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Theorem bposlem1 27549

Description: An upper bound on the prime powers dividing a central binomial coefficient. (Contributed by Mario Carneiro, 9-Mar-2014.)

**Assertion**
Ref	Expression
bposlem1	⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃↑(𝑃 pCnt ((2 · 𝑁)C𝑁))) ≤ (2 · 𝑁))

Proof of Theorem bposlem1 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 14059	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (1...(2 · 𝑁)) ∈ Fin)
2		2nn 12360	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
3		nnmulcl 12303	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2 · 𝑁) ∈ ℕ)
4	2, 3	mpan 703	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (2 · 𝑁) ∈ ℕ)
5	4	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · 𝑁) ∈ ℕ)
6		prmnn 16786	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
7	6	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑃 ∈ ℕ)
8		elfznn 13630	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...(2 · 𝑁)) → 𝑘 ∈ ℕ)
9	8	adantl 487	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑘 ∈ ℕ)
10	9	nnnn0d 12611	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑘 ∈ ℕ₀)
11	7, 10	nnexpcld 14331	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃↑𝑘) ∈ ℕ)
12		nnrp 13076	. . . . . . . . . 10 ⊢ ((2 · 𝑁) ∈ ℕ → (2 · 𝑁) ∈ ℝ⁺)
13		nnrp 13076	. . . . . . . . . 10 ⊢ ((𝑃↑𝑘) ∈ ℕ → (𝑃↑𝑘) ∈ ℝ⁺)
14		rpdivcl 13091	. . . . . . . . . 10 ⊢ (((2 · 𝑁) ∈ ℝ⁺ ∧ (𝑃↑𝑘) ∈ ℝ⁺) → ((2 · 𝑁) / (𝑃↑𝑘)) ∈ ℝ⁺)
15	12, 13, 14	syl2an 608	. . . . . . . . 9 ⊢ (((2 · 𝑁) ∈ ℕ ∧ (𝑃↑𝑘) ∈ ℕ) → ((2 · 𝑁) / (𝑃↑𝑘)) ∈ ℝ⁺)
16	5, 11, 15	syl2anc 596	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) / (𝑃↑𝑘)) ∈ ℝ⁺)
17	16	rpred 13108	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) / (𝑃↑𝑘)) ∈ ℝ)
18	17	flcld 13881	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘((2 · 𝑁) / (𝑃↑𝑘))) ∈ ℤ)
19		2z 12672	. . . . . . 7 ⊢ 2 ∈ ℤ
20		simpll 779	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑁 ∈ ℕ)
21		nnrp 13076	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ⁺)
22		rpdivcl 13091	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℝ⁺ ∧ (𝑃↑𝑘) ∈ ℝ⁺) → (𝑁 / (𝑃↑𝑘)) ∈ ℝ⁺)
23	21, 13, 22	syl2an 608	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃↑𝑘) ∈ ℕ) → (𝑁 / (𝑃↑𝑘)) ∈ ℝ⁺)
24	20, 11, 23	syl2anc 596	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑁 / (𝑃↑𝑘)) ∈ ℝ⁺)
25	24	rpred 13108	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑁 / (𝑃↑𝑘)) ∈ ℝ)
26	25	flcld 13881	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℤ)
27		zmulcl 12689	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℤ) → (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))) ∈ ℤ)
28	19, 26, 27	sylancr 599	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))) ∈ ℤ)
29	18, 28	zsubcld 12752	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ∈ ℤ)
30	29	zred 12747	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ∈ ℝ)
31		1re 11254	. . . . . 6 ⊢ 1 ∈ ℝ
32		0re 11256	. . . . . 6 ⊢ 0 ∈ ℝ
33	31, 32	ifcli 4530	. . . . 5 ⊢ if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0) ∈ ℝ
34	33	a1i 11	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0) ∈ ℝ)
35	28	zred 12747	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))) ∈ ℝ)
36	17, 35	resubcld 11688	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ∈ ℝ)
37		2re 12361	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
38	37	a1i 11	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 2 ∈ ℝ)
39	18	zred 12747	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘((2 · 𝑁) / (𝑃↑𝑘))) ∈ ℝ)
40		flle 13882	. . . . . . . . . . 11 ⊢ (((2 · 𝑁) / (𝑃↑𝑘)) ∈ ℝ → (⌊‘((2 · 𝑁) / (𝑃↑𝑘))) ≤ ((2 · 𝑁) / (𝑃↑𝑘)))
