Theorem aks4d1p8d2 42407

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 16605	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ)
4	3	nnred 12164	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℝ)
5		aks4d1p8d2.1	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℕ)
6	1, 5	pccld 16782	. . 3 ⊢ (𝜑 → (𝑃 pCnt 𝑅) ∈ ℕ0)
7	4, 6	reexpcld 14090	. 2 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℝ)
8		aks4d1p8d2.4	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℙ)
9		prmnn 16605	. . . . 5 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℕ)
11	10	nnred 12164	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℝ)
12	7, 11	remulcld 11166	. 2 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∈ ℝ)
13	5	nnred 12164	. 2 ⊢ (𝜑 → 𝑅 ∈ ℝ)
14	7	recnd 11164	. . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℂ)
15	14	mulridd 11153	. . 3 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 1) = (𝑃↑(𝑃 pCnt 𝑅)))
16		1red 11137	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
17	3	nnrpd 12951	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℝ+)
18	6	nn0zd 12517	. . . . 5 ⊢ (𝜑 → (𝑃 pCnt 𝑅) ∈ ℤ)
19	17, 18	rpexpcld 14174	. . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℝ+)
20		prmgt1 16628	. . . . 5 ⊢ (𝑄 ∈ ℙ → 1 < 𝑄)
21	8, 20	syl 17	. . . 4 ⊢ (𝜑 → 1 < 𝑄)
22	16, 11, 19, 21	ltmul2dd 13009	. . 3 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 1) < ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄))
23	15, 22	eqbrtrrd 5123	. 2 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄))
24	3	nnzd 12518	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℤ)
25	24, 6	zexpcld 14014	. . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℤ)
26	10	nnzd 12518	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℤ)
27	5	nnzd 12518	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℤ)
28	25, 26	gcdcomd 16445	. . . . 5 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) gcd 𝑄) = (𝑄 gcd (𝑃↑(𝑃 pCnt 𝑅))))
29		0lt1 11663	. . . . . . . . . . . . . 14 ⊢ 0 < 1
30	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < 1)
31		0red 11139	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ ℝ)
32	31, 16	ltnled 11284	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 < 1 ↔ ¬ 1 ≤ 0))
33	30, 32	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 1 ≤ 0)
34	11	recnd 11164	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑄 ∈ ℂ)
35	34	exp1d 14068	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑄↑1) = 𝑄)
36	35	eqcomd 2743	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑄 = (𝑄↑1))
37	36	oveq2d 7376	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 pCnt 𝑄) = (𝑄 pCnt (𝑄↑1)))
38		1zzd 12526	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℤ)
39		pcid 16805	. . . . . . . . . . . . . . . 16 ⊢ ((𝑄 ∈ ℙ ∧ 1 ∈ ℤ) → (𝑄 pCnt (𝑄↑1)) = 1)
40	8, 38, 39	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 pCnt (𝑄↑1)) = 1)
41	37, 40	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄 pCnt 𝑄) = 1)
42		aks4d1p8d2.8	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑄 ∥ 𝑁)
43	42	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → 𝑄 ∥ 𝑁)
44		breq1 5102	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 = 𝑄 → (𝑃 ∥ 𝑁 ↔ 𝑄 ∥ 𝑁))
45	44	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (𝑃 ∥ 𝑁 ↔ 𝑄 ∥ 𝑁))
46	45	bicomd 223	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (𝑄 ∥ 𝑁 ↔ 𝑃 ∥ 𝑁))
47	46	biimpd 229	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (𝑄 ∥ 𝑁 → 𝑃 ∥ 𝑁))
48	43, 47	mpd 15	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → 𝑃 ∥ 𝑁)
49		aks4d1p8d2.7	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 𝑃 ∥ 𝑁)
50	49	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ¬ 𝑃 ∥ 𝑁)
51	48, 50	pm2.65da 817	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ¬ 𝑃 = 𝑄)
52	51	neqcomd 2747	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ 𝑄 = 𝑃)
53		aks4d1p8d2.5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∥ 𝑅)
54		pcelnn 16802	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 ∈ ℙ ∧ 𝑅 ∈ ℕ) → ((𝑃 pCnt 𝑅) ∈ ℕ ↔ 𝑃 ∥ 𝑅))
55	1, 5, 54	syl2anc 585	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑃 pCnt 𝑅) ∈ ℕ ↔ 𝑃 ∥ 𝑅))
56	53, 55	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑃 pCnt 𝑅) ∈ ℕ)
57		prmdvdsexpb 16647	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄 ∈ ℙ ∧ 𝑃 ∈ ℙ ∧ (𝑃 pCnt 𝑅) ∈ ℕ) → (𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ 𝑄 = 𝑃))
