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Theorem bposlem1 26432

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 13693	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (1...(2 · 𝑁)) ∈ Fin)
2		2nn 12046	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
3		nnmulcl 11997	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2 · 𝑁) ∈ ℕ)
4	2, 3	mpan 687	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (2 · 𝑁) ∈ ℕ)
5	4	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · 𝑁) ∈ ℕ)
6		prmnn 16379	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
7	6	ad2antlr 724	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑃 ∈ ℕ)
8		elfznn 13285	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...(2 · 𝑁)) → 𝑘 ∈ ℕ)
9	8	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑘 ∈ ℕ)
10	9	nnnn0d 12293	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑘 ∈ ℕ₀)
11	7, 10	nnexpcld 13960	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃↑𝑘) ∈ ℕ)
12		nnrp 12741	. . . . . . . . . 10 ⊢ ((2 · 𝑁) ∈ ℕ → (2 · 𝑁) ∈ ℝ⁺)
13		nnrp 12741	. . . . . . . . . 10 ⊢ ((𝑃↑𝑘) ∈ ℕ → (𝑃↑𝑘) ∈ ℝ⁺)
14		rpdivcl 12755	. . . . . . . . . 10 ⊢ (((2 · 𝑁) ∈ ℝ⁺ ∧ (𝑃↑𝑘) ∈ ℝ⁺) → ((2 · 𝑁) / (𝑃↑𝑘)) ∈ ℝ⁺)
15	12, 13, 14	syl2an 596	. . . . . . . . 9 ⊢ (((2 · 𝑁) ∈ ℕ ∧ (𝑃↑𝑘) ∈ ℕ) → ((2 · 𝑁) / (𝑃↑𝑘)) ∈ ℝ⁺)
16	5, 11, 15	syl2anc 584	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) / (𝑃↑𝑘)) ∈ ℝ⁺)
17	16	rpred 12772	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) / (𝑃↑𝑘)) ∈ ℝ)
18	17	flcld 13518	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘((2 · 𝑁) / (𝑃↑𝑘))) ∈ ℤ)
19		2z 12352	. . . . . . 7 ⊢ 2 ∈ ℤ
20		simpll 764	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑁 ∈ ℕ)
21		nnrp 12741	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ⁺)
22		rpdivcl 12755	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℝ⁺ ∧ (𝑃↑𝑘) ∈ ℝ⁺) → (𝑁 / (𝑃↑𝑘)) ∈ ℝ⁺)
23	21, 13, 22	syl2an 596	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃↑𝑘) ∈ ℕ) → (𝑁 / (𝑃↑𝑘)) ∈ ℝ⁺)
24	20, 11, 23	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑁 / (𝑃↑𝑘)) ∈ ℝ⁺)
25	24	rpred 12772	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑁 / (𝑃↑𝑘)) ∈ ℝ)
26	25	flcld 13518	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℤ)
27		zmulcl 12369	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℤ) → (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))) ∈ ℤ)
28	19, 26, 27	sylancr 587	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))) ∈ ℤ)
29	18, 28	zsubcld 12431	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ∈ ℤ)
30	29	zred 12426	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ∈ ℝ)
31		1re 10975	. . . . . 6 ⊢ 1 ∈ ℝ
32		0re 10977	. . . . . 6 ⊢ 0 ∈ ℝ
33	31, 32	ifcli 4506	. . . . 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 12426	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))) ∈ ℝ)
36	17, 35	resubcld 11403	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ∈ ℝ)
37		2re 12047	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
38	37	a1i 11	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 2 ∈ ℝ)
39	18	zred 12426	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘((2 · 𝑁) / (𝑃↑𝑘))) ∈ ℝ)
40		flle 13519	. . . . . . . . . . 11 ⊢ (((2 · 𝑁) / (𝑃↑𝑘)) ∈ ℝ → (⌊‘((2 · 𝑁) / (𝑃↑𝑘))) ≤ ((2 · 𝑁) / (𝑃↑𝑘)))
41	17, 40	syl 17	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘((2 · 𝑁) / (𝑃↑𝑘))) ≤ ((2 · 𝑁) / (𝑃↑𝑘)))
42	39, 17, 35, 41	lesub1dd 11591	. . . . . . . . 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 11285	. . . . . . . . . . . . 13 ⊢ (((𝑁 / (𝑃↑𝑘)) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑁 / (𝑃↑𝑘)) − 1) ∈ ℝ)
44	25, 31, 43	sylancl 586	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑁 / (𝑃↑𝑘)) − 1) ∈ ℝ)
45		remulcl 10956	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ ((𝑁 / (𝑃↑𝑘)) − 1) ∈ ℝ) → (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) ∈ ℝ)
46	37, 44, 45	sylancr 587	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) ∈ ℝ)
47		flltp1 13520	. . . . . . . . . . . . . 14 ⊢ ((𝑁 / (𝑃↑𝑘)) ∈ ℝ → (𝑁 / (𝑃↑𝑘)) < ((⌊‘(𝑁 / (𝑃↑𝑘))) + 1))
48	25, 47	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑁 / (𝑃↑𝑘)) < ((⌊‘(𝑁 / (𝑃↑𝑘))) + 1))
49		1red 10976	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 1 ∈ ℝ)
50	26	zred 12426	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℝ)