41	17, 40	syl 18	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘((2 · 𝑁) / (𝑃↑𝑘))) ≤ ((2 · 𝑁) / (𝑃↑𝑘)))
42	39, 17, 35, 41	lesub1dd 11876	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ (((2 · 𝑁) / (𝑃↑𝑘)) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))))
43		resubcl 11568	. . . . . . . . . . . . 13 ⊢ (((𝑁 / (𝑃↑𝑘)) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑁 / (𝑃↑𝑘)) − 1) ∈ ℝ)
44	25, 31, 43	sylancl 598	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑁 / (𝑃↑𝑘)) − 1) ∈ ℝ)
45		remulcl 11231	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ ((𝑁 / (𝑃↑𝑘)) − 1) ∈ ℝ) → (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) ∈ ℝ)
46	37, 44, 45	sylancr 599	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) ∈ ℝ)
47		flltp1 13883	. . . . . . . . . . . . . 14 ⊢ ((𝑁 / (𝑃↑𝑘)) ∈ ℝ → (𝑁 / (𝑃↑𝑘)) < ((⌊‘(𝑁 / (𝑃↑𝑘))) + 1))
48	25, 47	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑁 / (𝑃↑𝑘)) < ((⌊‘(𝑁 / (𝑃↑𝑘))) + 1))
49		1red 11255	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 1 ∈ ℝ)
50	26	zred 12747	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℝ)
51	25, 49, 50	ltsubaddd 11856	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((𝑁 / (𝑃↑𝑘)) − 1) < (⌊‘(𝑁 / (𝑃↑𝑘))) ↔ (𝑁 / (𝑃↑𝑘)) < ((⌊‘(𝑁 / (𝑃↑𝑘))) + 1)))
52	48, 51	mpbird 260	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑁 / (𝑃↑𝑘)) − 1) < (⌊‘(𝑁 / (𝑃↑𝑘))))
53		2pos 12391	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
54	37, 53	pm3.2i 476	. . . . . . . . . . . . . 14 ⊢ (2 ∈ ℝ ∧ 0 < 2)
55		ltmul2 12112	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 / (𝑃↑𝑘)) − 1) ∈ ℝ ∧ (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((𝑁 / (𝑃↑𝑘)) − 1) < (⌊‘(𝑁 / (𝑃↑𝑘))) ↔ (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) < (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))))
56	54, 55	mp3an3 1479	. . . . . . . . . . . . 13 ⊢ ((((𝑁 / (𝑃↑𝑘)) − 1) ∈ ℝ ∧ (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℝ) → (((𝑁 / (𝑃↑𝑘)) − 1) < (⌊‘(𝑁 / (𝑃↑𝑘))) ↔ (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) < (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))))
57	44, 50, 56	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((𝑁 / (𝑃↑𝑘)) − 1) < (⌊‘(𝑁 / (𝑃↑𝑘))) ↔ (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) < (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))))
58	52, 57	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) < (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))))
59	46, 35, 17, 58	ltsub2dd 11873	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) < (((2 · 𝑁) / (𝑃↑𝑘)) − (2 · ((𝑁 / (𝑃↑𝑘)) − 1))))
60		2cnd 12365	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 2 ∈ ℂ)
61		nncn 12287	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
62	61	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑁 ∈ ℂ)
63	11	nncnd 12295	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃↑𝑘) ∈ ℂ)
64	11	nnne0d 12332	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃↑𝑘) ≠ 0)
65	60, 62, 63, 64	divassd 12072	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) / (𝑃↑𝑘)) = (2 · (𝑁 / (𝑃↑𝑘))))
66	25	recnd 11283	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑁 / (𝑃↑𝑘)) ∈ ℂ)
67	60, 66	muls1d 11720	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) = ((2 · (𝑁 / (𝑃↑𝑘))) − 2))
68	65, 67	oveq12d 7433	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) − (2 · ((𝑁 / (𝑃↑𝑘)) − 1))) = ((2 · (𝑁 / (𝑃↑𝑘))) − ((2 · (𝑁 / (𝑃↑𝑘))) − 2)))
69		remulcl 11231	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℝ ∧ (𝑁 / (𝑃↑𝑘)) ∈ ℝ) → (2 · (𝑁 / (𝑃↑𝑘))) ∈ ℝ)
70	37, 25, 69	sylancr 599	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · (𝑁 / (𝑃↑𝑘))) ∈ ℝ)
