ztprmneprm - Mathbox for Alexander van der Vekens

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Theorem ztprmneprm 48979

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 12593	. . 3 ⊢ (𝑍 ∈ ℤ ↔ (𝑍 ∈ ℕ₀ ∨ (𝑍 ∈ ℝ ∧ -𝑍 ∈ ℕ)))
2		elnn0 12494	. . . . 5 ⊢ (𝑍 ∈ ℕ₀ ↔ (𝑍 ∈ ℕ ∨ 𝑍 = 0))
3		elnn1uz2 12937	. . . . . . 7 ⊢ (𝑍 ∈ ℕ ↔ (𝑍 = 1 ∨ 𝑍 ∈ (ℤ_≥‘2)))
4		oveq1 7407	. . . . . . . . . . . 12 ⊢ (𝑍 = 1 → (𝑍 · 𝐴) = (1 · 𝐴))
5	4	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑍 = 1 ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → (𝑍 · 𝐴) = (1 · 𝐴))
6	5	eqeq1d 2767	. . . . . . . . . 10 ⊢ ((𝑍 = 1 ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → ((𝑍 · 𝐴) = 𝐵 ↔ (1 · 𝐴) = 𝐵))
7		prmz 16721	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℤ)
8	7	zcnd 12689	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℂ)
9	8	mullidd 11215	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℙ → (1 · 𝐴) = 𝐴)
10	9	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (1 · 𝐴) = 𝐴)
11	10	eqeq1d 2767	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((1 · 𝐴) = 𝐵 ↔ 𝐴 = 𝐵))
12	11	biimpd 232	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((1 · 𝐴) = 𝐵 → 𝐴 = 𝐵))
13	12	adantl 486	. . . . . . . . . 10 ⊢ ((𝑍 = 1 ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → ((1 · 𝐴) = 𝐵 → 𝐴 = 𝐵))
14	6, 13	sylbid 243	. . . . . . . . 9 ⊢ ((𝑍 = 1 ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))
15	14	ex 417	. . . . . . . 8 ⊢ (𝑍 = 1 → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
16		prmuz2 16742	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ (ℤ_≥‘2))
17	16	adantr 485	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → 𝐴 ∈ (ℤ_≥‘2))
18		nprm 16734	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ (ℤ_≥‘2) ∧ 𝐴 ∈ (ℤ_≥‘2)) → ¬ (𝑍 · 𝐴) ∈ ℙ)
19	17, 18	sylan2 604	. . . . . . . . . 10 ⊢ ((𝑍 ∈ (ℤ_≥‘2) ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → ¬ (𝑍 · 𝐴) ∈ ℙ)
20		eleq1 2853	. . . . . . . . . . . . 13 ⊢ ((𝑍 · 𝐴) = 𝐵 → ((𝑍 · 𝐴) ∈ ℙ ↔ 𝐵 ∈ ℙ))
21	20	notbid 321	. . . . . . . . . . . 12 ⊢ ((𝑍 · 𝐴) = 𝐵 → (¬ (𝑍 · 𝐴) ∈ ℙ ↔ ¬ 𝐵 ∈ ℙ))
22		pm2.24 125	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℙ → (¬ 𝐵 ∈ ℙ → 𝐴 = 𝐵))
23	22	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (¬ 𝐵 ∈ ℙ → 𝐴 = 𝐵))
24	23	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝑍 ∈ (ℤ_≥‘2) ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → (¬ 𝐵 ∈ ℙ → 𝐴 = 𝐵))
25	24	com12 33	. . . . . . . . . . . 12 ⊢ (¬ 𝐵 ∈ ℙ → ((𝑍 ∈ (ℤ_≥‘2) ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → 𝐴 = 𝐵))
26	21, 25	biimtrdi 256	. . . . . . . . . . 11 ⊢ ((𝑍 · 𝐴) = 𝐵 → (¬ (𝑍 · 𝐴) ∈ ℙ → ((𝑍 ∈ (ℤ_≥‘2) ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → 𝐴 = 𝐵)))
27	26	com3l 90	. . . . . . . . . 10 ⊢ (¬ (𝑍 · 𝐴) ∈ ℙ → ((𝑍 ∈ (ℤ_≥‘2) ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
28	19, 27	mpcom 39	. . . . . . . . 9 ⊢ ((𝑍 ∈ (ℤ_≥‘2) ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))
29	28	ex 417	. . . . . . . 8 ⊢ (𝑍 ∈ (ℤ_≥‘2) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
