Theorem logdivbnd 25697

Description: A bound on a sum of logs, used in pntlemk 25747. This is not as precise as logdivsum 25674 in its asymptotic behavior, but it is valid for all 𝑁 and does not require a limit value. (Contributed by Mario Carneiro, 13-Apr-2016.)

Assertion

Ref	Expression
logdivbnd	⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ≤ ((((log‘𝑁) + 1)↑2) / 2))

Distinct variable group:   𝑛,𝑁

Proof of Theorem logdivbnd

Dummy variables 𝑖 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2re 11449	. . . 4 ⊢ 2 ∈ ℝ
2		fzfid 13091	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ∈ Fin)
3		elfzuz 12655	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ≥‘1))
4	3	adantl 475	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ (ℤ≥‘1))
5		nnuz 12029	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
6	4, 5	syl6eleqr 2870	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
7	6	nnrpd 12179	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℝ+)
8	7	relogcld 24806	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (log‘𝑛) ∈ ℝ)
9	8, 6	nndivred 11429	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((log‘𝑛) / 𝑛) ∈ ℝ)
10	2, 9	fsumrecl 14872	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ∈ ℝ)
11		remulcl 10357	. . . 4 ⊢ ((2 ∈ ℝ ∧ Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ∈ ℝ) → (2 · Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) ∈ ℝ)
12	1, 10, 11	sylancr 581	. . 3 ⊢ (𝑁 ∈ ℕ → (2 · Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) ∈ ℝ)
13		elfznn 12687	. . . . . . 7 ⊢ (𝑖 ∈ (1...𝑁) → 𝑖 ∈ ℕ)
14	13	adantl 475	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → 𝑖 ∈ ℕ)
15	14	nnrecred 11426	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → (1 / 𝑖) ∈ ℝ)
16	2, 15	fsumrecl 14872	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...𝑁)(1 / 𝑖) ∈ ℝ)
17	16	resqcld 13356	. . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) ∈ ℝ)
18		nnrp 12150	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+)
19	18	relogcld 24806	. . . . 5 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ∈ ℝ)
20		peano2re 10549	. . . . 5 ⊢ ((log‘𝑁) ∈ ℝ → ((log‘𝑁) + 1) ∈ ℝ)
21	19, 20	syl 17	. . . 4 ⊢ (𝑁 ∈ ℕ → ((log‘𝑁) + 1) ∈ ℝ)
22	21	resqcld 13356	. . 3 ⊢ (𝑁 ∈ ℕ → (((log‘𝑁) + 1)↑2) ∈ ℝ)
23	10	recnd 10405	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ∈ ℂ)
24	23	2timesd 11625	. . . 4 ⊢ (𝑁 ∈ ℕ → (2 · Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) = (Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) + Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)))
25		fzfid 13091	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1...𝑛) ∈ Fin)
26		elfznn 12687	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...𝑛) → 𝑖 ∈ ℕ)
27	26	adantl 475	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → 𝑖 ∈ ℕ)
28	27	nnrecred 11426	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → (1 / 𝑖) ∈ ℝ)
29	25, 28	fsumrecl 14872	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) ∈ ℝ)
30	29, 6	nndivred 11429	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) ∈ ℝ)
31	2, 30	fsumrecl 14872	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) ∈ ℝ)
32		fzfid 13091	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1...(𝑛 − 1)) ∈ Fin)
33		elfznn 12687	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑛 − 1)) → 𝑖 ∈ ℕ)
34	33	adantl 475	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...(𝑛 − 1))) → 𝑖 ∈ ℕ)
35	34	nnrecred 11426	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...(𝑛 − 1))) → (1 / 𝑖) ∈ ℝ)
36	32, 35	fsumrecl 14872	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) ∈ ℝ)
37	36, 6	nndivred 11429	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛) ∈ ℝ)
38	2, 37	fsumrecl 14872	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛) ∈ ℝ)
39	6	nncnd 11392	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℂ)
40		ax-1cn 10330	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
41		npcan 10632	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
42	39, 40, 41	sylancl 580	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((𝑛 − 1) + 1) = 𝑛)
43	42	fveq2d 6450	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (log‘((𝑛 − 1) + 1)) = (log‘𝑛))
44	43	oveq2d 6938	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘((𝑛 − 1) + 1))) = (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘𝑛)))
45		nnm1nn0 11685	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
46		harmonicbnd3 25186	. . . . . . . . . . . . 13 ⊢ ((𝑛 − 1) ∈ ℕ0 → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘((𝑛 − 1) + 1))) ∈ (0[,]γ))
47	6, 45, 46	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘((𝑛 − 1) + 1))) ∈ (0[,]γ))
48	44, 47	eqeltrrd 2860	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘𝑛)) ∈ (0[,]γ))
49		0re 10378	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
50		emre 25184	. . . . . . . . . . . . 13 ⊢ γ ∈ ℝ
51	49, 50	elicc2i 12551	. . . . . . . . . . . 12 ⊢ ((Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘𝑛)) ∈ (0[,]γ) ↔ ((Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘𝑛)) ∈ ℝ ∧ 0 ≤ (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘𝑛)) ∧ (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘𝑛)) ≤ γ))
