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Theorem bposlem1 26538

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 13794	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (1...(2 · 𝑁)) ∈ Fin)
2		2nn 12147	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
3		nnmulcl 12098	. . . . . . . . . . 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 16476	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
7	6	ad2antlr 724	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑃 ∈ ℕ)
8		elfznn 13386	. . . . . . . . . . . 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 12394	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑘 ∈ ℕ₀)
11	7, 10	nnexpcld 14061	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃↑𝑘) ∈ ℕ)
12		nnrp 12842	. . . . . . . . . 10 ⊢ ((2 · 𝑁) ∈ ℕ → (2 · 𝑁) ∈ ℝ⁺)
13		nnrp 12842	. . . . . . . . . 10 ⊢ ((𝑃↑𝑘) ∈ ℕ → (𝑃↑𝑘) ∈ ℝ⁺)
14		rpdivcl 12856	. . . . . . . . . 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 12873	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) / (𝑃↑𝑘)) ∈ ℝ)
18	17	flcld 13619	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘((2 · 𝑁) / (𝑃↑𝑘))) ∈ ℤ)
19		2z 12453	. . . . . . 7 ⊢ 2 ∈ ℤ
20		simpll 764	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑁 ∈ ℕ)
21		nnrp 12842	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ⁺)
22		rpdivcl 12856	. . . . . . . . . . 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 12873	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑁 / (𝑃↑𝑘)) ∈ ℝ)
26	25	flcld 13619	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℤ)
27		zmulcl 12470	. . . . . . 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 12532	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ∈ ℤ)
30	29	zred 12527	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ∈ ℝ)
31		1re 11076	. . . . . 6 ⊢ 1 ∈ ℝ
32		0re 11078	. . . . . 6 ⊢ 0 ∈ ℝ
33	31, 32	ifcli 4520	. . . . 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 12527	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))) ∈ ℝ)
36	17, 35	resubcld 11504	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ∈ ℝ)
37		2re 12148	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
38	37	a1i 11	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 2 ∈ ℝ)
39	18	zred 12527	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘((2 · 𝑁) / (𝑃↑𝑘))) ∈ ℝ)
40		flle 13620	. . . . . . . . . . 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 11692	. . . . . . . . 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 11386	. . . . . . . . . . . . 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 11057	. . . . . . . . . . . 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 13621	. . . . . . . . . . . . . 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 11077	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 1 ∈ ℝ)
50	26	zred 12527	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℝ)
51	25, 49, 50	ltsubaddd 11672	. . . . . . . . . . . . 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 12177	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
54	37, 53	pm3.2i 471	. . . . . . . . . . . . . 14 ⊢ (2 ∈ ℝ ∧ 0 < 2)
55		ltmul2 11927	. . . . . . . . . . . . . 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 11689	. . . . . . . . . 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 12152	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 2 ∈ ℂ)
61		nncn 12082	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
62	61	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑁 ∈ ℂ)
63	11	nncnd 12090	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃↑𝑘) ∈ ℂ)
64	11	nnne0d 12124	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃↑𝑘) ≠ 0)
65	60, 62, 63, 64	divassd 11887	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) / (𝑃↑𝑘)) = (2 · (𝑁 / (𝑃↑𝑘))))
66	25	recnd 11104	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑁 / (𝑃↑𝑘)) ∈ ℂ)
67	60, 66	muls1d 11536	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) = ((2 · (𝑁 / (𝑃↑𝑘))) − 2))
68	65, 67	oveq12d 7355	. . . . . . . . . . 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 11057	. . . . . . . . . . . . . 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 11104	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · (𝑁 / (𝑃↑𝑘))) ∈ ℂ)
72		2cn 12149	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
73		nncan 11351	. . . . . . . . . . . 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 2776	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) − (2 · ((𝑁 / (𝑃↑𝑘)) − 1))) = 2)
76	59, 75	breqtrd 5118	. . . . . . . . 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 11234	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) < 2)
78		df-2 12137	. . . . . . . 8 ⊢ 2 = (1 + 1)
79	77, 78	breqtrdi 5133	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) < (1 + 1))
80		1z 12451	. . . . . . . 8 ⊢ 1 ∈ ℤ
81		zleltp1 12472	. . . . . . . 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 4479	. . . . . . 7 ⊢ (𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0) = 1)
85	84	breq2d 5104	. . . . . 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 12122	. . . . . . . . . . 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 12089	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℝ)
91		prmuz2 16498	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
92	91	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ (ℤ_≥‘2))
93		eluz2gt1 12761	. . . . . . . . . . . . . 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 13357	. . . . . . . . . . . 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 12871	. . . . . . . . . . . 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 26535	. . . . . . . . . . 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 12526	. . . . . . . . . . 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 12089	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) ∈ ℝ)
107		1red 11077	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 ∈ ℝ)
108	37	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 2 ∈ ℝ)
109		1lt2 12245	. . . . . . . . . . . . . . . . . 18 ⊢ 1 < 2
110	109	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 < 2)
111		2t1e2 12237	. . . . . . . . . . . . . . . . . 18 ⊢ (2 · 1) = 2
112		nnre 12081	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
113	112	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ ℝ)
114		0le2 12176	. . . . . . . . . . . . . . . . . . . . 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 12102	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
118	117	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 ≤ 𝑁)
119		lemul2a 11931	. . . . . . . . . . . . . . . . . . 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 5128	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 2 ≤ (2 · 𝑁))
122	107, 108, 106, 110, 121	ltletrd 11236	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 < (2 · 𝑁))