71	70	recnd 11283	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · (𝑁 / (𝑃↑𝑘))) ∈ ℂ)
72		2cn 12362	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
73		nncan 11533	. . . . . . . . . . . 12 ⊢ (((2 · (𝑁 / (𝑃↑𝑘))) ∈ ℂ ∧ 2 ∈ ℂ) → ((2 · (𝑁 / (𝑃↑𝑘))) − ((2 · (𝑁 / (𝑃↑𝑘))) − 2)) = 2)
74	71, 72, 73	sylancl 598	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · (𝑁 / (𝑃↑𝑘))) − ((2 · (𝑁 / (𝑃↑𝑘))) − 2)) = 2)
75	68, 74	eqtrd 2795	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) − (2 · ((𝑁 / (𝑃↑𝑘)) − 1))) = 2)
76	59, 75	breqtrd 5131	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) < 2)
77	30, 36, 38, 42, 76	lelttrd 11414	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) < 2)
78		df-2 12349	. . . . . . . 8 ⊢ 2 = (1 + 1)
79	77, 78	breqtrdi 5146	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) < (1 + 1))
80		1z 12670	. . . . . . . 8 ⊢ 1 ∈ ℤ
81		zleltp1 12691	. . . . . . . 8 ⊢ ((((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ∈ ℤ ∧ 1 ∈ ℤ) → (((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ 1 ↔ ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) < (1 + 1)))
82	29, 80, 81	sylancl 598	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ 1 ↔ ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) < (1 + 1)))
83	79, 82	mpbird 260	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ 1)
84		iftrue 4488	. . . . . . 7 ⊢ (𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0) = 1)
85	84	breq2d 5115	. . . . . 6 ⊢ (𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → (((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0) ↔ ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ 1))
86	83, 85	syl5ibrcom 250	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0)))
87	9	nnge1d 12330	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 1 ≤ 𝑘)
88	87	biantrurd 542	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑘 ≤ (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ↔ (1 ≤ 𝑘 ∧ 𝑘 ≤ (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))))))
89	6	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℕ)
90	89	nnred 12294	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℝ)
91		prmuz2 16808	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
92	91	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ (ℤ_≥‘2))
93		eluz2gt1 12991	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 1 < 𝑃)
94	92, 93	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 < 𝑃)
95	90, 94	jca 521	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃 ∈ ℝ ∧ 1 < 𝑃))
96	95	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃 ∈ ℝ ∧ 1 < 𝑃))
97		elfzelz 13600	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...(2 · 𝑁)) → 𝑘 ∈ ℤ)
98	97	adantl 487	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑘 ∈ ℤ)
99	4	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) ∈ ℕ)
100	99	nnrpd 13106	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) ∈ ℝ⁺)
101	100	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · 𝑁) ∈ ℝ⁺)
102		efexple 27546	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℝ ∧ 1 < 𝑃) ∧ 𝑘 ∈ ℤ ∧ (2 · 𝑁) ∈ ℝ⁺) → ((𝑃↑𝑘) ≤ (2 · 𝑁) ↔ 𝑘 ≤ (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))))
103	96, 98, 101, 102	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑃↑𝑘) ≤ (2 · 𝑁) ↔ 𝑘 ≤ (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))))
104	9	nnzd 12663	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑘 ∈ ℤ)
105	80	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 1 ∈ ℤ)
106	99	nnred 12294	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) ∈ ℝ)
107		1red 11255	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 ∈ ℝ)
108	37	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 2 ∈ ℝ)
