Theorem nzss 41549

Description: The set of multiples of m, mℤ, is a subset of those of n, nℤ, iff n divides m. Lemma 2.1(a) of https://www.mscs.dal.ca/~selinger/3343/handouts/ideals.pdf p. 5, with mℤ and nℤ as images of the divides relation under m and n. (Contributed by Steve Rodriguez, 20-Jan-2020.)

Hypotheses

Ref	Expression
nzss.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
nzss.n	⊢ (𝜑 → 𝑁 ∈ 𝑉)

Assertion

Ref	Expression
nzss	⊢ (𝜑 → (( ∥ “ {𝑀}) ⊆ ( ∥ “ {𝑁}) ↔ 𝑁 ∥ 𝑀))

Proof of Theorem nzss

Dummy variables 𝑥 𝑦 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nzss.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2		nzss.n	. 2 ⊢ (𝜑 → 𝑁 ∈ 𝑉)
3		iddvds 15794	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∥ 𝑀)
4		breq2 5043	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (𝑀 ∥ 𝑥 ↔ 𝑀 ∥ 𝑀))
5	4	elabg 3574	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀 ∈ {𝑥 ∣ 𝑀 ∥ 𝑥} ↔ 𝑀 ∥ 𝑀))
6	3, 5	mpbird 260	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ {𝑥 ∣ 𝑀 ∥ 𝑥})
7		reldvds 41547	. . . . . . . . 9 ⊢ Rel ∥
8		relimasn 5937	. . . . . . . . 9 ⊢ (Rel ∥ → ( ∥ “ {𝑀}) = {𝑥 ∣ 𝑀 ∥ 𝑥})
9	7, 8	ax-mp 5	. . . . . . . 8 ⊢ ( ∥ “ {𝑀}) = {𝑥 ∣ 𝑀 ∥ 𝑥}
10	6, 9	eleqtrrdi 2842	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ( ∥ “ {𝑀}))
11		ssel 3880	. . . . . . 7 ⊢ (( ∥ “ {𝑀}) ⊆ ( ∥ “ {𝑁}) → (𝑀 ∈ ( ∥ “ {𝑀}) → 𝑀 ∈ ( ∥ “ {𝑁})))
12	10, 11	syl5 34	. . . . . 6 ⊢ (( ∥ “ {𝑀}) ⊆ ( ∥ “ {𝑁}) → (𝑀 ∈ ℤ → 𝑀 ∈ ( ∥ “ {𝑁})))
13		breq2 5043	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝑁 ∥ 𝑥 ↔ 𝑁 ∥ 𝑀))
14		relimasn 5937	. . . . . . . 8 ⊢ (Rel ∥ → ( ∥ “ {𝑁}) = {𝑥 ∣ 𝑁 ∥ 𝑥})
15	7, 14	ax-mp 5	. . . . . . 7 ⊢ ( ∥ “ {𝑁}) = {𝑥 ∣ 𝑁 ∥ 𝑥}
16	13, 15	elab2g 3578	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝑀 ∈ ( ∥ “ {𝑁}) ↔ 𝑁 ∥ 𝑀))
17	12, 16	mpbidi 244	. . . . 5 ⊢ (( ∥ “ {𝑀}) ⊆ ( ∥ “ {𝑁}) → (𝑀 ∈ ℤ → 𝑁 ∥ 𝑀))
18	17	com12 32	. . . 4 ⊢ (𝑀 ∈ ℤ → (( ∥ “ {𝑀}) ⊆ ( ∥ “ {𝑁}) → 𝑁 ∥ 𝑀))
19	18	adantr 484	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ 𝑉) → (( ∥ “ {𝑀}) ⊆ ( ∥ “ {𝑁}) → 𝑁 ∥ 𝑀))
20		ssid 3909	. . . . . . 7 ⊢ {0} ⊆ {0}
21		simpl 486	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∥ 𝑀 ∧ 𝑁 = 0) → 𝑁 ∥ 𝑀)