52	51	simp2bi 1137	. . . . . . . . . . 11 ⊢ ((Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘𝑛)) ∈ (0[,]γ) → 0 ≤ (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘𝑛)))
53	48, 52	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 0 ≤ (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘𝑛)))
54	36, 8	subge0d 10965	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (0 ≤ (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘𝑛)) ↔ (log‘𝑛) ≤ Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖)))
55	53, 54	mpbid 224	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (log‘𝑛) ≤ Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖))
56	8, 36, 7, 55	lediv1dd 12239	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((log‘𝑛) / 𝑛) ≤ (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛))
57	27	nnrpd 12179	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → 𝑖 ∈ ℝ+)
58	57	rpreccld 12191	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → (1 / 𝑖) ∈ ℝ+)
59	58	rpge0d 12185	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → 0 ≤ (1 / 𝑖))
60		elfzelz 12659	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
61	60	adantl 475	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℤ)
62		peano2zm 11772	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℤ → (𝑛 − 1) ∈ ℤ)
63	61, 62	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ ℤ)
64	6	nnred 11391	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℝ)
65	64	lem1d 11311	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑛)
66		eluz2 11998	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘(𝑛 − 1)) ↔ ((𝑛 − 1) ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ (𝑛 − 1) ≤ 𝑛))
67	63, 61, 65, 66	syl3anbrc 1400	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ (ℤ≥‘(𝑛 − 1)))
68		fzss2 12698	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘(𝑛 − 1)) → (1...(𝑛 − 1)) ⊆ (1...𝑛))
69	67, 68	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1...(𝑛 − 1)) ⊆ (1...𝑛))
70	25, 28, 59, 69	fsumless 14932	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) ≤ Σ𝑖 ∈ (1...𝑛)(1 / 𝑖))
71	6	nngt0d 11424	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 0 < 𝑛)
72		lediv1 11242	. . . . . . . . . 10 ⊢ ((Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) ∈ ℝ ∧ Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) ≤ Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) ↔ (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛) ≤ (Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛)))
73	36, 29, 64, 71, 72	syl112anc 1442	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) ≤ Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) ↔ (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛) ≤ (Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛)))
74	70, 73	mpbid 224	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛) ≤ (Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛))
75	9, 37, 30, 56, 74	letrd 10533	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((log‘𝑛) / 𝑛) ≤ (Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛))
76	2, 9, 30, 75	fsumle 14935	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ≤ Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛))
77	2, 9, 37, 56	fsumle 14935	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ≤ Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛))
78	10, 10, 31, 38, 76, 77	le2addd 10994	. . . . 5 ⊢ (𝑁 ∈ ℕ → (Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) + Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) ≤ (Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) + Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛)))
79		oveq1 6929	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
80	79	oveq2d 6938	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (1...(𝑚 − 1)) = (1...(𝑛 − 1)))
81	80	sumeq1d 14839	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖))
82	81, 81	jca 507	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) ∧ Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖)))
83		oveq1 6929	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 − 1) = ((𝑛 + 1) − 1))
84	83	oveq2d 6938	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (1...(𝑚 − 1)) = (1...((𝑛 + 1) − 1)))
85	84	sumeq1d 14839	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖))
86	85, 85	jca 507	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → (Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) ∧ Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖)))
87		oveq1 6929	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
88		1m1e0 11447	. . . . . . . . . . . . . 14 ⊢ (1 − 1) = 0
89	87, 88	syl6eq 2830	. . . . . . . . . . . . 13 ⊢ (𝑚 = 1 → (𝑚 − 1) = 0)
90	89	oveq2d 6938	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (1...(𝑚 − 1)) = (1...0))
91		fz10 12679	. . . . . . . . . . . 12 ⊢ (1...0) = ∅
92	90, 91	syl6eq 2830	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (1...(𝑚 − 1)) = ∅)
93	92	sumeq1d 14839	. . . . . . . . . 10 ⊢ (𝑚 = 1 → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ ∅ (1 / 𝑖))
94		sum0 14859	. . . . . . . . . 10 ⊢ Σ𝑖 ∈ ∅ (1 / 𝑖) = 0