58	8, 1, 56, 57	syl3anc 1374	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ 𝑄 = 𝑃))
59	58	notbid 318	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ ¬ 𝑄 = 𝑃))
60	52, 59	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)))
61	3, 6	nnexpcld 14172	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℕ)
62		pceq0 16803	. . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))) = 0 ↔ ¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅))))
64	60, 63	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))) = 0)
65	41, 64	breq12d 5112	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄 pCnt 𝑄) ≤ (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))) ↔ 1 ≤ 0))
66	65	notbid 318	. . . . . . . . . . . 12 ⊢ (𝜑 → (¬ (𝑄 pCnt 𝑄) ≤ (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))) ↔ ¬ 1 ≤ 0))
67	33, 66	mpbird 257	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑄 pCnt 𝑄) ≤ (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
68	67	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 = 𝑄) → ¬ (𝑄 pCnt 𝑄) ≤ (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
69		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 = 𝑄) → 𝑝 = 𝑄)
70	69	oveq1d 7375	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 = 𝑄) → (𝑝 pCnt 𝑄) = (𝑄 pCnt 𝑄))
71	69	oveq1d 7375	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 = 𝑄) → (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))) = (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
72	70, 71	breq12d 5112	. . . . . . . . . . 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 318	. . . . . . . . . 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 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 = 𝑄) → ¬ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
75	74, 8	rspcime 3582	. . . . . . . 8 ⊢ (𝜑 → ∃𝑝 ∈ ℙ ¬ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
76		rexnal 3089	. . . . . . . . 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 232	. . . . . . 7 ⊢ (𝜑 → ¬ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
79		pc2dvds 16811	. . . . . . . . 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 585	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅)))))
81	80	notbid 318	. . . . . . 7 ⊢ (𝜑 → (¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ ¬ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅)))))
82	78, 81	mpbird 257	. . . . . 6 ⊢ (𝜑 → ¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)))
83		coprm 16642	. . . . . . 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 585	. . . . . 6 ⊢ (𝜑 → (¬ 𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ (𝑄 gcd (𝑃↑(𝑃 pCnt 𝑅))) = 1))
85	82, 84	mpbid 232	. . . . 5 ⊢ (𝜑 → (𝑄 gcd (𝑃↑(𝑃 pCnt 𝑅))) = 1)
86	28, 85	eqtrd 2772	. . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) gcd 𝑄) = 1)
87		pcdvds 16796	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑅 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝑅)) ∥ 𝑅)
88	1, 5, 87	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∥ 𝑅)
89		aks4d1p8d2.6	. . . 4 ⊢ (𝜑 → 𝑄 ∥ 𝑅)
90	25, 26, 27, 86, 88, 89	coprmdvds2d 42323	. . 3 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∥ 𝑅)
91	25, 26	zmulcld 12606	. . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∈ ℤ)
92		dvdsle 16241	. . . 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 585	. . 3 ⊢ (𝜑 → (((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∥ 𝑅 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ≤ 𝑅))
94	90, 93	mpd 15	. 2 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ≤ 𝑅)
95	7, 12, 13, 23, 94	ltletrd 11297	1 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < 𝑅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3061   class class class wbr 5099  (class class class)co 7360  0cc0 11030  1c1 11031   · cmul 11035   < clt 11170   ≤ cle 11171  ℕcn 12149  ℤcz 12492  ↑cexp 13988   ∥ cdvds 16183   gcd cgcd 16425  ℙcprime 16602   pCnt cpc 16768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-sup 9349  df-inf 9350  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12150  df-2 12212  df-3 12213  df-n0 12406  df-z 12493  df-uz 12756  df-q 12866  df-rp 12910  df-fz 13428  df-fl 13716  df-mod 13794  df-seq 13929  df-exp 13989  df-cj 15026  df-re 15027  df-im 15028  df-sqrt 15162  df-abs 15163  df-dvds 16184  df-gcd 16426  df-prm 16603  df-pc 16769

This theorem is referenced by:  aks4d1p8  42409