Theorem nzerooringczr 21447

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 4306	. . 3 ⊢ (¬ (ZeroO‘𝐶) = ∅ ↔ ∃ℎ ℎ ∈ (ZeroO‘𝐶))
3		nzerooringczr.u	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		nzerooringczr.c	. . . . . . . . 9 ⊢ 𝐶 = (RingCat‘𝑈)
5	4	ringccat 20608	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
6	3, 5	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		iszeroi 17945	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ ℎ ∈ (ZeroO‘𝐶)) → (ℎ ∈ (Base‘𝐶) ∧ (ℎ ∈ (InitO‘𝐶) ∧ ℎ ∈ (TermO‘𝐶))))
8	6, 7	sylan 581	. . . . . 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 20620	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶))
12		nzerooringczr.i	. . . . . . . . . 10 ⊢ (𝜑 → ℤring ∈ 𝑈)
13	3, 12, 4	irinitoringc 21446	. . . . . . . . 9 ⊢ (𝜑 → ℤring ∈ (InitO‘𝐶))
14	6	ad2antrr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → 𝐶 ∈ Cat)
15		simplr 769	. . . . . . . . . . . . . . . . 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 17948	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ℎ ∈ (InitO‘𝐶)) ∧ ℤring ∈ (InitO‘𝐶)) → ℎ( ≃𝑐 ‘𝐶)ℤring)
18	6	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ℎ ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → ℎ ∈ (TermO‘𝐶))
21	18, 19, 20	termoeu1w 17955	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ℎ ∈ (TermO‘𝐶)) ∧ 𝑍 ∈ (TermO‘𝐶)) → 𝑍( ≃𝑐 ‘𝐶)ℎ)
22		cictr 17741	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐶 ∈ Cat ∧ 𝑍( ≃𝑐 ‘𝐶)ℎ ∧ ℎ( ≃𝑐 ‘𝐶)ℤring) → 𝑍( ≃𝑐 ‘𝐶)ℤring)
23	6, 22	syl3an1 1164	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑍( ≃𝑐 ‘𝐶)ℎ ∧ ℎ( ≃𝑐 ‘𝐶)ℤring) → 𝑍( ≃𝑐 ‘𝐶)ℤring)
24		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
25		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Base‘𝐶) = (Base‘𝐶)
26	9	eldifad 3915	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑍 ∈ Ring)
27	10, 26	elind 4154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑍 ∈ (𝑈 ∩ Ring))
28	4, 25, 3	ringcbas 20595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring))
29	27, 28	eleqtrrd 2840	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
30		zringring 21416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ℤring ∈ Ring
31	30	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → ℤring ∈ Ring)
32	12, 31	elind 4154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ℤring ∈ (𝑈 ∩ Ring))
33	32, 28	eleqtrrd 2840	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ℤring ∈ (Base‘𝐶))
34	24, 25, 6, 29, 33	cic 17735	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑍( ≃𝑐 ‘𝐶)ℤring ↔ ∃𝑓 𝑓 ∈ (𝑍(Iso‘𝐶)ℤring)))
35		n0 4307	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑍(Iso‘𝐶)ℤring) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑍(Iso‘𝐶)ℤring))
36		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
37	25, 36, 24, 6, 29, 33	isohom 17712	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (𝑍(Iso‘𝐶)ℤring) ⊆ (𝑍(Hom ‘𝐶)ℤring))
38		ssn0 4358	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (𝑍(Hom ‘𝐶)ℤring) = (𝑍 RingHom ℤring))
40	39	neeq1d 2992	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → ((𝑍(Hom ‘𝐶)ℤring) ≠ ∅ ↔ (𝑍 RingHom ℤring) ≠ ∅))
41		zringnzr 21427	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ℤring ∈ NzRing
42		nrhmzr 20482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ ℤring ∈ NzRing) → (𝑍 RingHom ℤring) = ∅)
43	9, 41, 42	sylancl 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (𝑍 RingHom ℤring) = ∅)
44		eqneqall 2944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . . . . . . . . . . . . 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 1120	. . . . . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . 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 1932	. . 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 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  Hom chom 17200  Catccat 17599  Isociso 17682   ≃_𝑐 ccic 17731  InitOcinito 17917  TermOctermo 17918  ZeroOczeroo 17919  Ringcrg 20180   RingHom crh 20417  NzRingcnzr 20457  RingCatcringc 20590  ℤ_ringczring 21413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-addf 11117  ax-mulf 11118

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-supp 8113  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-er 8645  df-map 8777  df-pm 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-dju 9825  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-xnn0 12487  df-z 12501  df-dec 12620  df-uz 12764  df-fz 13436  df-seq 13937  df-hash 14266  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-mulr 17203  df-starv 17204  df-tset 17208  df-ple 17209  df-ds 17211  df-unif 17212  df-hom 17213  df-cco 17214  df-0g 17373  df-cat 17603  df-cid 17604  df-homf 17605  df-sect 17683  df-inv 17684  df-iso 17685  df-cic 17732  df-ssc 17746  df-resc 17747  df-subc 17748  df-inito 17920  df-termo 17921  df-zeroo 17922  df-estrc 18058  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-mhm 18720  df-grp 18878  df-minusg 18879  df-mulg 19010  df-subg 19065  df-ghm 19154  df-cmn 19723  df-abl 19724  df-mgp 20088  df-rng 20100  df-ur 20129  df-ring 20182  df-cring 20183  df-rhm 20420  df-nzr 20458  df-subrng 20491  df-subrg 20515  df-ringc 20591  df-cnfld 21322  df-zring 21414

This theorem is referenced by: (None)