ztprmneprm - Mathbox for Alexander van der Vekens

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Theorem ztprmneprm 47023

Description: A prime is not an integer multiple of another prime. (Contributed by AV, 23-May-2019.)

**Assertion**
Ref	Expression
ztprmneprm	⊢ ((𝑍 ∈ ℤ ∧ 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))

**Proof of Theorem ztprmneprm**
Step	Hyp	Ref	Expression
1		elznn0nn 12572	. . 3 ⊢ (𝑍 ∈ ℤ ↔ (𝑍 ∈ ℕ₀ ∨ (𝑍 ∈ ℝ ∧ -𝑍 ∈ ℕ)))
2		elnn0 12474	. . . . 5 ⊢ (𝑍 ∈ ℕ₀ ↔ (𝑍 ∈ ℕ ∨ 𝑍 = 0))
3		elnn1uz2 12909	. . . . . . 7 ⊢ (𝑍 ∈ ℕ ↔ (𝑍 = 1 ∨ 𝑍 ∈ (ℤ_≥‘2)))
4		oveq1 7416	. . . . . . . . . . . 12 ⊢ (𝑍 = 1 → (𝑍 · 𝐴) = (1 · 𝐴))
5	4	adantr 482	. . . . . . . . . . 11 ⊢ ((𝑍 = 1 ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → (𝑍 · 𝐴) = (1 · 𝐴))
6	5	eqeq1d 2735	. . . . . . . . . 10 ⊢ ((𝑍 = 1 ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → ((𝑍 · 𝐴) = 𝐵 ↔ (1 · 𝐴) = 𝐵))
7		prmz 16612	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℤ)
8	7	zcnd 12667	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℂ)
9	8	mullidd 11232	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℙ → (1 · 𝐴) = 𝐴)
10	9	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (1 · 𝐴) = 𝐴)
11	10	eqeq1d 2735	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((1 · 𝐴) = 𝐵 ↔ 𝐴 = 𝐵))
12	11	biimpd 228	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((1 · 𝐴) = 𝐵 → 𝐴 = 𝐵))
13	12	adantl 483	. . . . . . . . . 10 ⊢ ((𝑍 = 1 ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → ((1 · 𝐴) = 𝐵 → 𝐴 = 𝐵))
14	6, 13	sylbid 239	. . . . . . . . 9 ⊢ ((𝑍 = 1 ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))
15	14	ex 414	. . . . . . . 8 ⊢ (𝑍 = 1 → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
16		prmuz2 16633	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ (ℤ_≥‘2))
17	16	adantr 482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → 𝐴 ∈ (ℤ_≥‘2))
18		nprm 16625	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ (ℤ_≥‘2) ∧ 𝐴 ∈ (ℤ_≥‘2)) → ¬ (𝑍 · 𝐴) ∈ ℙ)
19	17, 18	sylan2 594	. . . . . . . . . 10 ⊢ ((𝑍 ∈ (ℤ_≥‘2) ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → ¬ (𝑍 · 𝐴) ∈ ℙ)
20		eleq1 2822	. . . . . . . . . . . . 13 ⊢ ((𝑍 · 𝐴) = 𝐵 → ((𝑍 · 𝐴) ∈ ℙ ↔ 𝐵 ∈ ℙ))
21	20	notbid 318	. . . . . . . . . . . 12 ⊢ ((𝑍 · 𝐴) = 𝐵 → (¬ (𝑍 · 𝐴) ∈ ℙ ↔ ¬ 𝐵 ∈ ℙ))
22		pm2.24 124	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℙ → (¬ 𝐵 ∈ ℙ → 𝐴 = 𝐵))
23	22	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (¬ 𝐵 ∈ ℙ → 𝐴 = 𝐵))
24	23	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝑍 ∈ (ℤ_≥‘2) ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → (¬ 𝐵 ∈ ℙ → 𝐴 = 𝐵))
25	24	com12 32	. . . . . . . . . . . 12 ⊢ (¬ 𝐵 ∈ ℙ → ((𝑍 ∈ (ℤ_≥‘2) ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → 𝐴 = 𝐵))
26	21, 25	syl6bi 253	. . . . . . . . . . 11 ⊢ ((𝑍 · 𝐴) = 𝐵 → (¬ (𝑍 · 𝐴) ∈ ℙ → ((𝑍 ∈ (ℤ_≥‘2) ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → 𝐴 = 𝐵)))
27	26	com3l 89	. . . . . . . . . 10 ⊢ (¬ (𝑍 · 𝐴) ∈ ℙ → ((𝑍 ∈ (ℤ_≥‘2) ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
