ztprmneprm - Mathbox for Alexander van der Vekens

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Theorem ztprmneprm 49662

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 12731	. . 3 ⊢ (𝑍 ∈ ℤ ↔ (𝑍 ∈ ℕ₀ ∨ (𝑍 ∈ ℝ ∧ -𝑍 ∈ ℕ)))
2		elnn0 12632	. . . . 5 ⊢ (𝑍 ∈ ℕ₀ ↔ (𝑍 ∈ ℕ ∨ 𝑍 = 0))
3		elnn1uz2 13076	. . . . . . 7 ⊢ (𝑍 ∈ ℕ ↔ (𝑍 = 1 ∨ 𝑍 ∈ (ℤ_≥‘2)))
4		oveq1 7447	. . . . . . . . . . . 12 ⊢ (𝑍 = 1 → (𝑍 · 𝐴) = (1 · 𝐴))
5	4	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑍 = 1 ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → (𝑍 · 𝐴) = (1 · 𝐴))
6	5	eqeq1d 2765	. . . . . . . . . 10 ⊢ ((𝑍 = 1 ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → ((𝑍 · 𝐴) = 𝐵 ↔ (1 · 𝐴) = 𝐵))
7		prmz 16876	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℤ)
8	7	zcnd 12828	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℂ)
9	8	mullidd 11351	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℙ → (1 · 𝐴) = 𝐴)
10	9	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (1 · 𝐴) = 𝐴)
11	10	eqeq1d 2765	. . . . . . . . . . . 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 487	. . . . . . . . . 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 418	. . . . . . . 8 ⊢ (𝑍 = 1 → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
16		prmuz2 16897	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ (ℤ_≥‘2))
17	16	adantr 486	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → 𝐴 ∈ (ℤ_≥‘2))
18		nprm 16889	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ (ℤ_≥‘2) ∧ 𝐴 ∈ (ℤ_≥‘2)) → ¬ (𝑍 · 𝐴) ∈ ℙ)
19	17, 18	sylan2 605	. . . . . . . . . 10 ⊢ ((𝑍 ∈ (ℤ_≥‘2) ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → ¬ (𝑍 · 𝐴) ∈ ℙ)
20		eleq1 2852	. . . . . . . . . . . . 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 487	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (¬ 𝐵 ∈ ℙ → 𝐴 = 𝐵))
24	23	adantl 487	. . . . . . . . . . . . 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 418	. . . . . . . 8 ⊢ (𝑍 ∈ (ℤ_≥‘2) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
30	15, 29	jaoi 871	. . . . . . 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 7447	. . . . . . . . 9 ⊢ (𝑍 = 0 → (𝑍 · 𝐴) = (0 · 𝐴))
33	32	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑍 = 0 → ((𝑍 · 𝐴) = 𝐵 ↔ (0 · 𝐴) = 𝐵))
34		prmnn 16875	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
35	34	nnred 12374	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℝ)
36		mul02lem2 11511	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → (0 · 𝐴) = 0)
37	35, 36	syl 18	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℙ → (0 · 𝐴) = 0)
38	37	adantr 486	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (0 · 𝐴) = 0)
39	38	eqeq1d 2765	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((0 · 𝐴) = 𝐵 ↔ 0 = 𝐵))
40		prmnn 16875	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℙ → 𝐵 ∈ ℕ)
41		elnnne0 12644	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ ↔ (𝐵 ∈ ℕ₀ ∧ 𝐵 ≠ 0))
42		eqneqall 2970	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 = 0 → (𝐵 ≠ 0 → 𝐴 = 𝐵))
43	42	eqcoms 2771	. . . . . . . . . . . . . . 15 ⊢ (0 = 𝐵 → (𝐵 ≠ 0 → 𝐴 = 𝐵))
44	43	com12 33	. . . . . . . . . . . . . 14 ⊢ (𝐵 ≠ 0 → (0 = 𝐵 → 𝐴 = 𝐵))
45	44	adantl 487	. . . . . . . . . . . . 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 487	. . . . . . . . . 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 871	. . . . 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 12727	. . . . . 6 ⊢ ( -𝑍 ∈ ℕ ↔ ( -𝑍 ∈ ℤ ∧ 0 < -𝑍))
56		lt0neg1 11846	. . . . . . . 8 ⊢ (𝑍 ∈ ℝ → (𝑍 < 0 ↔ 0 < -𝑍))
57	34	nngt0d 12411	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℙ → 0 < 𝐴)
58	57	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → 0 < 𝐴)
59		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝑍 ∈ ℝ ∧ 𝑍 < 0) → 𝑍 < 0)
60	58, 59	anim12ci 626	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → (𝑍 < 0 ∧ 0 < 𝐴))
61	60	orcd 887	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → ((𝑍 < 0 ∧ 0 < 𝐴) ∨ (0 < 𝑍 ∧ 𝐴 < 0)))
62		simprl 783	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → 𝑍 ∈ ℝ)
63	35	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → 𝐴 ∈ ℝ)
64	63	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → 𝐴 ∈ ℝ)
65	62, 64	mul2lt0bi 13252	. . . . . . . . . . . 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 418	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 ∈ ℝ ∧ 𝑍 < 0) → (𝑍 · 𝐴) < 0))
68		breq1 5109	. . . . . . . . . . . . . 14 ⊢ ((𝑍 · 𝐴) = 𝐵 → ((𝑍 · 𝐴) < 0 ↔ 𝐵 < 0))
69	68	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 · 𝐴) = 𝐵) → ((𝑍 · 𝐴) < 0 ↔ 𝐵 < 0))
70		nnnn0 12637	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℕ₀)
71		nn0nlt0 12656	. . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (𝐵 < 0 → 𝐴 = 𝐵))
76	75	adantr 486	. . . . . . . . . . . . 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 418	. . . . . . . . . . 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 418	. . . . . . . 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 496	. . . . . 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 412	. . . 4 ⊢ ((𝑍 ∈ ℝ ∧ -𝑍 ∈ ℕ) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
86	54, 85	jaoi 871	. . 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 1134	1 ⊢ ((𝑍 ∈ ℤ ∧ 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 class class class wbr 5106 ‘cfv 6555 (class class class)co 7440 ℝcr 11223 0cc0 11224 1c1 11225 · cmul 11229 < clt 11367 -cneg 11566 ℕcn 12359 2c2 12421 ℕ₀cn0 12630 ℤcz 12717 ℤ_≥cuz 12989 ℙcprime 16872

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301 ax-pre-sup 11302

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-om 7899 df-2nd 8023 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-er 8742 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-sup 9458 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-div 11998 df-nn 12360 df-2 12429 df-3 12430 df-n0 12631 df-z 12718 df-uz 12990 df-rp 13145 df-seq 14169 df-exp 14229 df-cj 15292 df-re 15293 df-im 15294 df-sqrt 15428 df-abs 15429 df-dvds 16449 df-prm 16873

This theorem is used by: zlmodzxznm 49812

W3C validator