30	15, 29	jaoi 870	. . . . . . 7 ⊢ ((𝑍 = 1 ∨ 𝑍 ∈ (ℤ_≥‘2)) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
31	3, 30	sylbi 220	. . . . . 6 ⊢ (𝑍 ∈ ℕ → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
32		oveq1 7407	. . . . . . . . 9 ⊢ (𝑍 = 0 → (𝑍 · 𝐴) = (0 · 𝐴))
33	32	eqeq1d 2767	. . . . . . . 8 ⊢ (𝑍 = 0 → ((𝑍 · 𝐴) = 𝐵 ↔ (0 · 𝐴) = 𝐵))
34		prmnn 16720	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
35	34	nnred 12236	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℝ)
36		mul02lem2 11375	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → (0 · 𝐴) = 0)
37	35, 36	syl 18	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℙ → (0 · 𝐴) = 0)
38	37	adantr 485	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (0 · 𝐴) = 0)
39	38	eqeq1d 2767	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((0 · 𝐴) = 𝐵 ↔ 0 = 𝐵))
40		prmnn 16720	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℙ → 𝐵 ∈ ℕ)
41		elnnne0 12506	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ ↔ (𝐵 ∈ ℕ₀ ∧ 𝐵 ≠ 0))
42		eqneqall 2971	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 = 0 → (𝐵 ≠ 0 → 𝐴 = 𝐵))
43	42	eqcoms 2773	. . . . . . . . . . . . . . 15 ⊢ (0 = 𝐵 → (𝐵 ≠ 0 → 𝐴 = 𝐵))
44	43	com12 33	. . . . . . . . . . . . . 14 ⊢ (𝐵 ≠ 0 → (0 = 𝐵 → 𝐴 = 𝐵))
45	44	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℕ₀ ∧ 𝐵 ≠ 0) → (0 = 𝐵 → 𝐴 = 𝐵))
46	41, 45	sylbi 220	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ → (0 = 𝐵 → 𝐴 = 𝐵))
47	40, 46	syl 18	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℙ → (0 = 𝐵 → 𝐴 = 𝐵))
48	47	adantl 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (0 = 𝐵 → 𝐴 = 𝐵))
49	39, 48	sylbid 243	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((0 · 𝐴) = 𝐵 → 𝐴 = 𝐵))
50	49	com12 33	. . . . . . . 8 ⊢ ((0 · 𝐴) = 𝐵 → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → 𝐴 = 𝐵))
51	33, 50	biimtrdi 256	. . . . . . 7 ⊢ (𝑍 = 0 → ((𝑍 · 𝐴) = 𝐵 → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → 𝐴 = 𝐵)))
52	51	com23 87	. . . . . 6 ⊢ (𝑍 = 0 → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
53	31, 52	jaoi 870	. . . . 5 ⊢ ((𝑍 ∈ ℕ ∨ 𝑍 = 0) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
54	2, 53	sylbi 220	. . . 4 ⊢ (𝑍 ∈ ℕ₀ → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
55		elnnz 12589	. . . . . 6 ⊢ (-𝑍 ∈ ℕ ↔ (-𝑍 ∈ ℤ ∧ 0 < -𝑍))
56		lt0neg1 11708	. . . . . . . 8 ⊢ (𝑍 ∈ ℝ → (𝑍 < 0 ↔ 0 < -𝑍))
57	34	nngt0d 12273	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℙ → 0 < 𝐴)
58	57	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → 0 < 𝐴)
59		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((𝑍 ∈ ℝ ∧ 𝑍 < 0) → 𝑍 < 0)
60	58, 59	anim12ci 625	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → (𝑍 < 0 ∧ 0 < 𝐴))
61	60	orcd 886	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → ((𝑍 < 0 ∧ 0 < 𝐴) ∨ (0 < 𝑍 ∧ 𝐴 < 0)))
62		simprl 782	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → 𝑍 ∈ ℝ)
63	35	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → 𝐴 ∈ ℝ)
64	63	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → 𝐴 ∈ ℝ)