28	19, 27	mpcom 38	. . . . . . . . 9 ⊢ ((𝑍 ∈ (ℤ_≥‘2) ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))
29	28	ex 414	. . . . . . . 8 ⊢ (𝑍 ∈ (ℤ_≥‘2) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
30	15, 29	jaoi 856	. . . . . . 7 ⊢ ((𝑍 = 1 ∨ 𝑍 ∈ (ℤ_≥‘2)) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
31	3, 30	sylbi 216	. . . . . 6 ⊢ (𝑍 ∈ ℕ → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
32		oveq1 7416	. . . . . . . . 9 ⊢ (𝑍 = 0 → (𝑍 · 𝐴) = (0 · 𝐴))
33	32	eqeq1d 2735	. . . . . . . 8 ⊢ (𝑍 = 0 → ((𝑍 · 𝐴) = 𝐵 ↔ (0 · 𝐴) = 𝐵))
34		prmnn 16611	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
35	34	nnred 12227	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℝ)
36		mul02lem2 11391	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → (0 · 𝐴) = 0)
37	35, 36	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℙ → (0 · 𝐴) = 0)
38	37	adantr 482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (0 · 𝐴) = 0)
39	38	eqeq1d 2735	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((0 · 𝐴) = 𝐵 ↔ 0 = 𝐵))
40		prmnn 16611	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℙ → 𝐵 ∈ ℕ)
41		elnnne0 12486	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ ↔ (𝐵 ∈ ℕ₀ ∧ 𝐵 ≠ 0))
42		eqneqall 2952	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 = 0 → (𝐵 ≠ 0 → 𝐴 = 𝐵))
43	42	eqcoms 2741	. . . . . . . . . . . . . . 15 ⊢ (0 = 𝐵 → (𝐵 ≠ 0 → 𝐴 = 𝐵))
44	43	com12 32	. . . . . . . . . . . . . 14 ⊢ (𝐵 ≠ 0 → (0 = 𝐵 → 𝐴 = 𝐵))
45	44	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℕ₀ ∧ 𝐵 ≠ 0) → (0 = 𝐵 → 𝐴 = 𝐵))
46	41, 45	sylbi 216	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ → (0 = 𝐵 → 𝐴 = 𝐵))
47	40, 46	syl 17	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℙ → (0 = 𝐵 → 𝐴 = 𝐵))
48	47	adantl 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (0 = 𝐵 → 𝐴 = 𝐵))
49	39, 48	sylbid 239	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((0 · 𝐴) = 𝐵 → 𝐴 = 𝐵))
50	49	com12 32	. . . . . . . 8 ⊢ ((0 · 𝐴) = 𝐵 → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → 𝐴 = 𝐵))
51	33, 50	syl6bi 253	. . . . . . 7 ⊢ (𝑍 = 0 → ((𝑍 · 𝐴) = 𝐵 → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → 𝐴 = 𝐵)))
52	51	com23 86	. . . . . 6 ⊢ (𝑍 = 0 → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
53	31, 52	jaoi 856	. . . . 5 ⊢ ((𝑍 ∈ ℕ ∨ 𝑍 = 0) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
54	2, 53	sylbi 216	. . . 4 ⊢ (𝑍 ∈ ℕ₀ → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
55		elnnz 12568	. . . . . 6 ⊢ (-𝑍 ∈ ℕ ↔ (-𝑍 ∈ ℤ ∧ 0 < -𝑍))
56		lt0neg1 11720	. . . . . . . 8 ⊢ (𝑍 ∈ ℝ → (𝑍 < 0 ↔ 0 < -𝑍))
57	34	nngt0d 12261	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℙ → 0 < 𝐴)
58	57	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → 0 < 𝐴)
59		simpr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑍 ∈ ℝ ∧ 𝑍 < 0) → 𝑍 < 0)
60	58, 59	anim12ci 615	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → (𝑍 < 0 ∧ 0 < 𝐴))
