Theorem logdivbnd 27535

Description: A bound on a sum of logs, used in pntlemk 27585. This is not as precise as logdivsum 27512 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 12231	. . . 4 ⊢ 2 ∈ ℝ
2		fzfid 13908	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ∈ Fin)
3		elfzuz 13448	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ≥‘1))
4	3	adantl 481	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ (ℤ≥‘1))
5		nnuz 12802	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
6	4, 5	eleqtrrdi 2848	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
7	6	nnrpd 12959	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℝ+)
8	7	relogcld 26600	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (log‘𝑛) ∈ ℝ)
9	8, 6	nndivred 12211	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((log‘𝑛) / 𝑛) ∈ ℝ)
10	2, 9	fsumrecl 15669	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ∈ ℝ)
11		remulcl 11123	. . . 4 ⊢ ((2 ∈ ℝ ∧ Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ∈ ℝ) → (2 · Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) ∈ ℝ)
12	1, 10, 11	sylancr 588	. . 3 ⊢ (𝑁 ∈ ℕ → (2 · Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) ∈ ℝ)
13		elfznn 13481	. . . . . . 7 ⊢ (𝑖 ∈ (1...𝑁) → 𝑖 ∈ ℕ)
14	13	adantl 481	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → 𝑖 ∈ ℕ)
15	14	nnrecred 12208	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → (1 / 𝑖) ∈ ℝ)
16	2, 15	fsumrecl 15669	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...𝑁)(1 / 𝑖) ∈ ℝ)
17	16	resqcld 14060	. . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) ∈ ℝ)
18		nnrp 12929	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+)
19	18	relogcld 26600	. . . . 5 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ∈ ℝ)
20		peano2re 11318	. . . . 5 ⊢ ((log‘𝑁) ∈ ℝ → ((log‘𝑁) + 1) ∈ ℝ)
21	19, 20	syl 17	. . . 4 ⊢ (𝑁 ∈ ℕ → ((log‘𝑁) + 1) ∈ ℝ)
22	21	resqcld 14060	. . 3 ⊢ (𝑁 ∈ ℕ → (((log‘𝑁) + 1)↑2) ∈ ℝ)
23	10	recnd 11172	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ∈ ℂ)
24	23	2timesd 12396	. . . 4 ⊢ (𝑁 ∈ ℕ → (2 · Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) = (Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) + Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)))
25		fzfid 13908	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1...𝑛) ∈ Fin)
26		elfznn 13481	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...𝑛) → 𝑖 ∈ ℕ)
27	26	adantl 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → 𝑖 ∈ ℕ)
28	27	nnrecred 12208	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → (1 / 𝑖) ∈ ℝ)
29	25, 28	fsumrecl 15669	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) ∈ ℝ)
30	29, 6	nndivred 12211	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) ∈ ℝ)
31	2, 30	fsumrecl 15669	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) ∈ ℝ)
32		fzfid 13908	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1...(𝑛 − 1)) ∈ Fin)
33		elfznn 13481	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑛 − 1)) → 𝑖 ∈ ℕ)
34	33	adantl 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...(𝑛 − 1))) → 𝑖 ∈ ℕ)
35	34	nnrecred 12208	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...(𝑛 − 1))) → (1 / 𝑖) ∈ ℝ)
36	32, 35	fsumrecl 15669	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) ∈ ℝ)
37	36, 6	nndivred 12211	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛) ∈ ℝ)
38	2, 37	fsumrecl 15669	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛) ∈ ℝ)
39	6	nncnd 12173	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℂ)
40		ax-1cn 11096	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
41		npcan 11401	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
42	39, 40, 41	sylancl 587	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((𝑛 − 1) + 1) = 𝑛)
43	42	fveq2d 6846	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (log‘((𝑛 − 1) + 1)) = (log‘𝑛))
44	43	oveq2d 7384	. . . . . . . . . . . 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 12454	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
46		harmonicbnd3 26986	. . . . . . . . . . . . 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 2838	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘𝑛)) ∈ (0[,]γ))
49		0re 11146	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
50		emre 26984	. . . . . . . . . . . . 13 ⊢ γ ∈ ℝ
51	49, 50	elicc2i 13340	. . . . . . . . . . . 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 1147	. . . . . . . . . . 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 11739	. . . . . . . . . 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 232	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (log‘𝑛) ≤ Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖))
56	8, 36, 7, 55	lediv1dd 13019	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((log‘𝑛) / 𝑛) ≤ (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛))
57	27	nnrpd 12959	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → 𝑖 ∈ ℝ+)
58	57	rpreccld 12971	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → (1 / 𝑖) ∈ ℝ+)
59	58	rpge0d 12965	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → 0 ≤ (1 / 𝑖))
60		elfzelz 13452	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
61	60	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℤ)
62		peano2zm 12546	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℤ → (𝑛 − 1) ∈ ℤ)
