Theorem aks4d1p8d2 42177

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 16585	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ)
4	3	nnred 12140	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℝ)
5		aks4d1p8d2.1	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℕ)
6	1, 5	pccld 16762	. . 3 ⊢ (𝜑 → (𝑃 pCnt 𝑅) ∈ ℕ0)
7	4, 6	reexpcld 14070	. 2 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℝ)
8		aks4d1p8d2.4	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℙ)
9		prmnn 16585	. . . . 5 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℕ)
11	10	nnred 12140	. . 3 ⊢ (𝜑 → 𝑄 ∈ ℝ)
12	7, 11	remulcld 11142	. 2 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∈ ℝ)
13	5	nnred 12140	. 2 ⊢ (𝜑 → 𝑅 ∈ ℝ)
14	7	recnd 11140	. . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℂ)
15	14	mulridd 11129	. . 3 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 1) = (𝑃↑(𝑃 pCnt 𝑅)))
16		1red 11113	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
17	3	nnrpd 12932	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℝ+)
18	6	nn0zd 12494	. . . . 5 ⊢ (𝜑 → (𝑃 pCnt 𝑅) ∈ ℤ)
19	17, 18	rpexpcld 14154	. . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℝ+)
20		prmgt1 16608	. . . . 5 ⊢ (𝑄 ∈ ℙ → 1 < 𝑄)
21	8, 20	syl 17	. . . 4 ⊢ (𝜑 → 1 < 𝑄)
22	16, 11, 19, 21	ltmul2dd 12990	. . 3 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 1) < ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄))
23	15, 22	eqbrtrrd 5113	. 2 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄))
24	3	nnzd 12495	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℤ)
25	24, 6	zexpcld 13994	. . . 4 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℤ)
26	10	nnzd 12495	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℤ)
27	5	nnzd 12495	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℤ)
28	25, 26	gcdcomd 16425	. . . . 5 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) gcd 𝑄) = (𝑄 gcd (𝑃↑(𝑃 pCnt 𝑅))))
29		0lt1 11639	. . . . . . . . . . . . . 14 ⊢ 0 < 1
30	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < 1)
31		0red 11115	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ ℝ)
32	31, 16	ltnled 11260	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 < 1 ↔ ¬ 1 ≤ 0))
33	30, 32	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 1 ≤ 0)
34	11	recnd 11140	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑄 ∈ ℂ)
35	34	exp1d 14048	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑄↑1) = 𝑄)
36	35	eqcomd 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑄 = (𝑄↑1))
37	36	oveq2d 7362	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 pCnt 𝑄) = (𝑄 pCnt (𝑄↑1)))
38		1zzd 12503	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℤ)
39		pcid 16785	. . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . 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 5092	. . . . . . . . . . . . . . . . . . . . . 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 816	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ¬ 𝑃 = 𝑄)
52	51	neqcomd 2741	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ 𝑄 = 𝑃)
53		aks4d1p8d2.5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∥ 𝑅)
54		pcelnn 16782	. . . . . . . . . . . . . . . . . . . 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 16627	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄 ∈ ℙ ∧ 𝑃 ∈ ℙ ∧ (𝑃 pCnt 𝑅) ∈ ℕ) → (𝑄 ∥ (𝑃↑(𝑃 pCnt 𝑅)) ↔ 𝑄 = 𝑃))
58	8, 1, 56, 57	syl3anc 1373	. . . . . . . . . . . . . . . . 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 14152	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℕ)
62		pceq0 16783	. . . . . . . . . . . . . . . 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 5102	. . . . . . . . . . . . 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 7361	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 = 𝑄) → (𝑝 pCnt 𝑄) = (𝑄 pCnt 𝑄))
71	69	oveq1d 7361	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 = 𝑄) → (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))) = (𝑄 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
72	70, 71	breq12d 5102	. . . . . . . . . . 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 3577	. . . . . . . 8 ⊢ (𝜑 → ∃𝑝 ∈ ℙ ¬ (𝑝 pCnt 𝑄) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝑅))))
76		rexnal 3084	. . . . . . . . 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 16791	. . . . . . . . 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 16622	. . . . . . 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 2766	. . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) gcd 𝑄) = 1)
87		pcdvds 16776	. . . . 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 42093	. . 3 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∥ 𝑅)
91	25, 26	zmulcld 12583	. . . 4 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) · 𝑄) ∈ ℤ)
92		dvdsle 16221	. . . 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 11273	1 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < 𝑅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   class class class wbr 5089  (class class class)co 7346  0cc0 11006  1c1 11007   · cmul 11011   < clt 11146   ≤ cle 11147  ℕcn 12125  ℤcz 12468  ↑cexp 13968   ∥ cdvds 16163   gcd cgcd 16405  ℙcprime 16582   pCnt cpc 16748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-pre-sup 11084

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-inf 9327  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-2 12188  df-3 12189  df-n0 12382  df-z 12469  df-uz 12733  df-q 12847  df-rp 12891  df-fz 13408  df-fl 13696  df-mod 13774  df-seq 13909  df-exp 13969  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-dvds 16164  df-gcd 16406  df-prm 16583  df-pc 16749

This theorem is referenced by:  aks4d1p8  42179