aks4d1p8d2 - Mathbox for metakunt

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Theorem aks4d1p8d2 43326

Description: Any prime power dividing a positive integer is less than that integer if that integer has another prime factor. (Contributed by metakunt, 13-Nov-2024.)

**Hypotheses**
Ref	Expression
aks4d1p8d2.1	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks4d1p8d2.2	⊢ (𝜑 → 𝑁 ∈ ℕ)
aks4d1p8d2.3	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks4d1p8d2.4	⊢ (𝜑 → 𝑄 ∈ ℙ)
aks4d1p8d2.5	⊢ (𝜑 → 𝑃 ∥ 𝑅)
aks4d1p8d2.6	⊢ (𝜑 → 𝑄 ∥ 𝑅)
aks4d1p8d2.7	⊢ (𝜑 → ¬ 𝑃 ∥ 𝑁)
aks4d1p8d2.8	⊢ (𝜑 → 𝑄 ∥ 𝑁)

**Assertion**
Ref	Expression
aks4d1p8d2	⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < 𝑅)

Proof of Theorem aks4d1p8d2 Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aks4d1p8d2.3	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℙ)
2		prmnn 16875	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
3	1, 2	syl 18	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ)
4	3	nnred 12374	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℝ)
5		aks4d1p8d2.1	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℕ)
6	1, 5	pccld 17054	. . 3 ⊢ (𝜑 → (𝑃 pCnt 𝑅) ∈ ℕ₀)
7	4, 6	reexpcld 14330	. 2 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℝ)
8		aks4d1p8d2.4	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℙ)
9		prmnn 16875	. . . . 5 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
10	8, 9	syl 18	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℕ)
11	10	nnred 12374	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℝ)
12	7, 11	remulcld 11363	. 2 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∈ ℝ)
13	5	nnred 12374	. 2 ⊢ (𝜑 → 𝑅 ∈ ℝ)
14	7	recnd 11361	. . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℂ)
15	14	mulridd 11350	. . 3 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 1) = (𝑃↑(𝑃 pCnt 𝑅)))
16		1red 11333	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
17	3	nnrpd 13186	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℝ⁺)
18	6	nn0zd 12742	. . . . 5 ⊢ (𝜑 → (𝑃 pCnt 𝑅) ∈ ℤ)
19	17, 18	rpexpcld 14415	. . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℝ⁺)
20		prmgt1 16899	. . . . 5 ⊢ (𝑄 ∈ ℙ → 1 < 𝑄)
21	8, 20	syl 18	. . . 4 ⊢ (𝜑 → 1 < 𝑄)
22	16, 11, 19, 21	ltmul2dd 13244	. . 3 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 1) < ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄))
23	15, 22	eqbrtrrd 5132	. 2 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄))
24	3	nnzd 12743	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℤ)
25	24, 6	zexpcld 14254	. . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℤ)
26	10	nnzd 12743	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℤ)
27	5	nnzd 12743	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℤ)
28	25, 26	gcdcomd 16712	. . . . 5 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) gcd 𝑄) = (𝑄 gcd (𝑃↑(𝑃 pCnt 𝑅))))
29		0lt1 11862	. . . . . . . . . . . . . 14 ⊢ 0 < 1
30	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < 1)
31		0red 11335	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ ℝ)
32	31, 16	ltnled 11481	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 < 1 ↔ ¬ 1 ≤ 0))
33	30, 32	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 1 ≤ 0)
34	11	recnd 11361	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑄 ∈ ℂ)
35	34	exp1d 14308	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑄↑1) = 𝑄)
36	35	eqcomd 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑄 = (𝑄↑1))
37	36	oveq2d 7456	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 pCnt 𝑄) = (𝑄 pCnt (𝑄↑1)))
38		1zzd 12751	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℤ)
39		pcid 17077	. . . . . . . . . . . . . . . 16 ⊢ ((𝑄 ∈ ℙ ∧ 1 ∈ ℤ) → (𝑄 pCnt (𝑄↑1)) = 1)
40	8, 38, 39	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 pCnt (𝑄↑1)) = 1)
41	37, 40	eqtrd 2798	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄 pCnt 𝑄) = 1)
42		aks4d1p8d2.8	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑄 ∥ 𝑁)
43	42	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → 𝑄 ∥ 𝑁)
44		breq1 5109	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 = 𝑄 → (𝑃 ∥ 𝑁 ↔ 𝑄 ∥ 𝑁))
45	44	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (𝑃 ∥ 𝑁 ↔ 𝑄 ∥ 𝑁))
46	45	bicomd 226	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (𝑄 ∥ 𝑁 ↔ 𝑃 ∥ 𝑁))
47	46	biimpd 232	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (𝑄 ∥ 𝑁 → 𝑃 ∥ 𝑁))
48	43, 47	mpd 16	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → 𝑃 ∥ 𝑁)
49		aks4d1p8d2.7	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 𝑃 ∥ 𝑁)
50	49	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ¬ 𝑃 ∥ 𝑁)
51	48, 50	pm2.65da 829	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ¬ 𝑃 = 𝑄)
52	51	neqcomd 2773	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ 𝑄 = 𝑃)
53		aks4d1p8d2.5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∥ 𝑅)
54		pcelnn 17074	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 ∈ ℙ ∧ 𝑅 ∈ ℕ) → ((𝑃 pCnt 𝑅) ∈ ℕ ↔ 𝑃 ∥ 𝑅))
55	1, 5, 54	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑃 pCnt 𝑅) ∈ ℕ ↔ 𝑃 ∥ 𝑅))
56	53, 55	mpbird 260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑃 pCnt 𝑅) ∈ ℕ)
57		prmdvdsexpb 16918	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄 ∈ ℙ ∧ 𝑃 ∈ ℙ ∧ (𝑃 pCnt 𝑅) ∈ ℕ) → (𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ 𝑄 = 𝑃))