109		1lt2 12459	. . . . . . . . . . . . . . . . . 18 ⊢ 1 < 2
110	109	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 < 2)
111		2t1e2 12449	. . . . . . . . . . . . . . . . . 18 ⊢ (2 · 1) = 2
112		nnre 12286	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
113	112	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ ℝ)
114		0le2 12389	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ≤ 2
115	37, 114	pm3.2i 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2 ∈ ℝ ∧ 0 ≤ 2)
116	115	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 ∈ ℝ ∧ 0 ≤ 2))
117		nnge1 12310	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
118	117	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 ≤ 𝑁)
119		lemul2a 12116	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 ≤ 2)) ∧ 1 ≤ 𝑁) → (2 · 1) ≤ (2 · 𝑁))
120	107, 113, 116, 118, 119	syl31anc 1400	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 1) ≤ (2 · 𝑁))
121	111, 120	eqbrtrrid 5141	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 2 ≤ (2 · 𝑁))
122	107, 108, 106, 110, 121	ltletrd 11416	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 < (2 · 𝑁))
123	106, 122	rplogcld 26895	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (log‘(2 · 𝑁)) ∈ ℝ⁺)
124	90, 94	rplogcld 26895	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (log‘𝑃) ∈ ℝ⁺)
125	123, 124	rpdivcld 13125	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((log‘(2 · 𝑁)) / (log‘𝑃)) ∈ ℝ⁺)
126	125	rpred 13108	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((log‘(2 · 𝑁)) / (log‘𝑃)) ∈ ℝ)
127	126	flcld 13881	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℤ)
128	127	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℤ)
129		elfz 13589	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℤ ∧ 1 ∈ ℤ ∧ (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℤ) → (𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ↔ (1 ≤ 𝑘 ∧ 𝑘 ≤ (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))))))
130	104, 105, 128, 129	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ↔ (1 ≤ 𝑘 ∧ 𝑘 ≤ (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))))))
131	88, 103, 130	3bitr4rd 315	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ↔ (𝑃↑𝑘) ≤ (2 · 𝑁)))
132	131	notbid 321	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (¬ 𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ↔ ¬ (𝑃↑𝑘) ≤ (2 · 𝑁)))
133	106	adantr 486	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · 𝑁) ∈ ℝ)
134	11	nnred 12294	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃↑𝑘) ∈ ℝ)
135	133, 134	ltnled 11403	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) < (𝑃↑𝑘) ↔ ¬ (𝑃↑𝑘) ≤ (2 · 𝑁)))
136	132, 135	bitr4d 285	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (¬ 𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ↔ (2 · 𝑁) < (𝑃↑𝑘)))
137	16	rpge0d 13112	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 0 ≤ ((2 · 𝑁) / (𝑃↑𝑘)))
138	137	adantrr 730	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → 0 ≤ ((2 · 𝑁) / (𝑃↑𝑘)))
139	11	nngt0d 12331	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 0 < (𝑃↑𝑘))
140		ltdivmul 12136	. . . . . . . . . . . . . . . . 17 ⊢ (((2 · 𝑁) ∈ ℝ ∧ 1 ∈ ℝ ∧ ((𝑃↑𝑘) ∈ ℝ ∧ 0 < (𝑃↑𝑘))) → (((2 · 𝑁) / (𝑃↑𝑘)) < 1 ↔ (2 · 𝑁) < ((𝑃↑𝑘) · 1)))
141	133, 49, 134, 139, 140	syl112anc 1401	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) < 1 ↔ (2 · 𝑁) < ((𝑃↑𝑘) · 1)))
142	63	mulridd 11272	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑃↑𝑘) · 1) = (𝑃↑𝑘))
143	142	breq2d 5115	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) < ((𝑃↑𝑘) · 1) ↔ (2 · 𝑁) < (𝑃↑𝑘)))
144	141, 143	bitrd 282	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) < 1 ↔ (2 · 𝑁) < (𝑃↑𝑘)))
