≠ ∅ → ((𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring) → (𝜑 → (ZeroO‘𝐶) = ∅)))
49	48	com13 89	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ((𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring) → ((𝑍(Iso‘𝐶)ℤring) ≠ ∅ → (ZeroO‘𝐶) = ∅)))
50	37, 49	mpd 16	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ((𝑍(Iso‘𝐶)ℤring) ≠ ∅ → (ZeroO‘𝐶) = ∅))
51	35, 50	biimtrrid 246	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (∃𝑓 𝑓 ∈ (𝑍(Iso‘𝐶)ℤring) → (ZeroO‘𝐶) = ∅))
52	34, 51	sylbid 243	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑍( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅))
53	52	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑍( ≃𝑐 ‘𝐶)ℎ ∧ ℎ( ≃𝑐 ‘𝐶)ℤring) → (𝑍( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅))
54	23, 53	mpd 16	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑍( ≃𝑐 ‘𝐶)ℎ ∧ ℎ( ≃𝑐 ‘𝐶)ℤring) → (ZeroO‘𝐶) = ∅)
55	54	3exp 1137	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑍( ≃𝑐 ‘𝐶)ℎ → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅)))
56	55	a1dd 51	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑍( ≃𝑐 ‘𝐶)ℎ → (ℎ ∈ (Base‘𝐶) → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅))))
57	56	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ℎ ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → (ℎ ∈ (Base‘𝐶) → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅)))
59	58	exp31 425	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℎ ∈ (TermO‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℎ ∈ (Base‘𝐶) → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅)))))
60	59	com34 92	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℎ ∈ (TermO‘𝐶) → (ℎ ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅)))))
61	60	com25 100	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ℎ ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℎ ∈ (TermO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
62	61	ad2antrr 739	. . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → (ℎ ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℎ ∈ (TermO‘𝐶) → (ZeroO‘𝐶) = ∅))))
64	63	ex 418	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) → (ℤring ∈ (InitO‘𝐶) → (ℎ ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℎ ∈ (TermO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
65	64	com25 100	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) → (ℎ ∈ (TermO‘𝐶) → (ℎ ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℤring ∈ (InitO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
66	65	expimpd 459	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶)) → (ℎ ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℤring ∈ (InitO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
67	66	com23 87	. . . . . . . . . . 11 ⊢ (𝜑 → (ℎ ∈ (Base‘𝐶) → ((ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶)) → (𝑍 ∈ (TermO‘𝐶) → (ℤring ∈ (InitO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
68	67	impd 416	. . . . . . . . . 10 ⊢ (𝜑 → ((ℎ ∈ (Base‘𝐶) ∧ (ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶))) → (𝑍 ∈ (TermO‘𝐶) → (ℤring ∈ (InitO‘𝐶) → (ZeroO‘𝐶) = ∅))))
69	68	com24 96	. . . . . . . . 9 ⊢ (𝜑 → (ℤring ∈ (InitO‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → ((ℎ ∈ (Base‘𝐶) ∧ (ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶))) → (ZeroO‘𝐶) = ∅))))
70	13, 69	mpd 16	. . . . . . . 8 ⊢ (𝜑 → (𝑍 ∈ (TermO‘𝐶) → ((ℎ ∈ (Base‘𝐶) ∧ (ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶))) → (ZeroO‘𝐶) = ∅)))
71	11, 70	mpd 16	. . . . . . 7 ⊢ (𝜑 → ((ℎ ∈ (Base‘𝐶) ∧ (ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶))) → (ZeroO‘𝐶) = ∅))
72	71	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ (ZeroO‘𝐶)) → ((ℎ ∈ (Base‘𝐶) ∧ (ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶))) → (ZeroO‘𝐶) = ∅))
73	8, 72	mpd 16	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ (ZeroO‘𝐶)) → (ZeroO‘𝐶) = ∅)
74	73	expcom 419	. . . 4 ⊢ (ℎ ∈ (ZeroO‘𝐶) → (𝜑 → (ZeroO‘𝐶) = ∅))
75	74	exlimiv 1963	. . 3 ⊢ (∃ℎ ℎ ∈ (ZeroO‘𝐶) → (𝜑 → (ZeroO‘𝐶) = ∅))
76	2, 75	sylbi 220	. 2 ⊢ (¬ (ZeroO‘𝐶) = ∅ → (𝜑 → (ZeroO‘𝐶) = ∅))
77	1, 76	pm2.61i 184	1 ⊢ (𝜑 → (ZeroO‘𝐶) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2956   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  Hom chom 17432  Catccat 17831  Isociso 17914   ≃_𝑐 ccic 17963  InitOcinito 18149  TermOctermo 18150  ZeroOczeroo 18151  Ringcrg 20452   RingHom crh 20692  NzRingcnzr 20755  RingCatcringc 20890  ℤ_ringczring 21745

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270  ax-addf 11272  ax-mulf 11273

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-supp 8171  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-oadd 8473  df-er 8710  df-map 8842  df-pm 8843  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-dju 9975  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-xnn0 12673  df-z 12687  df-dec 12808  df-uz 12959  df-fz 13633  df-seq 14138  df-hash 14468  df-struct 17318  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-ress 17402  df-plusg 17434  df-mulr 17435  df-starv 17436  df-tset 17440  df-ple 17441  df-ds 17443  df-unif 17444  df-hom 17445  df-cco 17446  df-0g 17605  df-cat 17835  df-cid 17836  df-homf 17837  df-sect 17915  df-inv 17916  df-iso 17917  df-cic 17964  df-ssc 17978  df-resc 17979  df-subc 17980  df-inito 18152  df-termo 18153  df-zeroo 18154  df-estrc 18290  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-mhm 18971  df-grp 19140  df-minusg 19141  df-mulg 19271  df-subg 19326  df-ghm 19421  df-cmn 19989  df-abl 19990  df-mgp 20354  df-rng 20368  df-ur 20401  df-ring 20454  df-cring 20455  df-rhm 20695  df-nzr 20756  df-subrng 20791  df-subrg 20815  df-ringc 20891  df-cnfld 21672  df-zring 21746

This theorem is used by: (None)