22		breq1 5042	. . . . . . . . . . . . . 14 ⊢ (𝑁 = 0 → (𝑁 ∥ 𝑀 ↔ 0 ∥ 𝑀))
23		dvdszrcl 15783	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∥ 𝑀 → (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ))
24	23	simprd 499	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∥ 𝑀 → 𝑀 ∈ ℤ)
25		0dvds 15801	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℤ → (0 ∥ 𝑀 ↔ 𝑀 = 0))
26	24, 25	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∥ 𝑀 → (0 ∥ 𝑀 ↔ 𝑀 = 0))
27	22, 26	sylan9bbr 514	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∥ 𝑀 ∧ 𝑁 = 0) → (𝑁 ∥ 𝑀 ↔ 𝑀 = 0))
28	21, 27	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝑁 ∥ 𝑀 ∧ 𝑁 = 0) → 𝑀 = 0)
29	28	breq1d 5049	. . . . . . . . . . 11 ⊢ ((𝑁 ∥ 𝑀 ∧ 𝑁 = 0) → (𝑀 ∥ 𝑥 ↔ 0 ∥ 𝑥))
30		0dvds 15801	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℤ → (0 ∥ 𝑥 ↔ 𝑥 = 0))
31	29, 30	sylan9bb 513	. . . . . . . . . 10 ⊢ (((𝑁 ∥ 𝑀 ∧ 𝑁 = 0) ∧ 𝑥 ∈ ℤ) → (𝑀 ∥ 𝑥 ↔ 𝑥 = 0))
32	31	rabbidva 3378	. . . . . . . . 9 ⊢ ((𝑁 ∥ 𝑀 ∧ 𝑁 = 0) → {𝑥 ∈ ℤ ∣ 𝑀 ∥ 𝑥} = {𝑥 ∈ ℤ ∣ 𝑥 = 0})
33		0z 12152	. . . . . . . . . 10 ⊢ 0 ∈ ℤ
34		rabsn 4623	. . . . . . . . . 10 ⊢ (0 ∈ ℤ → {𝑥 ∈ ℤ ∣ 𝑥 = 0} = {0})
35	33, 34	ax-mp 5	. . . . . . . . 9 ⊢ {𝑥 ∈ ℤ ∣ 𝑥 = 0} = {0}
36	32, 35	eqtrdi 2787	. . . . . . . 8 ⊢ ((𝑁 ∥ 𝑀 ∧ 𝑁 = 0) → {𝑥 ∈ ℤ ∣ 𝑀 ∥ 𝑥} = {0})
37		breq1 5042	. . . . . . . . . . 11 ⊢ (𝑁 = 0 → (𝑁 ∥ 𝑥 ↔ 0 ∥ 𝑥))
38	37	rabbidv 3380	. . . . . . . . . 10 ⊢ (𝑁 = 0 → {𝑥 ∈ ℤ ∣ 𝑁 ∥ 𝑥} = {𝑥 ∈ ℤ ∣ 0 ∥ 𝑥})
39	30	rabbiia 3372	. . . . . . . . . . 11 ⊢ {𝑥 ∈ ℤ ∣ 0 ∥ 𝑥} = {𝑥 ∈ ℤ ∣ 𝑥 = 0}
40	39, 35	eqtri 2759	. . . . . . . . . 10 ⊢ {𝑥 ∈ ℤ ∣ 0 ∥ 𝑥} = {0}
41	38, 40	eqtrdi 2787	. . . . . . . . 9 ⊢ (𝑁 = 0 → {𝑥 ∈ ℤ ∣ 𝑁 ∥ 𝑥} = {0})
42	41	adantl 485	. . . . . . . 8 ⊢ ((𝑁 ∥ 𝑀 ∧ 𝑁 = 0) → {𝑥 ∈ ℤ ∣ 𝑁 ∥ 𝑥} = {0})
43	36, 42	sseq12d 3920	. . . . . . 7 ⊢ ((𝑁 ∥ 𝑀 ∧ 𝑁 = 0) → ({𝑥 ∈ ℤ ∣ 𝑀 ∥ 𝑥} ⊆ {𝑥 ∈ ℤ ∣ 𝑁 ∥ 𝑥} ↔ {0} ⊆ {0}))
44	20, 43	mpbiri 261	. . . . . 6 ⊢ ((𝑁 ∥ 𝑀 ∧ 𝑁 = 0) → {𝑥 ∈ ℤ ∣ 𝑀 ∥ 𝑥} ⊆ {𝑥 ∈ ℤ ∣ 𝑁 ∥ 𝑥})
45	24	zcnd 12248	. . . . . . . . . . . 12 ⊢ (𝑁 ∥ 𝑀 → 𝑀 ∈ ℂ)
46	45	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝑁 ∥ 𝑀 ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℤ) → 𝑀 ∈ ℂ)