123	106, 122	rplogcld 25890	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (log‘(2 · 𝑁)) ∈ ℝ⁺)
124	90, 94	rplogcld 25890	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (log‘𝑃) ∈ ℝ⁺)
125	123, 124	rpdivcld 12890	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((log‘(2 · 𝑁)) / (log‘𝑃)) ∈ ℝ⁺)
126	125	rpred 12873	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((log‘(2 · 𝑁)) / (log‘𝑃)) ∈ ℝ)
127	126	flcld 13619	. . . . . . . . . . . 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 13346	. . . . . . . . . . 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 311	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ↔ (𝑃↑𝑘) ≤ (2 · 𝑁)))
132	131	notbid 317	. . . . . . . 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 12089	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃↑𝑘) ∈ ℝ)
135	133, 134	ltnled 11223	. . . . . . . 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 12877	. . . . . . . . . . . . 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 12123	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 0 < (𝑃↑𝑘))
140		ltdivmul 11951	. . . . . . . . . . . . . . . . 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 11093	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑃↑𝑘) · 1) = (𝑃↑𝑘))
143	142	breq2d 5104	. . . . . . . . . . . . . . . 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 12196	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
148	146, 147	breqtrrdi 5134	. . . . . . . . . . . 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 12431	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
151		flbi 13637	. . . . . . . . . . . . 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 12877	. . . . . . . . . . . . . . 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 12907	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 < (𝑁 + 𝑁))
157	61	2timesd 12317	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → (2 · 𝑁) = (𝑁 + 𝑁))
158	156, 157	breqtrrd 5120	. . . . . . . . . . . . . . . . . . 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 11152	. . . . . . . . . . . . . . . . . . 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 11951	. . . . . . . . . . . . . . . . . . 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 5104	. . . . . . . . . . . . . . . . . 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 5134	. . . . . . . . . . . . . 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 13637	. . . . . . . . . . . . . . 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 7353	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))) = (2 · 0))
176		2t0e0 12243	. . . . . . . . . . . 12 ⊢ (2 · 0) = 0
177	175, 176	eqtrdi 2792	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))) = 0)
178	153, 177	oveq12d 7355	. . . . . . . . . 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 12194	. . . . . . . . . 10 ⊢ (0 − 0) = 0
180	178, 179	eqtrdi 2792	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) = 0)
181		0le0 12175	. . . . . . . . 9 ⊢ 0 ≤ 0
182	180, 181	eqbrtrdi 5131	. . . . . . . 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 4482	. . . . . . . 8 ⊢ (¬ 𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0) = 0)
186	185	eqcomd 2742	. . . . . . 7 ⊢ (¬ 𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → 0 = if(𝑘 ∈ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0))
187	186	breq2d 5104	. . . . . 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 15610	. . 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 26530	. . 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 12527	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℝ)
193		flle 13620	. . . . . . . . 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 12394	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) ∈ ℕ₀)
196	89, 195	nnexpcld 14061	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃↑(2 · 𝑁)) ∈ ℕ)
197	196	nnred 12089	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃↑(2 · 𝑁)) ∈ ℝ)
198		bernneq3 14047	. . . . . . . . . . . . 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 11224	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) ≤ (𝑃↑(2 · 𝑁)))
201	100	reeflogd 25885	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (exp‘(log‘(2 · 𝑁))) = (2 · 𝑁))
202	89	nnrpd 12871	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℝ⁺)
203	99	nnzd 12526	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 · 𝑁) ∈ ℤ)
204		reexplog 25856	. . . . . . . . . . . . 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 2742	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (exp‘((2 · 𝑁) · (log‘𝑃))) = (𝑃↑(2 · 𝑁)))
207	200, 201, 206	3brtr4d 5124	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (exp‘(log‘(2 · 𝑁))) ≤ (exp‘((2 · 𝑁) · (log‘𝑃))))
208	100	relogcld 25884	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (log‘(2 · 𝑁)) ∈ ℝ)
209	124	rpred 12873	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (log‘𝑃) ∈ ℝ)
210	106, 209	remulcld 11106	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((2 · 𝑁) · (log‘𝑃)) ∈ ℝ)
211		efle 15926	. . . . . . . . . . 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 12927	. . . . . . . . 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 11233	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ≤ (2 · 𝑁))
217		eluz 12697	. . . . . . . 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 13397	. . . . . 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 16694	. . . . 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 12880	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (((log‘(2 · 𝑁)) / (log‘𝑃)) ∈ ℝ ∧ 0 ≤ ((log‘(2 · 𝑁)) / (log‘𝑃))))
225		flge0nn0 13641	. . . . 5 ⊢ ((((log‘(2 · 𝑁)) / (log‘𝑃)) ∈ ℝ ∧ 0 ≤ ((log‘(2 · 𝑁)) / (log‘𝑃))) → (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℕ₀)
226		hashfz1 14161	. . . . 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 2777	. . 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 5124	. 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 12341	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
232		fzctr 13469	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (0...(2 · 𝑁)))
233		bccl2 14138	. . . . . . 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 16648	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ₀)
237	236	nn0zd 12525	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((2 · 𝑁)C𝑁)) ∈ ℤ)
238		efexple 26535	. . 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 1540 ∈ wcel 2105 ⊆ wss 3898 ifcif 4473 class class class wbr 5092 ‘cfv 6479 (class class class)co 7337 Fincfn 8804 ℂcc 10970 ℝcr 10971 0cc0 10972 1c1 10973 + caddc 10975 · cmul 10977 < clt 11110 ≤ cle 11111 − cmin 11306 / cdiv 11733 ℕcn 12074 2c2 12129 ℕ₀cn0 12334 ℤcz 12420 ℤ_≥cuz 12683 ℝ⁺crp 12831 ...cfz 13340 ⌊cfl 13611 ↑cexp 13883 Ccbc 14117 ♯chash 14145 Σcsu 15496 expce 15870 ℙcprime 16473 pCnt cpc 16634 logclog 25816

