ztprmneprm - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ztprmneprm		Structured version Visualization version GIF version

Theorem ztprmneprm 44386

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 11987	. . 3 ⊢ (𝑍 ∈ ℤ ↔ (𝑍 ∈ ℕ₀ ∨ (𝑍 ∈ ℝ ∧ -𝑍 ∈ ℕ)))
2		elnn0 11891	. . . . 5 ⊢ (𝑍 ∈ ℕ₀ ↔ (𝑍 ∈ ℕ ∨ 𝑍 = 0))
3		elnn1uz2 12317	. . . . . . 7 ⊢ (𝑍 ∈ ℕ ↔ (𝑍 = 1 ∨ 𝑍 ∈ (ℤ_≥‘2)))
4		oveq1 7155	. . . . . . . . . . . 12 ⊢ (𝑍 = 1 → (𝑍 · 𝐴) = (1 · 𝐴))
5	4	adantr 483	. . . . . . . . . . 11 ⊢ ((𝑍 = 1 ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → (𝑍 · 𝐴) = (1 · 𝐴))
6	5	eqeq1d 2821	. . . . . . . . . 10 ⊢ ((𝑍 = 1 ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → ((𝑍 · 𝐴) = 𝐵 ↔ (1 · 𝐴) = 𝐵))
7		prmz 16011	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℤ)
8	7	zcnd 12080	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℂ)
9	8	mulid2d 10651	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℙ → (1 · 𝐴) = 𝐴)
10	9	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (1 · 𝐴) = 𝐴)
11	10	eqeq1d 2821	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((1 · 𝐴) = 𝐵 ↔ 𝐴 = 𝐵))
12	11	biimpd 231	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((1 · 𝐴) = 𝐵 → 𝐴 = 𝐵))
13	12	adantl 484	. . . . . . . . . 10 ⊢ ((𝑍 = 1 ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → ((1 · 𝐴) = 𝐵 → 𝐴 = 𝐵))
14	6, 13	sylbid 242	. . . . . . . . 9 ⊢ ((𝑍 = 1 ∧ (𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ)) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))
15	14	ex 415	. . . . . . . 8 ⊢ (𝑍 = 1 → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
16		prmuz2 16032	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ (ℤ_≥‘2))
17	16	adantr 483	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → 𝐴 ∈ (ℤ_≥‘2))
18		nprm 16024	. . . . . . . . . . 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 2898	. . . . . . . . . . . . 13 ⊢ ((𝑍 · 𝐴) = 𝐵 → ((𝑍 · 𝐴) ∈ ℙ ↔ 𝐵 ∈ ℙ))
21	20	notbid 320	. . . . . . . . . . . 12 ⊢ ((𝑍 · 𝐴) = 𝐵 → (¬ (𝑍 · 𝐴) ∈ ℙ ↔ ¬ 𝐵 ∈ ℙ))
22		pm2.24 124	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℙ → (¬ 𝐵 ∈ ℙ → 𝐴 = 𝐵))
23	22	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (¬ 𝐵 ∈ ℙ → 𝐴 = 𝐵))
24	23	adantl 484	. . . . . . . . . . . . 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 255	. . . . . . . . . . 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 415	. . . . . . . 8 ⊢ (𝑍 ∈ (ℤ_≥‘2) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
30	15, 29	jaoi 853	. . . . . . 7 ⊢ ((𝑍 = 1 ∨ 𝑍 ∈ (ℤ_≥‘2)) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
31	3, 30	sylbi 219	. . . . . 6 ⊢ (𝑍 ∈ ℕ → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
32		oveq1 7155	. . . . . . . . 9 ⊢ (𝑍 = 0 → (𝑍 · 𝐴) = (0 · 𝐴))
33	32	eqeq1d 2821	. . . . . . . 8 ⊢ (𝑍 = 0 → ((𝑍 · 𝐴) = 𝐵 ↔ (0 · 𝐴) = 𝐵))
34		prmnn 16010	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
35	34	nnred 11645	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℝ)
36		mul02lem2 10809	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → (0 · 𝐴) = 0)
37	35, 36	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℙ → (0 · 𝐴) = 0)
38	37	adantr 483	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (0 · 𝐴) = 0)
39	38	eqeq1d 2821	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((0 · 𝐴) = 𝐵 ↔ 0 = 𝐵))
40		prmnn 16010	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℙ → 𝐵 ∈ ℕ)
41		elnnne0 11903	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ ↔ (𝐵 ∈ ℕ₀ ∧ 𝐵 ≠ 0))
42		eqneqall 3025	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 = 0 → (𝐵 ≠ 0 → 𝐴 = 𝐵))
43	42	eqcoms 2827	. . . . . . . . . . . . . . 15 ⊢ (0 = 𝐵 → (𝐵 ≠ 0 → 𝐴 = 𝐵))
44	43	com12 32	. . . . . . . . . . . . . 14 ⊢ (𝐵 ≠ 0 → (0 = 𝐵 → 𝐴 = 𝐵))