47	23	simpld 498	. . . . . . . . . . . . 13 ⊢ (𝑁 ∥ 𝑀 → 𝑁 ∈ ℤ)
48	47	zcnd 12248	. . . . . . . . . . . 12 ⊢ (𝑁 ∥ 𝑀 → 𝑁 ∈ ℂ)
49	48	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝑁 ∥ 𝑀 ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℤ) → 𝑁 ∈ ℂ)
50		simplr 769	. . . . . . . . . . 11 ⊢ (((𝑁 ∥ 𝑀 ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℤ) → 𝑁 ≠ 0)
51	46, 49, 50	divcan2d 11575	. . . . . . . . . 10 ⊢ (((𝑁 ∥ 𝑀 ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℤ) → (𝑁 · (𝑀 / 𝑁)) = 𝑀)
52	51	breq1d 5049	. . . . . . . . 9 ⊢ (((𝑁 ∥ 𝑀 ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℤ) → ((𝑁 · (𝑀 / 𝑁)) ∥ 𝑛 ↔ 𝑀 ∥ 𝑛))
53	47	adantr 484	. . . . . . . . . . 11 ⊢ ((𝑁 ∥ 𝑀 ∧ 𝑁 ≠ 0) → 𝑁 ∈ ℤ)
54		dvdsval2 15781	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ∧ 𝑀 ∈ ℤ) → (𝑁 ∥ 𝑀 ↔ (𝑀 / 𝑁) ∈ ℤ))
55	54	biimpd 232	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ∧ 𝑀 ∈ ℤ) → (𝑁 ∥ 𝑀 → (𝑀 / 𝑁) ∈ ℤ))
56	55	3com23 1128	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑁 ∥ 𝑀 → (𝑀 / 𝑁) ∈ ℤ))
57	56	3expa 1120	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑁 ≠ 0) → (𝑁 ∥ 𝑀 → (𝑀 / 𝑁) ∈ ℤ))
58	23, 57	sylan 583	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∥ 𝑀 ∧ 𝑁 ≠ 0) → (𝑁 ∥ 𝑀 → (𝑀 / 𝑁) ∈ ℤ))
59	58	imp 410	. . . . . . . . . . . 12 ⊢ (((𝑁 ∥ 𝑀 ∧ 𝑁 ≠ 0) ∧ 𝑁 ∥ 𝑀) → (𝑀 / 𝑁) ∈ ℤ)
60	59	anabss1 666	. . . . . . . . . . 11 ⊢ ((𝑁 ∥ 𝑀 ∧ 𝑁 ≠ 0) → (𝑀 / 𝑁) ∈ ℤ)
61	53, 60	jca 515	. . . . . . . . . 10 ⊢ ((𝑁 ∥ 𝑀 ∧ 𝑁 ≠ 0) → (𝑁 ∈ ℤ ∧ (𝑀 / 𝑁) ∈ ℤ))
62		muldvds1 15805	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ (𝑀 / 𝑁) ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑁 · (𝑀 / 𝑁)) ∥ 𝑛 → 𝑁 ∥ 𝑛))
63	62	3expa 1120	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℤ ∧ (𝑀 / 𝑁) ∈ ℤ) ∧ 𝑛 ∈ ℤ) → ((𝑁 · (𝑀 / 𝑁)) ∥ 𝑛 → 𝑁 ∥ 𝑛))
64	61, 63	sylan 583	. . . . . . . . 9 ⊢ (((𝑁 ∥ 𝑀 ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℤ) → ((𝑁 · (𝑀 / 𝑁)) ∥ 𝑛 → 𝑁 ∥ 𝑛))
65	52, 64	sylbird 263	. . . . . . . 8 ⊢ (((𝑁 ∥ 𝑀 ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℤ) → (𝑀 ∥ 𝑛 → 𝑁 ∥ 𝑛))
66	65	ss2rabdv 3975	. . . . . . 7 ⊢ ((𝑁 ∥ 𝑀 ∧ 𝑁 ≠ 0) → {𝑛 ∈ ℤ ∣ 𝑀 ∥ 𝑛} ⊆ {𝑛 ∈ ℤ ∣ 𝑁 ∥ 𝑛})
67		breq2 5043	. . . . . . . 8 ⊢ (𝑛 = 𝑥 → (𝑀 ∥ 𝑛 ↔ 𝑀 ∥ 𝑥))