58	8, 1, 56, 57	syl3anc 1399	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ 𝑄 = 𝑃))
59	58	notbid 321	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ ¬ 𝑄 = 𝑃))
60	52, 59	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)))
61	3, 6	nnexpcld 14413	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℕ)
62		pceq0 17075	. . . . . . . . . . . . . . . 16 ⊢ ((𝑄 ∈ ℙ ∧ (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℕ) → ((𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))) = 0 ↔ ¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅))))
63	8, 61, 62	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))) = 0 ↔ ¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅))))
64	60, 63	mpbird 260	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))) = 0)
65	41, 64	breq12d 5119	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄 pCnt 𝑄) ≤ (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))) ↔ 1 ≤ 0))
66	65	notbid 321	. . . . . . . . . . . 12 ⊢ (𝜑 → (¬ (𝑄 pCnt 𝑄) ≤ (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))) ↔ ¬ 1 ≤ 0))
67	33, 66	mpbird 260	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑄 pCnt 𝑄) ≤ (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
68	67	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 = 𝑄) → ¬ (𝑄 pCnt 𝑄) ≤ (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
69		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 = 𝑄) → 𝑝 = 𝑄)
70	69	oveq1d 7455	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 = 𝑄) → (𝑝 pCnt 𝑄) = (𝑄 pCnt 𝑄))
71	69	oveq1d 7455	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 = 𝑄) → (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))) = (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
72	70, 71	breq12d 5119	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 = 𝑄) → ((𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))) ↔ (𝑄 pCnt 𝑄) ≤ (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅)))))
73	72	notbid 321	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 = 𝑄) → (¬ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))) ↔ ¬ (𝑄 pCnt 𝑄) ≤ (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅)))))
74	68, 73	mpbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 = 𝑄) → ¬ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
75	74, 8	rspcime 3585	. . . . . . . 8 ⊢ (𝜑 → ∃𝑝 ∈ ℙ ¬ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
76		rexnal 3118	. . . . . . . . 9 ⊢ (∃𝑝 ∈ ℙ ¬ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))) ↔ ¬ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
77	76	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (∃𝑝 ∈ ℙ ¬ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))) ↔ ¬ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅)))))
78	75, 77	mpbid 235	. . . . . . 7 ⊢ (𝜑 → ¬ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
79		pc2dvds 17083	. . . . . . . . 9 ⊢ ((𝑄 ∈ ℤ ∧ (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℤ) → (𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅)))))
80	26, 25, 79	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅)))))
81	80	notbid 321	. . . . . . 7 ⊢ (𝜑 → (¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ ¬ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅)))))
82	78, 81	mpbird 260	. . . . . 6 ⊢ (𝜑 → ¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)))
83		coprm 16913	. . . . . . 7 ⊢ ((𝑄 ∈ ℙ ∧ (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℤ) → (¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ (𝑄 gcd (𝑃↑(𝑃 pCnt 𝑅))) = 1))
84	8, 25, 83	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ (𝑄 gcd (𝑃↑(𝑃 pCnt 𝑅))) = 1))
85	82, 84	mpbid 235	. . . . 5 ⊢ (𝜑 → (𝑄 gcd (𝑃↑(𝑃 pCnt 𝑅))) = 1)
86	28, 85	eqtrd 2798	. . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) gcd 𝑄) = 1)
87		pcdvds 17068	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑅 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝑅)) ∥ 𝑅)
88	1, 5, 87	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∥ 𝑅)
89		aks4d1p8d2.6	. . . 4 ⊢ (𝜑 → 𝑄 ∥ 𝑅)
90	25, 26, 27, 86, 88, 89	coprmdvds2d 43242	. . 3 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∥ 𝑅)
91	25, 26	zmulcld 12833	. . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∈ ℤ)
92		dvdsle 16506	. . . 4 ⊢ ((((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∈ ℤ ∧ 𝑅 ∈ ℕ) → (((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∥ 𝑅 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ≤ 𝑅))
93	91, 5, 92	syl2anc 596	. . 3 ⊢ (𝜑 → (((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∥ 𝑅 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ≤ 𝑅))
94	90, 93	mpd 16	. 2 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ≤ 𝑅)
95	7, 12, 13, 23, 94	ltletrd 11494	1 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < 𝑅)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ∃wrex 3090 class class class wbr 5106 (class class class)co 7440 0cc0 11224 1c1 11225 · cmul 11229 < clt 11367 ≤ cle 11368 ℕcn 12359 ℤcz 12717 ↑cexp 14228 ∥ cdvds 16448 gcd cgcd 16690 ℙcprime 16872 pCnt cpc 17040

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-1st 8022 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-inf 9459 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-q 13100 df-rp 13145 df-fz 13664 df-fl 13956 df-mod 14034 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-gcd 16691 df-prm 16873 df-pc 17041

This theorem is used by: aks4d1p8 43328

W3C validator