51	25, 49, 50	ltsubaddd 11571	. . . . . . . . . . . . 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 256	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑁 / (𝑃↑𝑘)) − 1) < (⌊‘(𝑁 / (𝑃↑𝑘))))
53		2pos 12076	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
54	37, 53	pm3.2i 471	. . . . . . . . . . . . . 14 ⊢ (2 ∈ ℝ ∧ 0 < 2)
55		ltmul2 11826	. . . . . . . . . . . . . 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 1449	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) < (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))))
59	46, 35, 17, 58	ltsub2dd 11588	. . . . . . . . . 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 12051	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 2 ∈ ℂ)
61		nncn 11981	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
62	61	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑁 ∈ ℂ)
63	11	nncnd 11989	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃↑𝑘) ∈ ℂ)
64	11	nnne0d 12023	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃↑𝑘) ≠ 0)
65	60, 62, 63, 64	divassd 11786	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) / (𝑃↑𝑘)) = (2 · (𝑁 / (𝑃↑𝑘))))
66	25	recnd 11003	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑁 / (𝑃↑𝑘)) ∈ ℂ)
67	60, 66	muls1d 11435	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) = ((2 · (𝑁 / (𝑃↑𝑘))) − 2))
68	65, 67	oveq12d 7293	. . . . . . . . . . 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 10956	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℝ ∧ (𝑁 / (𝑃↑𝑘)) ∈ ℝ) → (2 · (𝑁 / (𝑃↑𝑘))) ∈ ℝ)
70	37, 25, 69	sylancr 587	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · (𝑁 / (𝑃↑𝑘))) ∈ ℝ)
71	70	recnd 11003	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · (𝑁 / (𝑃↑𝑘))) ∈ ℂ)
72		2cn 12048	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
73		nncan 11250	. . . . . . . . . . . 12 ⊢ (((2 · (𝑁 / (𝑃↑𝑘))) ∈ ℂ ∧ 2 ∈ ℂ) → ((2 · (𝑁 / (𝑃↑𝑘))) − ((2 · (𝑁 / (𝑃↑𝑘))) − 2)) = 2)
74	71, 72, 73	sylancl 586	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · (𝑁 / (𝑃↑𝑘))) − ((2 · (𝑁 / (𝑃↑𝑘))) − 2)) = 2)
75	68, 74	eqtrd 2778	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) − (2 · ((𝑁 / (𝑃↑𝑘)) − 1))) = 2)
76	59, 75	breqtrd 5100	. . . . . . . . 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 11133	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) < 2)
78		df-2 12036	. . . . . . . 8 ⊢ 2 = (1 + 1)
79	77, 78	breqtrdi 5115	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) < (1 + 1))
80		1z 12350	. . . . . . . 8 ⊢ 1 ∈ ℤ
81		zleltp1 12371	. . . . . . . 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 586	. . . . . . 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 256	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ 1)
84		iftrue 4465	. . . . . . 7 ⊢ (𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0) = 1)
85	84	breq2d 5086	. . . . . 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 246	. . . . 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 12021	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 1 ≤ 𝑘)
88	87	biantrurd 533	. . . . . . . . . 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 482	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℕ)
90	89	nnred 11988	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℝ)
91		prmuz2 16401	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
92	91	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ (ℤ_≥‘2))
93		eluz2gt1 12660	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 1 < 𝑃)
94	92, 93	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 < 𝑃)
95	90, 94	jca 512	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃 ∈ ℝ ∧ 1 < 𝑃))
96	95	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃 ∈ ℝ ∧ 1 < 𝑃))
97		elfzelz 13256	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...(2 · 𝑁)) → 𝑘 ∈ ℤ)
98	97	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑘 ∈ ℤ)
99	4	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) ∈ ℕ)
100	99	nnrpd 12770	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) ∈ ℝ⁺)
101	100	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · 𝑁) ∈ ℝ⁺)
102		efexple 26429	. . . . . . . . . . 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 1370	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑃↑𝑘) ≤ (2 · 𝑁) ↔ 𝑘 ≤ (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))))
104	9	nnzd 12425	. . . . . . . . . . 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 11988	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) ∈ ℝ)