68	67	cbvrabv 3392	. . . . . . 7 ⊢ {𝑛 ∈ ℤ ∣ 𝑀 ∥ 𝑛} = {𝑥 ∈ ℤ ∣ 𝑀 ∥ 𝑥}
69		breq2 5043	. . . . . . . 8 ⊢ (𝑛 = 𝑥 → (𝑁 ∥ 𝑛 ↔ 𝑁 ∥ 𝑥))
70	69	cbvrabv 3392	. . . . . . 7 ⊢ {𝑛 ∈ ℤ ∣ 𝑁 ∥ 𝑛} = {𝑥 ∈ ℤ ∣ 𝑁 ∥ 𝑥}
71	66, 68, 70	3sstr3g 3931	. . . . . 6 ⊢ ((𝑁 ∥ 𝑀 ∧ 𝑁 ≠ 0) → {𝑥 ∈ ℤ ∣ 𝑀 ∥ 𝑥} ⊆ {𝑥 ∈ ℤ ∣ 𝑁 ∥ 𝑥})
72	44, 71	pm2.61dane 3019	. . . . 5 ⊢ (𝑁 ∥ 𝑀 → {𝑥 ∈ ℤ ∣ 𝑀 ∥ 𝑥} ⊆ {𝑥 ∈ ℤ ∣ 𝑁 ∥ 𝑥})
73		breq1 5042	. . . . . . . . . 10 ⊢ (𝑛 = 𝑀 → (𝑛 ∥ 𝑥 ↔ 𝑀 ∥ 𝑥))
74	73	rabbidv 3380	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → {𝑥 ∈ ℤ ∣ 𝑛 ∥ 𝑥} = {𝑥 ∈ ℤ ∣ 𝑀 ∥ 𝑥})
75	73	abbidv 2800	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → {𝑥 ∣ 𝑛 ∥ 𝑥} = {𝑥 ∣ 𝑀 ∥ 𝑥})
76	74, 75	eqeq12d 2752	. . . . . . . 8 ⊢ (𝑛 = 𝑀 → ({𝑥 ∈ ℤ ∣ 𝑛 ∥ 𝑥} = {𝑥 ∣ 𝑛 ∥ 𝑥} ↔ {𝑥 ∈ ℤ ∣ 𝑀 ∥ 𝑥} = {𝑥 ∣ 𝑀 ∥ 𝑥}))
77		simpr 488	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∥ 𝑦) → 𝑛 ∥ 𝑦)
78		dvdszrcl 15783	. . . . . . . . . . . . 13 ⊢ (𝑛 ∥ 𝑦 → (𝑛 ∈ ℤ ∧ 𝑦 ∈ ℤ))
79	78	simprd 499	. . . . . . . . . . . 12 ⊢ (𝑛 ∥ 𝑦 → 𝑦 ∈ ℤ)
80	79	ancri 553	. . . . . . . . . . 11 ⊢ (𝑛 ∥ 𝑦 → (𝑦 ∈ ℤ ∧ 𝑛 ∥ 𝑦))
81	77, 80	impbii 212	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∥ 𝑦) ↔ 𝑛 ∥ 𝑦)
82		breq2 5043	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑛 ∥ 𝑥 ↔ 𝑛 ∥ 𝑦))
83	82	elrab 3591	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ ℤ ∣ 𝑛 ∥ 𝑥} ↔ (𝑦 ∈ ℤ ∧ 𝑛 ∥ 𝑦))
84		vex 3402	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
85	84, 82	elab 3576	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∣ 𝑛 ∥ 𝑥} ↔ 𝑛 ∥ 𝑦)
86	81, 83, 85	3bitr4i 306	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∈ ℤ ∣ 𝑛 ∥ 𝑥} ↔ 𝑦 ∈ {𝑥 ∣ 𝑛 ∥ 𝑥})
87	86	eqriv 2733	. . . . . . . 8 ⊢ {𝑥 ∈ ℤ ∣ 𝑛 ∥ 𝑥} = {𝑥 ∣ 𝑛 ∥ 𝑥}
88	76, 87	vtoclg 3471	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → {𝑥 ∈ ℤ ∣ 𝑀 ∥ 𝑥} = {𝑥 ∣ 𝑀 ∥ 𝑥})
89	88	adantr 484	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ 𝑉) → {𝑥 ∈ ℤ ∣ 𝑀 ∥ 𝑥} = {𝑥 ∣ 𝑀 ∥ 𝑥})
90		breq1 5042	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑛 ∥ 𝑥 ↔ 𝑁 ∥ 𝑥))
91	90	rabbidv 3380	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → {𝑥 ∈ ℤ ∣ 𝑛 ∥ 𝑥} = {𝑥 ∈ ℤ ∣ 𝑁 ∥ 𝑥})