95	93, 94	syl6eq 2830	. . . . . . . . 9 ⊢ (𝑚 = 1 → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = 0)
96	95, 95	jca 507	. . . . . . . 8 ⊢ (𝑚 = 1 → (Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = 0 ∧ Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = 0))
97		oveq1 6929	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑁 + 1) → (𝑚 − 1) = ((𝑁 + 1) − 1))
98	97	oveq2d 6938	. . . . . . . . . 10 ⊢ (𝑚 = (𝑁 + 1) → (1...(𝑚 − 1)) = (1...((𝑁 + 1) − 1)))
99	98	sumeq1d 14839	. . . . . . . . 9 ⊢ (𝑚 = (𝑁 + 1) → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖))
100	99, 99	jca 507	. . . . . . . 8 ⊢ (𝑚 = (𝑁 + 1) → (Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖) ∧ Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖)))
101		peano2nn 11388	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
102	101, 5	syl6eleq 2869	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ (ℤ≥‘1))
103		fzfid 13091	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → (1...(𝑚 − 1)) ∈ Fin)
104		elfznn 12687	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (1...(𝑚 − 1)) → 𝑖 ∈ ℕ)
105	104	adantl 475	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) ∧ 𝑖 ∈ (1...(𝑚 − 1))) → 𝑖 ∈ ℕ)
106	105	nnrecred 11426	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) ∧ 𝑖 ∈ (1...(𝑚 − 1))) → (1 / 𝑖) ∈ ℝ)
107	103, 106	fsumrecl 14872	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) ∈ ℝ)
108	107	recnd 10405	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) ∈ ℂ)
109	82, 86, 96, 100, 102, 108, 108	fsumparts 14942	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1..^(𝑁 + 1))(Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) · (Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) − Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖))) = (((Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖) · Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖)) − (0 · 0)) − Σ𝑛 ∈ (1..^(𝑁 + 1))((Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) − Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖)) · Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖))))
110		nnz 11751	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
111		fzval3 12856	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (1...𝑁) = (1..^(𝑁 + 1)))
112	110, 111	syl 17	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (1...𝑁) = (1..^(𝑁 + 1)))
113	112	eqcomd 2784	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (1..^(𝑁 + 1)) = (1...𝑁))
114	36	recnd 10405	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) ∈ ℂ)
115	6	nnrecred 11426	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1 / 𝑛) ∈ ℝ)
116	115	recnd 10405	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1 / 𝑛) ∈ ℂ)
117		pncan 10628	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
118	39, 40, 117	sylancl 580	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((𝑛 + 1) − 1) = 𝑛)
119	118	oveq2d 6938	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1...((𝑛 + 1) − 1)) = (1...𝑛))
120	119	sumeq1d 14839	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...𝑛)(1 / 𝑖))
121	28	recnd 10405	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → (1 / 𝑖) ∈ ℂ)
122		oveq2 6930	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑛 → (1 / 𝑖) = (1 / 𝑛))
123	4, 121, 122	fsumm1 14887	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) = (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) + (1 / 𝑛)))
124	120, 123	eqtrd 2814	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) = (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) + (1 / 𝑛)))
125	114, 116, 124	mvrladdd 10788	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) − Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖)) = (1 / 𝑛))
126	125	oveq2d 6938	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) · (Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) − Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖))) = (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) · (1 / 𝑛)))
127	6	nnne0d 11425	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ≠ 0)
128	114, 39, 127	divrecd 11154	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛) = (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) · (1 / 𝑛)))
129	126, 128	eqtr4d 2817	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) · (Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) − Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖))) = (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛))
130	113, 129	sumeq12rdv 14845	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1..^(𝑁 + 1))(Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) · (Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) − Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖))) = Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛))
131		nncn 11383	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
132		pncan 10628	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
133	131, 40, 132	sylancl 580	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) − 1) = 𝑁)
134	133	oveq2d 6938	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (1...((𝑁 + 1) − 1)) = (1...𝑁))