107		1red 10976	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 ∈ ℝ)
108	37	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 2 ∈ ℝ)
109		1lt2 12144	. . . . . . . . . . . . . . . . . 18 ⊢ 1 < 2
110	109	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 < 2)
111		2t1e2 12136	. . . . . . . . . . . . . . . . . 18 ⊢ (2 · 1) = 2
112		nnre 11980	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
113	112	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ ℝ)
114		0le2 12075	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ≤ 2
115	37, 114	pm3.2i 471	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2 ∈ ℝ ∧ 0 ≤ 2)
116	115	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 ∈ ℝ ∧ 0 ≤ 2))
117		nnge1 12001	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
118	117	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 ≤ 𝑁)
119		lemul2a 11830	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 ≤ 2)) ∧ 1 ≤ 𝑁) → (2 · 1) ≤ (2 · 𝑁))
120	107, 113, 116, 118, 119	syl31anc 1372	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 1) ≤ (2 · 𝑁))
121	111, 120	eqbrtrrid 5110	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 2 ≤ (2 · 𝑁))
122	107, 108, 106, 110, 121	ltletrd 11135	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 < (2 · 𝑁))
123	106, 122	rplogcld 25784	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (log‘(2 · 𝑁)) ∈ ℝ⁺)
124	90, 94	rplogcld 25784	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (log‘𝑃) ∈ ℝ⁺)
125	123, 124	rpdivcld 12789	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((log‘(2 · 𝑁)) / (log‘𝑃)) ∈ ℝ⁺)
126	125	rpred 12772	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((log‘(2 · 𝑁)) / (log‘𝑃)) ∈ ℝ)
127	126	flcld 13518	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℤ)
128	127	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℤ)
129		elfz 13245	. . . . . . . . . . 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 1370	. . . . . . . . . 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 312	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ↔ (𝑃↑𝑘) ≤ (2 · 𝑁)))
132	131	notbid 318	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (¬ 𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ↔ ¬ (𝑃↑𝑘) ≤ (2 · 𝑁)))
133	106	adantr 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · 𝑁) ∈ ℝ)
134	11	nnred 11988	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃↑𝑘) ∈ ℝ)
135	133, 134	ltnled 11122	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) < (𝑃↑𝑘) ↔ ¬ (𝑃↑𝑘) ≤ (2 · 𝑁)))
136	132, 135	bitr4d 281	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (¬ 𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ↔ (2 · 𝑁) < (𝑃↑𝑘)))
137	16	rpge0d 12776	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 0 ≤ ((2 · 𝑁) / (𝑃↑𝑘)))
138	137	adantrr 714	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → 0 ≤ ((2 · 𝑁) / (𝑃↑𝑘)))
139	11	nngt0d 12022	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 0 < (𝑃↑𝑘))
140		ltdivmul 11850	. . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) < 1 ↔ (2 · 𝑁) < ((𝑃↑𝑘) · 1)))
142	63	mulid1d 10992	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑃↑𝑘) · 1) = (𝑃↑𝑘))
143	142	breq2d 5086	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) < ((𝑃↑𝑘) · 1) ↔ (2 · 𝑁) < (𝑃↑𝑘)))
144	141, 143	bitrd 278	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) < 1 ↔ (2 · 𝑁) < (𝑃↑𝑘)))
145	144	biimprd 247	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) < (𝑃↑𝑘) → ((2 · 𝑁) / (𝑃↑𝑘)) < 1))
146	145	impr 455	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((2 · 𝑁) / (𝑃↑𝑘)) < 1)
147		0p1e1 12095	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
148	146, 147	breqtrrdi 5116	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((2 · 𝑁) / (𝑃↑𝑘)) < (0 + 1))
149	17	adantrr 714	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((2 · 𝑁) / (𝑃↑𝑘)) ∈ ℝ)
150		0z 12330	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
151		flbi 13536	. . . . . . . . . . . . 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 586	. . . . . . . . . . . 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 710	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (⌊‘((2 · 𝑁) / (𝑃↑𝑘))) = 0)
154	24	rpge0d 12776	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 0 ≤ (𝑁 / (𝑃↑𝑘)))
155	154	adantrr 714	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → 0 ≤ (𝑁 / (𝑃↑𝑘)))