65	62, 64	mul2lt0bi 13112	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → ((𝑍 · 𝐴) < 0 ↔ ((𝑍 < 0 ∧ 0 < 𝐴) ∨ (0 < 𝑍 ∧ 𝐴 < 0))))
66	61, 65	mpbird 260	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → (𝑍 · 𝐴) < 0)
67	66	ex 417	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 ∈ ℝ ∧ 𝑍 < 0) → (𝑍 · 𝐴) < 0))
68		breq1 5107	. . . . . . . . . . . . . 14 ⊢ ((𝑍 · 𝐴) = 𝐵 → ((𝑍 · 𝐴) < 0 ↔ 𝐵 < 0))
69	68	adantl 486	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 · 𝐴) = 𝐵) → ((𝑍 · 𝐴) < 0 ↔ 𝐵 < 0))
70		nnnn0 12499	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℕ₀)
71		nn0nlt0 12518	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ ℕ₀ → ¬ 𝐵 < 0)
72	71	pm2.21d 122	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ ℕ₀ → (𝐵 < 0 → 𝐴 = 𝐵))
73	70, 72	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℕ → (𝐵 < 0 → 𝐴 = 𝐵))
74	40, 73	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℙ → (𝐵 < 0 → 𝐴 = 𝐵))
75	74	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (𝐵 < 0 → 𝐴 = 𝐵))
76	75	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 · 𝐴) = 𝐵) → (𝐵 < 0 → 𝐴 = 𝐵))
77	69, 76	sylbid 243	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 · 𝐴) = 𝐵) → ((𝑍 · 𝐴) < 0 → 𝐴 = 𝐵))
78	77	ex 417	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → ((𝑍 · 𝐴) < 0 → 𝐴 = 𝐵)))
79	78	com23 87	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) < 0 → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
80	67, 79	syldc 49	. . . . . . . . 9 ⊢ ((𝑍 ∈ ℝ ∧ 𝑍 < 0) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
81	80	ex 417	. . . . . . . 8 ⊢ (𝑍 ∈ ℝ → (𝑍 < 0 → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))))
82	56, 81	sylbird 263	. . . . . . 7 ⊢ (𝑍 ∈ ℝ → (0 < -𝑍 → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))))
83	82	adantld 495	. . . . . 6 ⊢ (𝑍 ∈ ℝ → ((-𝑍 ∈ ℤ ∧ 0 < -𝑍) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))))
84	55, 83	biimtrid 245	. . . . 5 ⊢ (𝑍 ∈ ℝ → (-𝑍 ∈ ℕ → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))))
85	84	imp 411	. . . 4 ⊢ ((𝑍 ∈ ℝ ∧ -𝑍 ∈ ℕ) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
86	54, 85	jaoi 870	. . 3 ⊢ ((𝑍 ∈ ℕ₀ ∨ (𝑍 ∈ ℝ ∧ -𝑍 ∈ ℕ)) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
87	1, 86	sylbi 220	. 2 ⊢ (𝑍 ∈ ℤ → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
88	87	3impib 1132	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 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ≠ wne 2960 class class class wbr 5104 ‘cfv 6525 (class class class)co 7400 ℝcr 11087 0cc0 11088 1c1 11089 · cmul 11093 < clt 11231 -cneg 11430 ℕcn 12221 2c2 12283 ℕ₀cn0 12492 ℤcz 12579 ℤ_≥cuz 12850 ℙcprime 16717

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5105 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6291 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-sup 9390 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12222 df-2 12291 df-3 12292 df-n0 12493 df-z 12580 df-uz 12851 df-rp 13005 df-seq 14026 df-exp 14086 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16299 df-prm 16718

This theorem is referenced by: zlmodzxznm 49129

W3C validator