63	61, 62	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ ℤ)
64	6	nnred 12172	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℝ)
65	64	lem1d 12087	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑛)
66		eluz2 12769	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘(𝑛 − 1)) ↔ ((𝑛 − 1) ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ (𝑛 − 1) ≤ 𝑛))
67	63, 61, 65, 66	syl3anbrc 1345	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ (ℤ≥‘(𝑛 − 1)))
68		fzss2 13492	. . . . . . . . . . 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 15731	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) ≤ Σ𝑖 ∈ (1...𝑛)(1 / 𝑖))
71	6	nngt0d 12206	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 0 < 𝑛)
72		lediv1 12019	. . . . . . . . . 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 1377	. . . . . . . . 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 232	. . . . . . . 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 11302	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((log‘𝑛) / 𝑛) ≤ (Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛))
76	2, 9, 30, 75	fsumle 15734	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ≤ Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛))
77	2, 9, 37, 56	fsumle 15734	. . . . . 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 11768	. . . . 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 7375	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
80	79	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (1...(𝑚 − 1)) = (1...(𝑛 − 1)))
81	80	sumeq1d 15635	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖))
82	81, 81	jca 511	. . . . . . . 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 7375	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 − 1) = ((𝑛 + 1) − 1))
84	83	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (1...(𝑚 − 1)) = (1...((𝑛 + 1) − 1)))
85	84	sumeq1d 15635	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖))
86	85, 85	jca 511	. . . . . . . 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 7375	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
88		1m1e0 12229	. . . . . . . . . . . . . 14 ⊢ (1 − 1) = 0
89	87, 88	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ (𝑚 = 1 → (𝑚 − 1) = 0)
90	89	oveq2d 7384	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (1...(𝑚 − 1)) = (1...0))
91		fz10 13473	. . . . . . . . . . . 12 ⊢ (1...0) = ∅
92	90, 91	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (1...(𝑚 − 1)) = ∅)
93	92	sumeq1d 15635	. . . . . . . . . 10 ⊢ (𝑚 = 1 → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ ∅ (1 / 𝑖))
94		sum0 15656	. . . . . . . . . 10 ⊢ Σ𝑖 ∈ ∅ (1 / 𝑖) = 0
95	93, 94	eqtrdi 2788	. . . . . . . . 9 ⊢ (𝑚 = 1 → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = 0)
96	95, 95	jca 511	. . . . . . . 8 ⊢ (𝑚 = 1 → (Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = 0 ∧ Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = 0))
97		oveq1 7375	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑁 + 1) → (𝑚 − 1) = ((𝑁 + 1) − 1))
98	97	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝑚 = (𝑁 + 1) → (1...(𝑚 − 1)) = (1...((𝑁 + 1) − 1)))
99	98	sumeq1d 15635	. . . . . . . . 9 ⊢ (𝑚 = (𝑁 + 1) → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖))
100	99, 99	jca 511	. . . . . . . 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 12169	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
102	101, 5	eleqtrdi 2847	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ (ℤ≥‘1))
103		fzfid 13908	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → (1...(𝑚 − 1)) ∈ Fin)
104		elfznn 13481	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (1...(𝑚 − 1)) → 𝑖 ∈ ℕ)
105	104	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) ∧ 𝑖 ∈ (1...(𝑚 − 1))) → 𝑖 ∈ ℕ)
106	105	nnrecred 12208	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) ∧ 𝑖 ∈ (1...(𝑚 − 1))) → (1 / 𝑖) ∈ ℝ)
107	103, 106	fsumrecl 15669	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) ∈ ℝ)
108	107	recnd 11172	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) ∈ ℂ)
109	82, 86, 96, 100, 102, 108, 108	fsumparts 15741	. . . . . . 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 12521	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
111		fzval3 13662	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (1...𝑁) = (1..^(𝑁 + 1)))
112	110, 111	syl 17	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (1...𝑁) = (1..^(𝑁 + 1)))
113	112	eqcomd 2743	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (1..^(𝑁 + 1)) = (1...𝑁))
114	36	recnd 11172	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) ∈ ℂ)
115	6	nnrecred 12208	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1 / 𝑛) ∈ ℝ)
116	115	recnd 11172	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1 / 𝑛) ∈ ℂ)
117		pncan 11398	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
118	39, 40, 117	sylancl 587	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((𝑛 + 1) − 1) = 𝑛)
119	118	oveq2d 7384	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1...((𝑛 + 1) − 1)) = (1...𝑛))
120	119	sumeq1d 15635	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...𝑛)(1 / 𝑖))
121	28	recnd 11172	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → (1 / 𝑖) ∈ ℂ)
122		oveq2 7376	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑛 → (1 / 𝑖) = (1 / 𝑛))