61	60	orcd 872	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → ((𝑍 < 0 ∧ 0 < 𝐴) ∨ (0 < 𝑍 ∧ 𝐴 < 0)))
62		simprl 770	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → 𝑍 ∈ ℝ)
63	35	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → 𝐴 ∈ ℝ)
64	63	adantr 482	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → 𝐴 ∈ ℝ)
65	62, 64	mul2lt0bi 13080	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → ((𝑍 · 𝐴) < 0 ↔ ((𝑍 < 0 ∧ 0 < 𝐴) ∨ (0 < 𝑍 ∧ 𝐴 < 0))))
66	61, 65	mpbird 257	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → (𝑍 · 𝐴) < 0)
67	66	ex 414	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 ∈ ℝ ∧ 𝑍 < 0) → (𝑍 · 𝐴) < 0))
68		breq1 5152	. . . . . . . . . . . . . 14 ⊢ ((𝑍 · 𝐴) = 𝐵 → ((𝑍 · 𝐴) < 0 ↔ 𝐵 < 0))
69	68	adantl 483	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 · 𝐴) = 𝐵) → ((𝑍 · 𝐴) < 0 ↔ 𝐵 < 0))
70		nnnn0 12479	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℕ₀)
71		nn0nlt0 12498	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ ℕ₀ → ¬ 𝐵 < 0)
72	71	pm2.21d 121	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ ℕ₀ → (𝐵 < 0 → 𝐴 = 𝐵))
73	70, 72	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℕ → (𝐵 < 0 → 𝐴 = 𝐵))
74	40, 73	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℙ → (𝐵 < 0 → 𝐴 = 𝐵))
75	74	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (𝐵 < 0 → 𝐴 = 𝐵))
76	75	adantr 482	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 · 𝐴) = 𝐵) → (𝐵 < 0 → 𝐴 = 𝐵))
77	69, 76	sylbid 239	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 · 𝐴) = 𝐵) → ((𝑍 · 𝐴) < 0 → 𝐴 = 𝐵))
78	77	ex 414	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → ((𝑍 · 𝐴) < 0 → 𝐴 = 𝐵)))
79	78	com23 86	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) < 0 → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
80	67, 79	syldc 48	. . . . . . . . 9 ⊢ ((𝑍 ∈ ℝ ∧ 𝑍 < 0) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
81	80	ex 414	. . . . . . . 8 ⊢ (𝑍 ∈ ℝ → (𝑍 < 0 → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))))
82	56, 81	sylbird 260	. . . . . . 7 ⊢ (𝑍 ∈ ℝ → (0 < -𝑍 → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))))
83	82	adantld 492	. . . . . 6 ⊢ (𝑍 ∈ ℝ → ((-𝑍 ∈ ℤ ∧ 0 < -𝑍) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))))
84	55, 83	biimtrid 241	. . . . 5 ⊢ (𝑍 ∈ ℝ → (-𝑍 ∈ ℕ → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))))
85	84	imp 408	. . . 4 ⊢ ((𝑍 ∈ ℝ ∧ -𝑍 ∈ ℕ) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
86	54, 85	jaoi 856	. . 3 ⊢ ((𝑍 ∈ ℕ₀ ∨ (𝑍 ∈ ℝ ∧ -𝑍 ∈ ℕ)) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
87	1, 86	sylbi 216	. 2 ⊢ (𝑍 ∈ ℤ → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
88	87	3impib 1117	1 ⊢ ((𝑍 ∈ ℤ ∧ 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℝcr 11109 0cc0 11110 1c1 11111 · cmul 11115 < clt 11248 -cneg 11445 ℕcn 12212 2c2 12267 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 ℙcprime 16608

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-dvds 16198 df-prm 16609

This theorem is referenced by: zlmodzxznm 47178

W3C validator