Theorem nzerooringczr 21417

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 20578	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
6	3, 5	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		iszeroi 17916	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ ℎ ∈ (ZeroO‘𝐶)) → (ℎ ∈ (Base‘𝐶) ∧ (ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶))))
8	6, 7	sylan 580	. . . . . 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 20590	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶))
12		nzerooringczr.i	. . . . . . . . . 10 ⊢ (𝜑 → ℤring ∈ 𝑈)
13	3, 12, 4	irinitoringc 21416	. . . . . . . . 9 ⊢ (𝜑 → ℤring ∈ (InitO‘𝐶))
14	6	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → 𝐶 ∈ Cat)
15		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → ℎ ∈ (InitO‘𝐶))
16		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → ℤring ∈ (InitO‘𝐶))
17	14, 15, 16	initoeu1w 17919	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → ℎ( ≃𝑐 ‘𝐶)ℤring)
18	6	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ℎ ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → 𝐶 ∈ Cat)
19		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ℎ ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → 𝑍 ∈ (TermO‘𝐶))
20		simplr 768	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ℎ ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → ℎ ∈ (TermO‘𝐶))
21	18, 19, 20	termoeu1w 17926	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ℎ ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → 𝑍( ≃𝑐 ‘𝐶)ℎ)
22		cictr 17712	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐶 ∈ Cat ∧ 𝑍( ≃𝑐 ‘𝐶)ℎ ∧ ℎ( ≃𝑐 ‘𝐶)ℤring) → 𝑍( ≃𝑐 ‘𝐶)ℤring)
23	6, 22	syl3an1 1163	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑍( ≃𝑐 ‘𝐶)ℎ ∧ ℎ( ≃𝑐 ‘𝐶)ℤring) → 𝑍( ≃𝑐 ‘𝐶)ℤring)
24		eqid 2731	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
25		eqid 2731	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Base‘𝐶) = (Base‘𝐶)
26	9	eldifad 3909	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑍 ∈ Ring)
27	10, 26	elind 4147	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑍 ∈ (𝑈 ∩ Ring))
28	4, 25, 3	ringcbas 20565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring))
29	27, 28	eleqtrrd 2834	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
30		zringring 21386	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ℤring ∈ Ring
31	30	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → ℤring ∈ Ring)
32	12, 31	elind 4147	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ℤring ∈ (𝑈 ∩ Ring))
33	32, 28	eleqtrrd 2834	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ℤring ∈ (Base‘𝐶))
34	24, 25, 6, 29, 33	cic 17706	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
37	25, 36, 24, 6, 29, 33	isohom 17683	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring))
38		ssn0 4351	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20567	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (𝑍(Hom ‘𝐶)ℤring) = (𝑍 RingHom ℤring))
40	39	neeq1d 2987	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → ((𝑍(Hom ‘𝐶)ℤring) ≠ ∅ ↔ (𝑍 RingHom ℤring) ≠ ∅))
41		zringnzr 21397	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ℤring ∈ NzRing
42		nrhmzr 20452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ ℤring ∈ NzRing) → (𝑍 RingHom ℤring) = ∅)
43	9, 41, 42	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (𝑍 RingHom ℤring) = ∅)
44		eqneqall 2939	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑍(Iso‘𝐶)ℤring) ≠ ∅ → ((𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring) → (𝜑 → (ZeroO‘𝐶) = ∅)))
49	48	com13 88	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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	biimtrrid 243	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (∃𝑓 𝑓 ∈ (𝑍(Iso‘𝐶)ℤring) → (ZeroO‘𝐶) = ∅))
52	34, 51	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑍( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅))