156	112, 21	ltaddrp2d 12806	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 < (𝑁 + 𝑁))
157	61	2timesd 12216	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → (2 · 𝑁) = (𝑁 + 𝑁))
158	156, 157	breqtrrd 5102	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 < (2 · 𝑁))
159	158	ad2antrr 723	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑁 < (2 · 𝑁))
160	112	ad2antrr 723	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑁 ∈ ℝ)
161		lttr 11051	. . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑁 < (2 · 𝑁) ∧ (2 · 𝑁) < (𝑃↑𝑘)) → 𝑁 < (𝑃↑𝑘)))
163	159, 162	mpand 692	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) < (𝑃↑𝑘) → 𝑁 < (𝑃↑𝑘)))
164		ltdivmul 11850	. . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑁 / (𝑃↑𝑘)) < 1 ↔ 𝑁 < ((𝑃↑𝑘) · 1)))
166	142	breq2d 5086	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑁 < ((𝑃↑𝑘) · 1) ↔ 𝑁 < (𝑃↑𝑘)))
167	165, 166	bitrd 278	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑁 / (𝑃↑𝑘)) < 1 ↔ 𝑁 < (𝑃↑𝑘)))
168	163, 167	sylibrd 258	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) < (𝑃↑𝑘) → (𝑁 / (𝑃↑𝑘)) < 1))
169	168	impr 455	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (𝑁 / (𝑃↑𝑘)) < 1)
170	169, 147	breqtrrdi 5116	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (𝑁 / (𝑃↑𝑘)) < (0 + 1))
171	25	adantrr 714	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (𝑁 / (𝑃↑𝑘)) ∈ ℝ)
172		flbi 13536	. . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . 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 710	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (⌊‘(𝑁 / (𝑃↑𝑘))) = 0)
175	174	oveq2d 7291	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))) = (2 · 0))
176		2t0e0 12142	. . . . . . . . . . . 12 ⊢ (2 · 0) = 0
177	175, 176	eqtrdi 2794	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))) = 0)
178	153, 177	oveq12d 7293	. . . . . . . . . 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 12093	. . . . . . . . . 10 ⊢ (0 − 0) = 0
180	178, 179	eqtrdi 2794	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) = 0)
181		0le0 12074	. . . . . . . . 9 ⊢ 0 ≤ 0
182	180, 181	eqbrtrdi 5113	. . . . . . . 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 457	. . . . . . 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 239	. . . . . 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 4468	. . . . . . . 8 ⊢ (¬ 𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0) = 0)
186	185	eqcomd 2744	. . . . . . 7 ⊢ (¬ 𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → 0 = if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0))
187	186	breq2d 5086	. . . . . 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 240	. . . . 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 179	. . . 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 15511	. . 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 26424	. . 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 12426	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℝ)
193		flle 13519	. . . . . . . . 9 ⊢ (((log‘(2 · 𝑁)) / (log‘𝑃)) ∈ ℝ → (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ≤ ((log‘(2 · 𝑁)) / (log‘𝑃)))
194	126, 193	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ≤ ((log‘(2 · 𝑁)) / (log‘𝑃)))
195	99	nnnn0d 12293	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) ∈ ℕ₀)
196	89, 195	nnexpcld 13960	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃↑(2 · 𝑁)) ∈ ℕ)
197	196	nnred 11988	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃↑(2 · 𝑁)) ∈ ℝ)
198		bernneq3 13946	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (2 · 𝑁) ∈ ℕ₀) → (2 · 𝑁) < (𝑃↑(2 · 𝑁)))
199	92, 195, 198	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) < (𝑃↑(2 · 𝑁)))
200	106, 197, 199	ltled 11123	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) ≤ (𝑃↑(2 · 𝑁)))
201	100	reeflogd 25779	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (exp‘(log‘(2 · 𝑁))) = (2 · 𝑁))
202	89	nnrpd 12770	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℝ⁺)
203	99	nnzd 12425	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) ∈ ℤ)
204		reexplog 25750	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℝ⁺ ∧ (2 · 𝑁) ∈ ℤ) → (𝑃↑(2 · 𝑁)) = (exp‘((2 · 𝑁) · (log‘𝑃))))
205	202, 203, 204	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃↑(2 · 𝑁)) = (exp‘((2 · 𝑁) · (log‘𝑃))))