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-inf2 9498 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-pre-sup 11050 ax-addf 11051 ax-mulf 11052

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-se 5576 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-isom 6488 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-of 7595 df-om 7781 df-1st 7899 df-2nd 7900 df-supp 8048 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-2o 8368 df-er 8569 df-map 8688 df-pm 8689 df-ixp 8757 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-fsupp 9227 df-fi 9268 df-sup 9299 df-inf 9300 df-oi 9367 df-card 9796 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-div 11734 df-nn 12075 df-2 12137 df-3 12138 df-4 12139 df-5 12140 df-6 12141 df-7 12142 df-8 12143 df-9 12144 df-n0 12335 df-z 12421 df-dec 12539 df-uz 12684 df-q 12790 df-rp 12832 df-xneg 12949 df-xadd 12950 df-xmul 12951 df-ioo 13184 df-ioc 13185 df-ico 13186 df-icc 13187 df-fz 13341 df-fzo 13484 df-fl 13613 df-mod 13691 df-seq 13823 df-exp 13884 df-fac 14089 df-bc 14118 df-hash 14146 df-shft 14877 df-cj 14909 df-re 14910 df-im 14911 df-sqrt 15045 df-abs 15046 df-limsup 15279 df-clim 15296 df-rlim 15297 df-sum 15497 df-ef 15876 df-sin 15878 df-cos 15879 df-pi 15881 df-dvds 16063 df-gcd 16301 df-prm 16474 df-pc 16635 df-struct 16945 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-mulr 17073 df-starv 17074 df-sca 17075 df-vsca 17076 df-ip 17077 df-tset 17078 df-ple 17079 df-ds 17081 df-unif 17082 df-hom 17083 df-cco 17084 df-rest 17230 df-topn 17231 df-0g 17249 df-gsum 17250 df-topgen 17251 df-pt 17252 df-prds 17255 df-xrs 17310 df-qtop 17315 df-imas 17316 df-xps 17318 df-mre 17392 df-mrc 17393 df-acs 17395 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-submnd 18528 df-mulg 18797 df-cntz 19019 df-cmn 19483 df-psmet 20695 df-xmet 20696 df-met 20697 df-bl 20698 df-mopn 20699 df-fbas 20700 df-fg 20701 df-cnfld 20704 df-top 22149 df-topon 22166 df-topsp 22188 df-bases 22202 df-cld 22276 df-ntr 22277 df-cls 22278 df-nei 22355 df-lp 22393 df-perf 22394 df-cn 22484 df-cnp 22485 df-haus 22572 df-tx 22819 df-hmeo 23012 df-fil 23103 df-fm 23195 df-flim 23196 df-flf 23197 df-xms 23579 df-ms 23580 df-tms 23581 df-cncf 24147 df-limc 25136 df-dv 25137 df-log 25818

This theorem is referenced by: bposlem5 26542 bposlem6 26543 chebbnd1lem1 26723

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