Theorem aks4d1p8d2 42952

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 16765	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
3	1, 2	syl 18	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ)
4	3	nnred 12273	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℝ)
5		aks4d1p8d2.1	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℕ)
6	1, 5	pccld 16943	. . 3 ⊢ (𝜑 → (𝑃 pCnt 𝑅) ∈ ℕ0)
7	4, 6	reexpcld 14228	. 2 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℝ)
8		aks4d1p8d2.4	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℙ)
9		prmnn 16765	. . . . 5 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
10	8, 9	syl 18	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℕ)
11	10	nnred 12273	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℝ)
12	7, 11	remulcld 11264	. 2 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∈ ℝ)
13	5	nnred 12273	. 2 ⊢ (𝜑 → 𝑅 ∈ ℝ)
14	7	recnd 11262	. . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℂ)
15	14	mulridd 11251	. . 3 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 1) = (𝑃↑(𝑃 pCnt 𝑅)))
16		1red 11234	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
17	3	nnrpd 13085	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℝ+)
18	6	nn0zd 12641	. . . . 5 ⊢ (𝜑 → (𝑃 pCnt 𝑅) ∈ ℤ)
19	17, 18	rpexpcld 14312	. . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℝ+)
20		prmgt1 16789	. . . . 5 ⊢ (𝑄 ∈ ℙ → 1 < 𝑄)
21	8, 20	syl 18	. . . 4 ⊢ (𝜑 → 1 < 𝑄)
22	16, 11, 19, 21	ltmul2dd 13143	. . 3 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 1) < ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄))
23	15, 22	eqbrtrrd 5129	. 2 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄))
24	3	nnzd 12642	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℤ)
25	24, 6	zexpcld 14152	. . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℤ)
26	10	nnzd 12642	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℤ)
27	5	nnzd 12642	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℤ)
28	25, 26	gcdcomd 16605	. . . . 5 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) gcd 𝑄) = (𝑄 gcd (𝑃↑(𝑃 pCnt 𝑅))))
29		0lt1 11761	. . . . . . . . . . . . . 14 ⊢ 0 < 1
30	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < 1)
31		0red 11236	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ ℝ)
32	31, 16	ltnled 11382	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 < 1 ↔ ¬ 1 ≤ 0))
33	30, 32	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 1 ≤ 0)
34	11	recnd 11262	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑄 ∈ ℂ)
35	34	exp1d 14206	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑄↑1) = 𝑄)
36	35	eqcomd 2766	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑄 = (𝑄↑1))
37	36	oveq2d 7430	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 pCnt 𝑄) = (𝑄 pCnt (𝑄↑1)))
38		1zzd 12650	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℤ)
39		pcid 16966	. . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . 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 5106	. . . . . . . . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ 𝑄 = 𝑃)
53		aks4d1p8d2.5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∥ 𝑅)
54		pcelnn 16963	. . . . . . . . . . . . . . . . . . . 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 16808	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄 ∈ ℙ ∧ 𝑃 ∈ ℙ ∧ (𝑃 pCnt 𝑅) ∈ ℕ) → (𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ 𝑄 = 𝑃))
58	8, 1, 56, 57	syl3anc 1398	. . . . . . . . . . . . . . . . 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 14310	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℕ)
62		pceq0 16964	. . . . . . . . . . . . . . . 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 5116	. . . . . . . . . . . . 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 7429	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 = 𝑄) → (𝑝 pCnt 𝑄) = (𝑄 pCnt 𝑄))
71	69	oveq1d 7429	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 = 𝑄) → (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))) = (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
72	70, 71	breq12d 5116	. . . . . . . . . . 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 3581	. . . . . . . 8 ⊢ (𝜑 → ∃𝑝 ∈ ℙ ¬ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
76		rexnal 3114	. . . . . . . . 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 16972	. . . . . . . . 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 16803	. . . . . . 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 2795	. . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) gcd 𝑄) = 1)
87		pcdvds 16957	. . . . 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 42868	. . 3 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∥ 𝑅)
91	25, 26	zmulcld 12732	. . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∈ ℤ)
92		dvdsle 16401	. . . 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 11395	1 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < 𝑅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   class class class wbr 5103  (class class class)co 7414  0cc0 11125  1c1 11126   · cmul 11130   < clt 11268   ≤ cle 11269  ℕcn 12258  ℤcz 12616  ↑cexp 14126   ∥ cdvds 16343   gcd cgcd 16585  ℙcprime 16762   pCnt cpc 16929

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202  ax-pre-sup 11203

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8456  df-2o 8457  df-er 8697  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-sup 9413  df-inf 9414  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-div 11897  df-nn 12259  df-2 12328  df-3 12329  df-n0 12530  df-z 12617  df-uz 12889  df-q 12999  df-rp 13044  df-fz 13563  df-fl 13854  df-mod 13932  df-seq 14067  df-exp 14127  df-cj 15187  df-re 15188  df-im 15189  df-sqrt 15323  df-abs 15324  df-dvds 16344  df-gcd 16586  df-prm 16763  df-pc 16930

This theorem is used by:  aks4d1p8  42954