145	144	biimprd 251	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) < (𝑃↑𝑘) → ((2 · 𝑁) / (𝑃↑𝑘)) < 1))
146	145	impr 460	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((2 · 𝑁) / (𝑃↑𝑘)) < 1)
147		0p1e1 12407	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
148	146, 147	breqtrrdi 5147	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((2 · 𝑁) / (𝑃↑𝑘)) < (0 + 1))
149	17	adantrr 730	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((2 · 𝑁) / (𝑃↑𝑘)) ∈ ℝ)
150		0z 12648	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
151		flbi 13899	. . . . . . . . . . . . 13 ⊢ ((((2 · 𝑁) / (𝑃↑𝑘)) ∈ ℝ ∧ 0 ∈ ℤ) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) = 0 ↔ (0 ≤ ((2 · 𝑁) / (𝑃↑𝑘)) ∧ ((2 · 𝑁) / (𝑃↑𝑘)) < (0 + 1))))
152	149, 150, 151	sylancl 598	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) = 0 ↔ (0 ≤ ((2 · 𝑁) / (𝑃↑𝑘)) ∧ ((2 · 𝑁) / (𝑃↑𝑘)) < (0 + 1))))
153	138, 148, 152	mpbir2and 726	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (⌊‘((2 · 𝑁) / (𝑃↑𝑘))) = 0)
154	24	rpge0d 13112	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 0 ≤ (𝑁 / (𝑃↑𝑘)))
155	154	adantrr 730	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → 0 ≤ (𝑁 / (𝑃↑𝑘)))
156	112, 21	ltaddrp2d 13142	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 < (𝑁 + 𝑁))
157	61	2timesd 12533	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → (2 · 𝑁) = (𝑁 + 𝑁))
158	156, 157	breqtrrd 5133	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 < (2 · 𝑁))
159	158	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑁 < (2 · 𝑁))
160	112	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑁 ∈ ℝ)
161		lttr 11332	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℝ ∧ (2 · 𝑁) ∈ ℝ ∧ (𝑃↑𝑘) ∈ ℝ) → ((𝑁 < (2 · 𝑁) ∧ (2 · 𝑁) < (𝑃↑𝑘)) → 𝑁 < (𝑃↑𝑘)))
162	160, 133, 134, 161	syl3anc 1398	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑁 < (2 · 𝑁) ∧ (2 · 𝑁) < (𝑃↑𝑘)) → 𝑁 < (𝑃↑𝑘)))
163	159, 162	mpand 708	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) < (𝑃↑𝑘) → 𝑁 < (𝑃↑𝑘)))
164		ltdivmul 12136	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℝ ∧ 1 ∈ ℝ ∧ ((𝑃↑𝑘) ∈ ℝ ∧ 0 < (𝑃↑𝑘))) → ((𝑁 / (𝑃↑𝑘)) < 1 ↔ 𝑁 < ((𝑃↑𝑘) · 1)))
165	160, 49, 134, 139, 164	syl112anc 1401	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑁 / (𝑃↑𝑘)) < 1 ↔ 𝑁 < ((𝑃↑𝑘) · 1)))
166	142	breq2d 5115	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑁 < ((𝑃↑𝑘) · 1) ↔ 𝑁 < (𝑃↑𝑘)))
167	165, 166	bitrd 282	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑁 / (𝑃↑𝑘)) < 1 ↔ 𝑁 < (𝑃↑𝑘)))
168	163, 167	sylibrd 262	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) < (𝑃↑𝑘) → (𝑁 / (𝑃↑𝑘)) < 1))
169	168	impr 460	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (𝑁 / (𝑃↑𝑘)) < 1)
170	169, 147	breqtrrdi 5147	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (𝑁 / (𝑃↑𝑘)) < (0 + 1))
171	25	adantrr 730	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (𝑁 / (𝑃↑𝑘)) ∈ ℝ)
172		flbi 13899	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 / (𝑃↑𝑘)) ∈ ℝ ∧ 0 ∈ ℤ) → ((⌊‘(𝑁 / (𝑃↑𝑘))) = 0 ↔ (0 ≤ (𝑁 / (𝑃↑𝑘)) ∧ (𝑁 / (𝑃↑𝑘)) < (0 + 1))))
173	171, 150, 172	sylancl 598	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((⌊‘(𝑁 / (𝑃↑𝑘))) = 0 ↔ (0 ≤ (𝑁 / (𝑃↑𝑘)) ∧ (𝑁 / (𝑃↑𝑘)) < (0 + 1))))
174	155, 170, 173	mpbir2and 726	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (⌊‘(𝑁 / (𝑃↑𝑘))) = 0)
175	174	oveq2d 7431	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))) = (2 · 0))
176		2t0e0 12457	. . . . . . . . . . . 12 ⊢ (2 · 0) = 0
177	175, 176	eqtrdi 2811	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))) = 0)
178	153, 177	oveq12d 7433	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) = (0 − 0))