92	90	abbidv 2800	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → {𝑥 ∣ 𝑛 ∥ 𝑥} = {𝑥 ∣ 𝑁 ∥ 𝑥})
93	91, 92	eqeq12d 2752	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ({𝑥 ∈ ℤ ∣ 𝑛 ∥ 𝑥} = {𝑥 ∣ 𝑛 ∥ 𝑥} ↔ {𝑥 ∈ ℤ ∣ 𝑁 ∥ 𝑥} = {𝑥 ∣ 𝑁 ∥ 𝑥}))
94	93, 87	vtoclg 3471	. . . . . . 7 ⊢ (𝑁 ∈ 𝑉 → {𝑥 ∈ ℤ ∣ 𝑁 ∥ 𝑥} = {𝑥 ∣ 𝑁 ∥ 𝑥})
95	94	adantl 485	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ 𝑉) → {𝑥 ∈ ℤ ∣ 𝑁 ∥ 𝑥} = {𝑥 ∣ 𝑁 ∥ 𝑥})
96	89, 95	sseq12d 3920	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ 𝑉) → ({𝑥 ∈ ℤ ∣ 𝑀 ∥ 𝑥} ⊆ {𝑥 ∈ ℤ ∣ 𝑁 ∥ 𝑥} ↔ {𝑥 ∣ 𝑀 ∥ 𝑥} ⊆ {𝑥 ∣ 𝑁 ∥ 𝑥}))
97	72, 96	syl5ib 247	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ 𝑉) → (𝑁 ∥ 𝑀 → {𝑥 ∣ 𝑀 ∥ 𝑥} ⊆ {𝑥 ∣ 𝑁 ∥ 𝑥}))
98	9, 15	sseq12i 3917	. . . 4 ⊢ (( ∥ “ {𝑀}) ⊆ ( ∥ “ {𝑁}) ↔ {𝑥 ∣ 𝑀 ∥ 𝑥} ⊆ {𝑥 ∣ 𝑁 ∥ 𝑥})
99	97, 98	syl6ibr 255	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ 𝑉) → (𝑁 ∥ 𝑀 → ( ∥ “ {𝑀}) ⊆ ( ∥ “ {𝑁})))
100	19, 99	impbid 215	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ 𝑉) → (( ∥ “ {𝑀}) ⊆ ( ∥ “ {𝑁}) ↔ 𝑁 ∥ 𝑀))
101	1, 2, 100	syl2anc 587	1 ⊢ (𝜑 → (( ∥ “ {𝑀}) ⊆ ( ∥ “ {𝑁}) ↔ 𝑁 ∥ 𝑀))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112  {cab 2714   ≠ wne 2932  {crab 3055   ⊆ wss 3853  {csn 4527   class class class wbr 5039   “ cima 5539  Rel wrel 5541  (class class class)co 7191  ℂcc 10692  0cc0 10694   · cmul 10699   / cdiv 11454  ℤcz 12141   ∥ cdvds 15778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501  ax-resscn 10751  ax-1cn 10752  ax-icn 10753  ax-addcl 10754  ax-addrcl 10755  ax-mulcl 10756  ax-mulrcl 10757  ax-mulcom 10758  ax-addass 10759  ax-mulass 10760  ax-distr 10761  ax-i2m1 10762  ax-1ne0 10763  ax-1rid 10764  ax-rnegex 10765  ax-rrecex 10766  ax-cnre 10767  ax-pre-lttri 10768  ax-pre-lttrn 10769  ax-pre-ltadd 10770  ax-pre-mulgt0 10771

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-ov 7194  df-oprab 7195  df-mpo 7196  df-om 7623  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-er 8369  df-en 8605  df-dom 8606  df-sdom 8607  df-pnf 10834  df-mnf 10835  df-xr 10836  df-ltxr 10837  df-le 10838  df-sub 11029  df-neg 11030  df-div 11455  df-nn 11796  df-n0 12056  df-z 12142  df-dvds 15779

This theorem is referenced by:  nzin  41550