135	134	sumeq1d 14839	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...𝑁)(1 / 𝑖))
136	135, 135	oveq12d 6940	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖) · Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖)) = (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖) · Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)))
137	16	recnd 10405	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...𝑁)(1 / 𝑖) ∈ ℂ)
138	137	sqvald 13324	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) = (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖) · Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)))
139	136, 138	eqtr4d 2817	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖) · Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖)) = (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2))
140		0cn 10368	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
141	140	mul01i 10566	. . . . . . . . . . 11 ⊢ (0 · 0) = 0
142	141	a1i 11	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (0 · 0) = 0)
143	139, 142	oveq12d 6940	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖) · Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖)) − (0 · 0)) = ((Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) − 0))
144	137	sqcld 13325	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) ∈ ℂ)
145	144	subid1d 10723	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) − 0) = (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2))
146	143, 145	eqtrd 2814	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖) · Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖)) − (0 · 0)) = (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2))
147	125, 120	oveq12d 6940	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) − Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖)) · Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖)) = ((1 / 𝑛) · Σ𝑖 ∈ (1...𝑛)(1 / 𝑖)))
148	29	recnd 10405	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) ∈ ℂ)
149	148, 39, 127	divrec2d 11155	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) = ((1 / 𝑛) · Σ𝑖 ∈ (1...𝑛)(1 / 𝑖)))
150	147, 149	eqtr4d 2817	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) − Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖)) · Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖)) = (Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛))
151	113, 150	sumeq12rdv 14845	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1..^(𝑁 + 1))((Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) − Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖)) · Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖)) = Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛))
152	146, 151	oveq12d 6940	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (((Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖) · Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖)) − (0 · 0)) − Σ𝑛 ∈ (1..^(𝑁 + 1))((Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) − Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖)) · Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖))) = ((Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) − Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛)))
153	109, 130, 152	3eqtr3rd 2823	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) − Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛)) = Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛))
154	31	recnd 10405	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) ∈ ℂ)
155	38	recnd 10405	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛) ∈ ℂ)
156	144, 154, 155	subaddd 10752	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (((Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) − Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛)) = Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛) ↔ (Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) + Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛)) = (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2)))
157	153, 156	mpbid 224	. . . . 5 ⊢ (𝑁 ∈ ℕ → (Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) + Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛)) = (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2))
158	78, 157	breqtrd 4912	. . . 4 ⊢ (𝑁 ∈ ℕ → (Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) + Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) ≤ (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2))
159	24, 158	eqbrtrd 4908	. . 3 ⊢ (𝑁 ∈ ℕ → (2 · Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) ≤ (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2))
160		flid 12928	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (⌊‘𝑁) = 𝑁)
161	110, 160	syl 17	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (⌊‘𝑁) = 𝑁)
162	161	oveq2d 6938	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1...(⌊‘𝑁)) = (1...𝑁))
163	162	sumeq1d 14839	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...(⌊‘𝑁))(1 / 𝑖) = Σ𝑖 ∈ (1...𝑁)(1 / 𝑖))
164		nnre 11382	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
165		nnge1 11404	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