45	44	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℕ₀ ∧ 𝐵 ≠ 0) → (0 = 𝐵 → 𝐴 = 𝐵))
46	41, 45	sylbi 219	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ → (0 = 𝐵 → 𝐴 = 𝐵))
47	40, 46	syl 17	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℙ → (0 = 𝐵 → 𝐴 = 𝐵))
48	47	adantl 484	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (0 = 𝐵 → 𝐴 = 𝐵))
49	39, 48	sylbid 242	. . . . . . . . 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 255	. . . . . . 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 853	. . . . 5 ⊢ ((𝑍 ∈ ℕ ∨ 𝑍 = 0) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
54	2, 53	sylbi 219	. . . 4 ⊢ (𝑍 ∈ ℕ₀ → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
55		elnnz 11983	. . . . . 6 ⊢ (-𝑍 ∈ ℕ ↔ (-𝑍 ∈ ℤ ∧ 0 < -𝑍))
56		lt0neg1 11138	. . . . . . . 8 ⊢ (𝑍 ∈ ℝ → (𝑍 < 0 ↔ 0 < -𝑍))
57	34	nngt0d 11678	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℙ → 0 < 𝐴)
58	57	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → 0 < 𝐴)
59		simpr 487	. . . . . . . . . . . . . 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 869	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → ((𝑍 < 0 ∧ 0 < 𝐴) ∨ (0 < 𝑍 ∧ 𝐴 < 0)))
62		simprl 769	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → 𝑍 ∈ ℝ)
63	35	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → 𝐴 ∈ ℝ)
64	63	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → 𝐴 ∈ ℝ)
65	62, 64	mul2lt0bi 12487	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → ((𝑍 · 𝐴) < 0 ↔ ((𝑍 < 0 ∧ 0 < 𝐴) ∨ (0 < 𝑍 ∧ 𝐴 < 0))))
66	61, 65	mpbird 259	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 ∈ ℝ ∧ 𝑍 < 0)) → (𝑍 · 𝐴) < 0)
67	66	ex 415	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 ∈ ℝ ∧ 𝑍 < 0) → (𝑍 · 𝐴) < 0))
68		breq1 5060	. . . . . . . . . . . . . 14 ⊢ ((𝑍 · 𝐴) = 𝐵 → ((𝑍 · 𝐴) < 0 ↔ 𝐵 < 0))
69	68	adantl 484	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 · 𝐴) = 𝐵) → ((𝑍 · 𝐴) < 0 ↔ 𝐵 < 0))
70		nnnn0 11896	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℕ₀)
71		nn0nlt0 11915	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → (𝐵 < 0 → 𝐴 = 𝐵))
76	75	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 · 𝐴) = 𝐵) → (𝐵 < 0 → 𝐴 = 𝐵))
77	69, 76	sylbid 242	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) ∧ (𝑍 · 𝐴) = 𝐵) → ((𝑍 · 𝐴) < 0 → 𝐴 = 𝐵))
78	77	ex 415	. . . . . . . . . . 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 415	. . . . . . . 8 ⊢ (𝑍 ∈ ℝ → (𝑍 < 0 → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))))
82	56, 81	sylbird 262	. . . . . . 7 ⊢ (𝑍 ∈ ℝ → (0 < -𝑍 → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))))
83	82	adantld 493	. . . . . 6 ⊢ (𝑍 ∈ ℝ → ((-𝑍 ∈ ℤ ∧ 0 < -𝑍) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))))
84	55, 83	syl5bi 244	. . . . 5 ⊢ (𝑍 ∈ ℝ → (-𝑍 ∈ ℕ → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))))
85	84	imp 409	. . . 4 ⊢ ((𝑍 ∈ ℝ ∧ -𝑍 ∈ ℕ) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
86	54, 85	jaoi 853	. . 3 ⊢ ((𝑍 ∈ ℕ₀ ∨ (𝑍 ∈ ℝ ∧ -𝑍 ∈ ℕ)) → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
87	1, 86	sylbi 219	. 2 ⊢ (𝑍 ∈ ℤ → ((𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵)))
88	87	3impib 1111	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 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 class class class wbr 5057 ‘cfv 6348 (class class class)co 7148 ℝcr 10528 0cc0 10529 1c1 10530 · cmul 10534 < clt 10667 -cneg 10863 ℕcn 11630 2c2 11684 ℕ₀cn0 11889 ℤcz 11973 ℤ_≥cuz 12235 ℙcprime 16007

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-sup 8898 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-n0 11890 df-z 11974 df-uz 12236 df-rp 12382 df-seq 13362 df-exp 13422 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-prm 16008

This theorem is referenced by: zlmodzxznm 44543

W3C validator