206	205	eqcomd 2744	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (exp‘((2 · 𝑁) · (log‘𝑃))) = (𝑃↑(2 · 𝑁)))
207	200, 201, 206	3brtr4d 5106	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (exp‘(log‘(2 · 𝑁))) ≤ (exp‘((2 · 𝑁) · (log‘𝑃))))
208	100	relogcld 25778	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (log‘(2 · 𝑁)) ∈ ℝ)
209	124	rpred 12772	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (log‘𝑃) ∈ ℝ)
210	106, 209	remulcld 11005	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((2 · 𝑁) · (log‘𝑃)) ∈ ℝ)
211		efle 15827	. . . . . . . . . . 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 584	. . . . . . . . . 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 256	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (log‘(2 · 𝑁)) ≤ ((2 · 𝑁) · (log‘𝑃)))
214	208, 106, 124	ledivmul2d 12826	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (((log‘(2 · 𝑁)) / (log‘𝑃)) ≤ (2 · 𝑁) ↔ (log‘(2 · 𝑁)) ≤ ((2 · 𝑁) · (log‘𝑃))))
215	213, 214	mpbird 256	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((log‘(2 · 𝑁)) / (log‘𝑃)) ≤ (2 · 𝑁))
216	192, 126, 106, 194, 215	letrd 11132	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ≤ (2 · 𝑁))
217		eluz 12596	. . . . . . . 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 584	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((2 · 𝑁) ∈ (ℤ_≥‘(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ↔ (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ≤ (2 · 𝑁)))
219	216, 218	mpbird 256	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) ∈ (ℤ_≥‘(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))))
220		fzss2 13296	. . . . . 6 ⊢ ((2 · 𝑁) ∈ (ℤ_≥‘(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ⊆ (1...(2 · 𝑁)))
221	219, 220	syl 17	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ⊆ (1...(2 · 𝑁)))
222		sumhash 16597	. . . . 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 584	. . . 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 12779	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (((log‘(2 · 𝑁)) / (log‘𝑃)) ∈ ℝ ∧ 0 ≤ ((log‘(2 · 𝑁)) / (log‘𝑃))))
225		flge0nn0 13540	. . . . 5 ⊢ ((((log‘(2 · 𝑁)) / (log‘𝑃)) ∈ ℝ ∧ 0 ≤ ((log‘(2 · 𝑁)) / (log‘𝑃))) → (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℕ₀)
226		hashfz1 14060	. . . . 5 ⊢ ((⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℕ₀ → (♯‘(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))))) = (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))))
227	224, 225, 226	3syl 18	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (♯‘(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))))) = (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))))
228	223, 227	eqtr2d 2779	. . 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 5106	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((2 · 𝑁)C𝑁)) ≤ (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))))
230		simpr 485	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℙ)
231		nnnn0 12240	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
232		fzctr 13368	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (0...(2 · 𝑁)))
233		bccl2 14037	. . . . . . 7 ⊢ (𝑁 ∈ (0...(2 · 𝑁)) → ((2 · 𝑁)C𝑁) ∈ ℕ)
234	231, 232, 233	3syl 18	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((2 · 𝑁)C𝑁) ∈ ℕ)
235	234	adantr 481	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((2 · 𝑁)C𝑁) ∈ ℕ)
236	230, 235	pccld 16551	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ₀)
237	236	nn0zd 12424	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((2 · 𝑁)C𝑁)) ∈ ℤ)
238		efexple 26429	. . 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 1375	. 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 256	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃↑(𝑃 pCnt ((2 · 𝑁)C𝑁))) ≤ (2 · 𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ⊆ wss 3887 ifcif 4459 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 / cdiv 11632 ℕcn 11973 2c2 12028 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ℝ⁺crp 12730 ...cfz 13239 ⌊cfl 13510 ↑cexp 13782 Ccbc 14016 ♯chash 14044 Σcsu 15397 expce 15771 ℙcprime 16376 pCnt cpc 16537 logclog 25710

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-dvds 15964 df-gcd 16202 df-prm 16377 df-pc 16538 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-log 25712

This theorem is referenced by: bposlem5 26436 bposlem6 26437 chebbnd1lem1 26617

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