Theorem aks4d1p8d2 42067

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 16708	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ)
4	3	nnred 12279	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℝ)
5		aks4d1p8d2.1	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℕ)
6	1, 5	pccld 16884	. . 3 ⊢ (𝜑 → (𝑃 pCnt 𝑅) ∈ ℕ0)
7	4, 6	reexpcld 14200	. 2 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℝ)
8		aks4d1p8d2.4	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℙ)
9		prmnn 16708	. . . . 5 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℕ)
11	10	nnred 12279	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℝ)
12	7, 11	remulcld 11289	. 2 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∈ ℝ)
13	5	nnred 12279	. 2 ⊢ (𝜑 → 𝑅 ∈ ℝ)
14	7	recnd 11287	. . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℂ)
15	14	mulridd 11276	. . 3 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 1) = (𝑃↑(𝑃 pCnt 𝑅)))
16		1red 11260	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
17	3	nnrpd 13073	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℝ+)
18	6	nn0zd 12637	. . . . 5 ⊢ (𝜑 → (𝑃 pCnt 𝑅) ∈ ℤ)
19	17, 18	rpexpcld 14283	. . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℝ+)
20		prmgt1 16731	. . . . 5 ⊢ (𝑄 ∈ ℙ → 1 < 𝑄)
21	8, 20	syl 17	. . . 4 ⊢ (𝜑 → 1 < 𝑄)
22	16, 11, 19, 21	ltmul2dd 13131	. . 3 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 1) < ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄))
23	15, 22	eqbrtrrd 5172	. 2 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄))
24	3	nnzd 12638	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℤ)
25	24, 6	zexpcld 14125	. . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℤ)
26	10	nnzd 12638	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℤ)
27	5	nnzd 12638	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℤ)
28	25, 26	gcdcomd 16548	. . . . 5 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) gcd 𝑄) = (𝑄 gcd (𝑃↑(𝑃 pCnt 𝑅))))
29		0lt1 11783	. . . . . . . . . . . . . 14 ⊢ 0 < 1
30	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < 1)
31		0red 11262	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ ℝ)
32	31, 16	ltnled 11406	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 < 1 ↔ ¬ 1 ≤ 0))
33	30, 32	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 1 ≤ 0)
34	11	recnd 11287	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑄 ∈ ℂ)
35	34	exp1d 14178	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑄↑1) = 𝑄)
36	35	eqcomd 2741	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑄 = (𝑄↑1))
37	36	oveq2d 7447	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 pCnt 𝑄) = (𝑄 pCnt (𝑄↑1)))
38		1zzd 12646	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℤ)
39		pcid 16907	. . . . . . . . . . . . . . . 16 ⊢ ((𝑄 ∈ ℙ ∧ 1 ∈ ℤ) → (𝑄 pCnt (𝑄↑1)) = 1)
40	8, 38, 39	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 pCnt (𝑄↑1)) = 1)
41	37, 40	eqtrd 2775	. . . . . . . . . . . . . 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 5151	. . . . . . . . . . . . . . . . . . . . . 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 2745	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ 𝑄 = 𝑃)
53		aks4d1p8d2.5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∥ 𝑅)
54		pcelnn 16904	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 ∈ ℙ ∧ 𝑅 ∈ ℕ) → ((𝑃 pCnt 𝑅) ∈ ℕ ↔ 𝑃 ∥ 𝑅))
55	1, 5, 54	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑃 pCnt 𝑅) ∈ ℕ ↔ 𝑃 ∥ 𝑅))
56	53, 55	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑃 pCnt 𝑅) ∈ ℕ)
57		prmdvdsexpb 16750	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄 ∈ ℙ ∧ 𝑃 ∈ ℙ ∧ (𝑃 pCnt 𝑅) ∈ ℕ) → (𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ 𝑄 = 𝑃))
58	8, 1, 56, 57	syl3anc 1370	. . . . . . . . . . . . . . . . 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 14281	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℕ)
62		pceq0 16905	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . 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 5161	. . . . . . . . . . . . 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 7446	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 = 𝑄) → (𝑝 pCnt 𝑄) = (𝑄 pCnt 𝑄))
71	69	oveq1d 7446	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 = 𝑄) → (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))) = (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
72	70, 71	breq12d 5161	. . . . . . . . . . 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 3627	. . . . . . . 8 ⊢ (𝜑 → ∃𝑝 ∈ ℙ ¬ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
76		rexnal 3098	. . . . . . . . 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 16913	. . . . . . . . 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 584	. . . . . . . 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 16745	. . . . . . 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 584	. . . . . 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 2775	. . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) gcd 𝑄) = 1)
87		pcdvds 16898	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑅 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝑅)) ∥ 𝑅)
88	1, 5, 87	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∥ 𝑅)
89		aks4d1p8d2.6	. . . 4 ⊢ (𝜑 → 𝑄 ∥ 𝑅)
90	25, 26, 27, 86, 88, 89	coprmdvds2d 41983	. . 3 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∥ 𝑅)
91	25, 26	zmulcld 12726	. . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∈ ℤ)
92		dvdsle 16344	. . . 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 584	. . 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 11419	1 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < 𝑅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   class class class wbr 5148  (class class class)co 7431  0cc0 11153  1c1 11154   · cmul 11158   < clt 11293   ≤ cle 11294  ℕcn 12264  ℤcz 12611  ↑cexp 14099   ∥ cdvds 16287   gcd cgcd 16528  ℙcprime 16705   pCnt cpc 16870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-fz 13545  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-dvds 16288  df-gcd 16529  df-prm 16706  df-pc 16871

This theorem is referenced by:  aks4d1p8  42069