179		0m0e0 12405	. . . . . . . . . 10 ⊢ (0 − 0) = 0
180	178, 179	eqtrdi 2811	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) = 0)
181		0le0 12388	. . . . . . . . 9 ⊢ 0 ≤ 0
182	180, 181	eqbrtrdi 5144	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ 0)
183	182	expr 462	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) < (𝑃↑𝑘) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ 0))
184	136, 183	sylbid 243	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (¬ 𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ 0))
185		iffalse 4491	. . . . . . . 8 ⊢ (¬ 𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0) = 0)
186	185	eqcomd 2766	. . . . . . 7 ⊢ (¬ 𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → 0 = if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0))
187	186	breq2d 5115	. . . . . 6 ⊢ (¬ 𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → (((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ 0 ↔ ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0)))
188	184, 187	mpbidi 244	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (¬ 𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0)))
189	86, 188	pm2.61d 181	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0))
190	1, 30, 34, 189	fsumle 15908	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → Σ𝑘 ∈ (1...(2 · 𝑁))((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ Σ𝑘 ∈ (1...(2 · 𝑁))if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0))
191		pcbcctr 27541	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((2 · 𝑁)C𝑁)) = Σ𝑘 ∈ (1...(2 · 𝑁))((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))))
192	127	zred 12747	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℝ)
193		flle 13882	. . . . . . . . 9 ⊢ (((log‘(2 · 𝑁)) / (log‘𝑃)) ∈ ℝ → (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ≤ ((log‘(2 · 𝑁)) / (log‘𝑃)))
194	126, 193	syl 18	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ≤ ((log‘(2 · 𝑁)) / (log‘𝑃)))
195	99	nnnn0d 12611	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) ∈ ℕ₀)
196	89, 195	nnexpcld 14331	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃↑(2 · 𝑁)) ∈ ℕ)
197	196	nnred 12294	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃↑(2 · 𝑁)) ∈ ℝ)
198		bernneq3 14317	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (2 · 𝑁) ∈ ℕ₀) → (2 · 𝑁) < (𝑃↑(2 · 𝑁)))
199	92, 195, 198	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) < (𝑃↑(2 · 𝑁)))
200	106, 197, 199	ltled 11404	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) ≤ (𝑃↑(2 · 𝑁)))
201	100	reeflogd 26890	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (exp‘(log‘(2 · 𝑁))) = (2 · 𝑁))
202	89	nnrpd 13106	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℝ⁺)
203	99	nnzd 12663	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) ∈ ℤ)
204		reexplog 26861	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℝ⁺ ∧ (2 · 𝑁) ∈ ℤ) → (𝑃↑(2 · 𝑁)) = (exp‘((2 · 𝑁) · (log‘𝑃))))
205	202, 203, 204	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃↑(2 · 𝑁)) = (exp‘((2 · 𝑁) · (log‘𝑃))))
206	205	eqcomd 2766	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (exp‘((2 · 𝑁) · (log‘𝑃))) = (𝑃↑(2 · 𝑁)))
207	200, 201, 206	3brtr4d 5137	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (exp‘(log‘(2 · 𝑁))) ≤ (exp‘((2 · 𝑁) · (log‘𝑃))))
208	100	relogcld 26889	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (log‘(2 · 𝑁)) ∈ ℝ)
209	124	rpred 13108	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (log‘𝑃) ∈ ℝ)
210	106, 209	remulcld 11285	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((2 · 𝑁) · (log‘𝑃)) ∈ ℝ)
211		efle 16228	. . . . . . . . . . 11 ⊢ (((log‘(2 · 𝑁)) ∈ ℝ ∧ ((2 · 𝑁) · (log‘𝑃)) ∈ ℝ) → ((log‘(2 · 𝑁)) ≤ ((2 · 𝑁) · (log‘𝑃)) ↔ (exp‘(log‘(2 · 𝑁))) ≤ (exp‘((2 · 𝑁) · (log‘𝑃)))))