166		harmonicubnd 25188	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ 1 ≤ 𝑁) → Σ𝑖 ∈ (1...(⌊‘𝑁))(1 / 𝑖) ≤ ((log‘𝑁) + 1))
167	164, 165, 166	syl2anc 579	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...(⌊‘𝑁))(1 / 𝑖) ≤ ((log‘𝑁) + 1))
168	163, 167	eqbrtrrd 4910	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...𝑁)(1 / 𝑖) ≤ ((log‘𝑁) + 1))
169	14	nnrpd 12179	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → 𝑖 ∈ ℝ+)
170	169	rpreccld 12191	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → (1 / 𝑖) ∈ ℝ+)
171	170	rpge0d 12185	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → 0 ≤ (1 / 𝑖))
172	2, 15, 171	fsumge0 14931	. . . . 5 ⊢ (𝑁 ∈ ℕ → 0 ≤ Σ𝑖 ∈ (1...𝑁)(1 / 𝑖))
173	49	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℝ)
174		log1 24769	. . . . . . 7 ⊢ (log‘1) = 0
175		1rp 12141	. . . . . . . . 9 ⊢ 1 ∈ ℝ+
176		logleb 24786	. . . . . . . . 9 ⊢ ((1 ∈ ℝ+ ∧ 𝑁 ∈ ℝ+) → (1 ≤ 𝑁 ↔ (log‘1) ≤ (log‘𝑁)))
177	175, 18, 176	sylancr 581	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (1 ≤ 𝑁 ↔ (log‘1) ≤ (log‘𝑁)))
178	165, 177	mpbid 224	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘1) ≤ (log‘𝑁))
179	174, 178	syl5eqbrr 4922	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 ≤ (log‘𝑁))
180	19	lep1d 11309	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ≤ ((log‘𝑁) + 1))
181	173, 19, 21, 179, 180	letrd 10533	. . . . 5 ⊢ (𝑁 ∈ ℕ → 0 ≤ ((log‘𝑁) + 1))
182	16, 21, 172, 181	le2sqd 13365	. . . 4 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖) ≤ ((log‘𝑁) + 1) ↔ (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) ≤ (((log‘𝑁) + 1)↑2)))
183	168, 182	mpbid 224	. . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) ≤ (((log‘𝑁) + 1)↑2))
184	12, 17, 22, 159, 183	letrd 10533	. 2 ⊢ (𝑁 ∈ ℕ → (2 · Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) ≤ (((log‘𝑁) + 1)↑2))
185	1	a1i 11	. . 3 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ)
186		2pos 11485	. . . 4 ⊢ 0 < 2
187	186	a1i 11	. . 3 ⊢ (𝑁 ∈ ℕ → 0 < 2)
188		lemuldiv2 11258	. . 3 ⊢ ((Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ∈ ℝ ∧ (((log‘𝑁) + 1)↑2) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((2 · Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) ≤ (((log‘𝑁) + 1)↑2) ↔ Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ≤ ((((log‘𝑁) + 1)↑2) / 2)))
189	10, 22, 185, 187, 188	syl112anc 1442	. 2 ⊢ (𝑁 ∈ ℕ → ((2 · Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) ≤ (((log‘𝑁) + 1)↑2) ↔ Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ≤ ((((log‘𝑁) + 1)↑2) / 2)))
190	184, 189	mpbid 224	1 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ≤ ((((log‘𝑁) + 1)↑2) / 2))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ⊆ wss 3792  ∅c0 4141   class class class wbr 4886  ‘cfv 6135  (class class class)co 6922  ℂcc 10270  ℝcr 10271  0cc0 10272  1c1 10273   + caddc 10275   · cmul 10277   < clt 10411   ≤ cle 10412   − cmin 10606   / cdiv 11032  ℕcn 11374  2c2 11430  ℕ₀cn0 11642  ℤcz 11728  ℤ_≥cuz 11992  ℝ⁺crp 12137  [,]cicc 12490  ...cfz 12643  ..^cfzo 12784  ⌊cfl 12910  ↑cexp 13178  Σcsu 14824  logclog 24738  γcem 25170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350  ax-addf 10351  ax-mulf 10352

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-iin 4756  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-om 7344  df-1st 7445  df-2nd 7446  df-supp 7577  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-ixp 8195  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-fsupp 8564  df-fi 8605  df-sup 8636  df-inf 8637  df-oi 8704  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-dec 11846  df-uz 11993  df-q 12096  df-rp 12138  df-xneg 12257  df-xadd 12258  df-xmul 12259  df-ioo 12491  df-ioc 12492  df-ico 12493  df-icc 12494  df-fz 12644  df-fzo 12785  df-fl 12912  df-mod 12988  df-seq 13120  df-exp 13179  df-fac 13379  df-bc 13408  df-hash 13436  df-shft 14214  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-limsup 14610  df-clim 14627  df-rlim 14628  df-sum 14825  df-ef 15200  df-e 15201  df-sin 15202  df-cos 15203  df-pi 15205  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-plusg 16351  df-mulr 16352  df-starv 16353  df-sca 16354  df-vsca 16355  df-ip 16356  df-tset 16357  df-ple 16358  df-ds 16360  df-unif 16361  df-hom 16362  df-cco 16363  df-rest 16469  df-topn 16470  df-0g 16488  df-gsum 16489  df-topgen 16490  df-pt 16491  df-prds 16494  df-xrs 16548  df-qtop 16553  df-imas 16554  df-xps 16556  df-mre 16632  df-mrc 16633  df-acs 16635  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-submnd 17722  df-mulg 17928  df-cntz 18133  df-cmn 18581  df-psmet 20134  df-xmet 20135  df-met 20136  df-bl 20137  df-mopn 20138  df-fbas 20139  df-fg 20140  df-cnfld 20143  df-top 21106  df-topon 21123  df-topsp 21145  df-bases 21158  df-cld 21231  df-ntr 21232  df-cls 21233  df-nei 21310  df-lp 21348  df-perf 21349  df-cn 21439  df-cnp 21440  df-haus 21527  df-tx 21774  df-hmeo 21967  df-fil 22058  df-fm 22150  df-flim 22151  df-flf 22152  df-xms 22533  df-ms 22534  df-tms 22535  df-cncf 23089  df-limc 24067  df-dv 24068  df-log 24740  df-em 25171

This theorem is referenced by:  pntlemk  25747