Theorem nzerooringczr 21433

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 4302	. . 3 ⊢ (¬ (ZeroO‘𝐶) = ∅ ↔ ∃ℎ ℎ ∈ (ZeroO‘𝐶))
3		nzerooringczr.u	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		nzerooringczr.c	. . . . . . . . 9 ⊢ 𝐶 = (RingCat‘𝑈)
5	4	ringccat 20594	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
6	3, 5	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		iszeroi 17931	. . . . . . 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 20606	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶))
12		nzerooringczr.i	. . . . . . . . . 10 ⊢ (𝜑 → ℤring ∈ 𝑈)
13	3, 12, 4	irinitoringc 21432	. . . . . . . . 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 17934	. . . . . . . . . . . . . . . 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 17941	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ℎ ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → 𝑍( ≃𝑐 ‘𝐶)ℎ)
22		cictr 17727	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
25		eqid 2734	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Base‘𝐶) = (Base‘𝐶)
26	9	eldifad 3911	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑍 ∈ Ring)
27	10, 26	elind 4150	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑍 ∈ (𝑈 ∩ Ring))
28	4, 25, 3	ringcbas 20581	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring))
29	27, 28	eleqtrrd 2837	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
30		zringring 21402	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ℤring ∈ Ring
31	30	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → ℤring ∈ Ring)
32	12, 31	elind 4150	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ℤring ∈ (𝑈 ∩ Ring))
33	32, 28	eleqtrrd 2837	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ℤring ∈ (Base‘𝐶))
34	24, 25, 6, 29, 33	cic 17721	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑍( ≃𝑐 ‘𝐶)ℤring ↔ ∃𝑓 𝑓 ∈ (𝑍(Iso‘𝐶)ℤring)))
35		n0 4303	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑍(Iso‘𝐶)ℤring) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑍(Iso‘𝐶)ℤring))
36		eqid 2734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
37	25, 36, 24, 6, 29, 33	isohom 17698	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring))
38		ssn0 4354	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (𝑍(Hom ‘𝐶)ℤring) = (𝑍 RingHom ℤring))
40	39	neeq1d 2989	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → ((𝑍(Hom ‘𝐶)ℤring) ≠ ∅ ↔ (𝑍 RingHom ℤring) ≠ ∅))
41		zringnzr 21413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ℤring ∈ NzRing
42		nrhmzr 20468	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ ℤring ∈ NzRing) → (𝑍 RingHom ℤring) = ∅)
43	9, 41, 42	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (𝑍 RingHom ℤring) = ∅)
44		eqneqall 2941	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2113   ≠ wne 2930   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4283   class class class wbr 5096  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  Hom chom 17186  Catccat 17585  Isociso 17668   ≃_𝑐 ccic 17717  InitOcinito 17903  TermOctermo 17904  ZeroOczeroo 17905  Ringcrg 20166   RingHom crh 20403  NzRingcnzr 20443  RingCatcringc 20576  ℤ_ringczring 21399

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-addf 11103  ax-mulf 11104

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-er 8633  df-map 8763  df-pm 8764  df-ixp 8834  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-dju 9811  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-9 12213  df-n0 12400  df-xnn0 12473  df-z 12487  df-dec 12606  df-uz 12750  df-fz 13422  df-seq 13923  df-hash 14252  df-struct 17072  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-ress 17156  df-plusg 17188  df-mulr 17189  df-starv 17190  df-tset 17194  df-ple 17195  df-ds 17197  df-unif 17198  df-hom 17199  df-cco 17200  df-0g 17359  df-cat 17589  df-cid 17590  df-homf 17591  df-sect 17669  df-inv 17670  df-iso 17671  df-cic 17718  df-ssc 17732  df-resc 17733  df-subc 17734  df-inito 17906  df-termo 17907  df-zeroo 17908  df-estrc 18044  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-mhm 18706  df-grp 18864  df-minusg 18865  df-mulg 18996  df-subg 19051  df-ghm 19140  df-cmn 19709  df-abl 19710  df-mgp 20074  df-rng 20086  df-ur 20115  df-ring 20168  df-cring 20169  df-rhm 20406  df-nzr 20444  df-subrng 20477  df-subrg 20501  df-ringc 20577  df-cnfld 21308  df-zring 21400

This theorem is referenced by: (None)