212	208, 210, 211	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((log‘(2 · 𝑁)) ≤ ((2 · 𝑁) · (log‘𝑃)) ↔ (exp‘(log‘(2 · 𝑁))) ≤ (exp‘((2 · 𝑁) · (log‘𝑃)))))
213	207, 212	mpbird 260	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (log‘(2 · 𝑁)) ≤ ((2 · 𝑁) · (log‘𝑃)))
214	208, 106, 124	ledivmul2d 13162	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (((log‘(2 · 𝑁)) / (log‘𝑃)) ≤ (2 · 𝑁) ↔ (log‘(2 · 𝑁)) ≤ ((2 · 𝑁) · (log‘𝑃))))
215	213, 214	mpbird 260	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((log‘(2 · 𝑁)) / (log‘𝑃)) ≤ (2 · 𝑁))
216	192, 126, 106, 194, 215	letrd 11413	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ≤ (2 · 𝑁))
217		eluz 12923	. . . . . . . 8 ⊢ (((⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℤ ∧ (2 · 𝑁) ∈ ℤ) → ((2 · 𝑁) ∈ (ℤ_≥‘(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ↔ (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ≤ (2 · 𝑁)))
218	127, 203, 217	syl2anc 596	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((2 · 𝑁) ∈ (ℤ_≥‘(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ↔ (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ≤ (2 · 𝑁)))
219	216, 218	mpbird 260	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) ∈ (ℤ_≥‘(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))))
220		fzss2 13641	. . . . . 6 ⊢ ((2 · 𝑁) ∈ (ℤ_≥‘(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ⊆ (1...(2 · 𝑁)))
221	219, 220	syl 18	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ⊆ (1...(2 · 𝑁)))
222		sumhash 17010	. . . . 5 ⊢ (((1...(2 · 𝑁)) ∈ Fin ∧ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ⊆ (1...(2 · 𝑁))) → Σ𝑘 ∈ (1...(2 · 𝑁))if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0) = (♯‘(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))))))
223	1, 221, 222	syl2anc 596	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → Σ𝑘 ∈ (1...(2 · 𝑁))if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0) = (♯‘(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))))))
224	125	rprege0d 13115	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (((log‘(2 · 𝑁)) / (log‘𝑃)) ∈ ℝ ∧ 0 ≤ ((log‘(2 · 𝑁)) / (log‘𝑃))))
225		flge0nn0 13903	. . . . 5 ⊢ ((((log‘(2 · 𝑁)) / (log‘𝑃)) ∈ ℝ ∧ 0 ≤ ((log‘(2 · 𝑁)) / (log‘𝑃))) → (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℕ₀)
226		hashfz1 14432	. . . . 5 ⊢ ((⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℕ₀ → (♯‘(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))))) = (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))))
227	224, 225, 226	3syl 19	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (♯‘(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))))) = (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))))
228	223, 227	eqtr2d 2796	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) = Σ𝑘 ∈ (1...(2 · 𝑁))if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0))
229	190, 191, 228	3brtr4d 5137	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((2 · 𝑁)C𝑁)) ≤ (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))))
230		simpr 490	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℙ)
231		nnnn0 12557	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
232		fzctr 13717	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (0...(2 · 𝑁)))
233		bccl2 14409	. . . . . . 7 ⊢ (𝑁 ∈ (0...(2 · 𝑁)) → ((2 · 𝑁)C𝑁) ∈ ℕ)
234	231, 232, 233	3syl 19	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((2 · 𝑁)C𝑁) ∈ ℕ)
235	234	adantr 486	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((2 · 𝑁)C𝑁) ∈ ℕ)
236	230, 235	pccld 16964	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ₀)