53	52	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . . . . . . . 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 1119	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑍( ≃𝑐 ‘𝐶)ℎ → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅)))
56	55	a1dd 50	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑍( ≃𝑐 ‘𝐶)ℎ → (ℎ ∈ (Base‘𝐶) → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅))))
57	56	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℎ ∈ (TermO‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℎ ∈ (Base‘𝐶) → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅)))))
60	59	com34 91	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℎ ∈ (TermO‘𝐶) → (ℎ ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ZeroO‘𝐶) = ∅)))))
61	60	com25 99	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℎ( ≃𝑐 ‘𝐶)ℤring → (ℎ ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℎ ∈ (TermO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
62	61	ad2antrr 726	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) → (ℤring ∈ (InitO‘𝐶) → (ℎ ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℎ ∈ (TermO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
65	64	com25 99	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) → (ℎ ∈ (TermO‘𝐶) → (ℎ ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℤring ∈ (InitO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
66	65	expimpd 453	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶)) → (ℎ ∈ (Base‘𝐶) → (𝑍 ∈ (TermO‘𝐶) → (ℤring ∈ (InitO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
67	66	com23 86	. . . . . . . . . . 11 ⊢ (𝜑 → (ℎ ∈ (Base‘𝐶) → ((ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶)) → (𝑍 ∈ (TermO‘𝐶) → (ℤring ∈ (InitO‘𝐶) → (ZeroO‘𝐶) = ∅)))))
68	67	impd 410	. . . . . . . . . 10 ⊢ (𝜑 → ((ℎ ∈ (Base‘𝐶) ∧ (ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶))) → (𝑍 ∈ (TermO‘𝐶) → (ℤring ∈ (InitO‘𝐶) → (ZeroO‘𝐶) = ∅))))
69	68	com24 95	. . . . . . . . 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 413	. . . 4 ⊢ (ℎ ∈ (ZeroO‘𝐶) → (𝜑 → (ZeroO‘𝐶) = ∅))
75	74	exlimiv 1931	. . 3 ⊢ (∃ℎ ℎ ∈ (ZeroO‘𝐶) → (𝜑 → (ZeroO‘𝐶) = ∅))
76	2, 75	sylbi 217	. 2 ⊢ (¬ (ZeroO‘𝐶) = ∅ → (𝜑 → (ZeroO‘𝐶) = ∅))
77	1, 76	pm2.61i 182	1 ⊢ (𝜑 → (ZeroO‘𝐶) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111   ≠ wne 2928   ∖ cdif 3894   ∩ cin 3896   ⊆ wss 3897  ∅c0 4280   class class class wbr 5089  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  Hom chom 17172  Catccat 17570  Isociso 17653   ≃_𝑐 ccic 17702  InitOcinito 17888  TermOctermo 17889  ZeroOczeroo 17890  Ringcrg 20151   RingHom crh 20387  NzRingcnzr 20427  RingCatcringc 20560  ℤ_ringczring 21383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-addf 11085  ax-mulf 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-oadd 8389  df-er 8622  df-map 8752  df-pm 8753  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-dju 9794  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-xnn0 12455  df-z 12469  df-dec 12589  df-uz 12733  df-fz 13408  df-seq 13909  df-hash 14238  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-starv 17176  df-tset 17180  df-ple 17181  df-ds 17183  df-unif 17184  df-hom 17185  df-cco 17186  df-0g 17345  df-cat 17574  df-cid 17575  df-homf 17576  df-sect 17654  df-inv 17655  df-iso 17656  df-cic 17703  df-ssc 17717  df-resc 17718  df-subc 17719  df-inito 17891  df-termo 17892  df-zeroo 17893  df-estrc 18029  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-mhm 18691  df-grp 18849  df-minusg 18850  df-mulg 18981  df-subg 19036  df-ghm 19125  df-cmn 19694  df-abl 19695  df-mgp 20059  df-rng 20071  df-ur 20100  df-ring 20153  df-cring 20154  df-rhm 20390  df-nzr 20428  df-subrng 20461  df-subrg 20485  df-ringc 20561  df-cnfld 21292  df-zring 21384

This theorem is referenced by: (None)