237	236	nn0zd 12662	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((2 · 𝑁)C𝑁)) ∈ ℤ)
238		efexple 27546	. . 3 ⊢ (((𝑃 ∈ ℝ ∧ 1 < 𝑃) ∧ (𝑃 pCnt ((2 · 𝑁)C𝑁)) ∈ ℤ ∧ (2 · 𝑁) ∈ ℝ⁺) → ((𝑃↑(𝑃 pCnt ((2 · 𝑁)C𝑁))) ≤ (2 · 𝑁) ↔ (𝑃 pCnt ((2 · 𝑁)C𝑁)) ≤ (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))))
239	90, 94, 237, 100, 238	syl211anc 1403	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((𝑃↑(𝑃 pCnt ((2 · 𝑁)C𝑁))) ≤ (2 · 𝑁) ↔ (𝑃 pCnt ((2 · 𝑁)C𝑁)) ≤ (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))))
240	229, 239	mpbird 260	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃↑(𝑃 pCnt ((2 · 𝑁)C𝑁))) ≤ (2 · 𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ⊆ wss 3899 ifcif 4482 class class class wbr 5103 ‘cfv 6534 (class class class)co 7415 Fincfn 8956 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 < clt 11289 ≤ cle 11290 − cmin 11487 / cdiv 11917 ℕcn 12279 2c2 12341 ℕ₀cn0 12550 ℤcz 12637 ℤ_≥cuz 12909 ℝ⁺crp 13064 ...cfz 13583 ⌊cfl 13873 ↑cexp 14147 Ccbc 14388 ♯chash 14416 Σcsu 15795 expce 16169 ℙcprime 16783 pCnt cpc 16950 logclog 26820

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-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7738 ax-inf2 9624 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225

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-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 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-se 5609 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 6300 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7372 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 df-supp 8161 df-frecs 8282 df-wrecs 8313 df-recs 8362 df-rdg 8401 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8909 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-fsupp 9336 df-fi 9385 df-sup 9416 df-inf 9417 df-oi 9486 df-card 9966 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11489 df-neg 11490 df-div 11918 df-nn 12280 df-2 12349 df-3 12350 df-4 12351 df-5 12352 df-6 12353 df-7 12354 df-8 12355 df-9 12356 df-n0 12551 df-z 12638 df-dec 12759 df-uz 12910 df-q 13020 df-rp 13065 df-xneg 13185 df-xadd 13186 df-xmul 13187 df-ioo 13424 df-ioc 13425 df-ico 13426 df-icc 13427 df-fz 13584 df-fzo 13732 df-fl 13875 df-mod 13953 df-seq 14088 df-exp 14148 df-fac 14360 df-bc 14389 df-hash 14417 df-shft 15162 df-cj 15208 df-re 15209 df-im 15210 df-sqrt 15344 df-abs 15345 df-limsup 15580 df-clim 15597 df-rlim 15598 df-sum 15796 df-ef 16175 df-sin 16177 df-cos 16178 df-pi 16180 df-dvds 16365 df-gcd 16607 df-prm 16784 df-pc 16951 df-struct 17261 df-sets 17278 df-slot 17296 df-ndx 17308 df-base 17324 df-ress 17345 df-plusg 17377 df-mulr 17378 df-starv 17379 df-sca 17380 df-vsca 17381 df-ip 17382 df-tset 17383 df-ple 17384 df-ds 17386 df-unif 17387 df-hom 17388 df-cco 17389 df-rest 17529 df-topn 17530 df-0g 17548 df-gsum 17549 df-topgen 17550 df-pt 17551 df-prds 17554 df-xrs 17610 df-qtop 17615 df-imas 17616 df-xps 17618 df-mre 17692 df-mrc 17693 df-acs 17695 df-mgm 18752 df-sgrp 18844 df-mnd 18860 df-submnd 18915 df-mulg 19214 df-cntz 19467 df-cmn 19932 df-psmet 21606 df-xmet 21607 df-met 21608 df-bl 21609 df-mopn 21610 df-fbas 21611 df-fg 21612 df-cnfld 21615 df-top 23148 df-topon 23165 df-topsp 23187 df-bases 23200 df-cld 23273 df-ntr 23274 df-cls 23275 df-nei 23352 df-lp 23390 df-perf 23391 df-cn 23481 df-cnp 23482 df-haus 23569 df-tx 23817 df-hmeo 24010 df-fil 24101 df-fm 24193 df-flim 24194 df-flf 24195 df-xms 24575 df-ms 24576 df-tms 24577 df-cncf 25135 df-limc 26122 df-dv 26123 df-log 26822

This theorem is used by: bposlem5 27553 bposlem6 27554 chebbnd1lem1 27734

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