123	4, 121, 122	fsumm1 15686	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) = (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) + (1 / 𝑛)))
124	120, 123	eqtrd 2772	. . . . . . . . . . 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 11562	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) − Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖)) = (1 / 𝑛))
126	125	oveq2d 7384	. . . . . . . . 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 12207	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ≠ 0)
128	114, 39, 127	divrecd 11932	. . . . . . . . 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 2775	. . . . . . . 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 15642	. . . . . . 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 12165	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
132		pncan 11398	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
133	131, 40, 132	sylancl 587	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) − 1) = 𝑁)
134	133	oveq2d 7384	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (1...((𝑁 + 1) − 1)) = (1...𝑁))
135	134	sumeq1d 15635	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...𝑁)(1 / 𝑖))
136	135, 135	oveq12d 7386	. . . . . . . . . . 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 11172	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...𝑁)(1 / 𝑖) ∈ ℂ)
138	137	sqvald 14078	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) = (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖) · Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)))
139	136, 138	eqtr4d 2775	. . . . . . . . . 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 11136	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
141	140	mul01i 11335	. . . . . . . . . . 11 ⊢ (0 · 0) = 0
142	141	a1i 11	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (0 · 0) = 0)
143	139, 142	oveq12d 7386	. . . . . . . . 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 14079	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) ∈ ℂ)
145	144	subid1d 11493	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) − 0) = (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2))
146	143, 145	eqtrd 2772	. . . . . . . 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 7386	. . . . . . . . . 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 11172	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) ∈ ℂ)
149	148, 39, 127	divrec2d 11933	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) = ((1 / 𝑛) · Σ𝑖 ∈ (1...𝑛)(1 / 𝑖)))
150	147, 149	eqtr4d 2775	. . . . . . . . 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 15642	. . . . . . . 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 7386	. . . . . . 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 2781	. . . . . 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 11172	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) ∈ ℂ)
155	38	recnd 11172	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛) ∈ ℂ)
156	144, 154, 155	subaddd 11522	. . . . . 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 232	. . . . 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 5126	. . . 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 5122	. . 3 ⊢ (𝑁 ∈ ℕ → (2 · Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) ≤ (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2))
160		flid 13740	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (⌊‘𝑁) = 𝑁)
161	110, 160	syl 17	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (⌊‘𝑁) = 𝑁)
162	161	oveq2d 7384	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1...(⌊‘𝑁)) = (1...𝑁))
163	162	sumeq1d 15635	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...(⌊‘𝑁))(1 / 𝑖) = Σ𝑖 ∈ (1...𝑁)(1 / 𝑖))
164		nnre 12164	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
165		nnge1 12185	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
166		harmonicubnd 26988	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ 1 ≤ 𝑁) → Σ𝑖 ∈ (1...(⌊‘𝑁))(1 / 𝑖) ≤ ((log‘𝑁) + 1))
167	164, 165, 166	syl2anc 585	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...(⌊‘𝑁))(1 / 𝑖) ≤ ((log‘𝑁) + 1))
168	163, 167	eqbrtrrd 5124	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...𝑁)(1 / 𝑖) ≤ ((log‘𝑁) + 1))
169	14	nnrpd 12959	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → 𝑖 ∈ ℝ+)
170	169	rpreccld 12971	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → (1 / 𝑖) ∈ ℝ+)
171	170	rpge0d 12965	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → 0 ≤ (1 / 𝑖))
172	2, 15, 171	fsumge0 15730	. . . . 5 ⊢ (𝑁 ∈ ℕ → 0 ≤ Σ𝑖 ∈ (1...𝑁)(1 / 𝑖))
173	49	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℝ)
174		log1 26562	. . . . . . 7 ⊢ (log‘1) = 0
175		1rp 12921	. . . . . . . . 9 ⊢ 1 ∈ ℝ+
176		logleb 26580	. . . . . . . . 9 ⊢ ((1 ∈ ℝ+ ∧ 𝑁 ∈ ℝ+) → (1 ≤ 𝑁 ↔ (log‘1) ≤ (log‘𝑁)))
177	175, 18, 176	sylancr 588	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (1 ≤ 𝑁 ↔ (log‘1) ≤ (log‘𝑁)))
178	165, 177	mpbid 232	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘1) ≤ (log‘𝑁))
179	174, 178	eqbrtrrid 5136	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 ≤ (log‘𝑁))
180	19	lep1d 12085	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ≤ ((log‘𝑁) + 1))
181	173, 19, 21, 179, 180	letrd 11302	. . . . 5 ⊢ (𝑁 ∈ ℕ → 0 ≤ ((log‘𝑁) + 1))
182	16, 21, 172, 181	le2sqd 14192	. . . 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 232	. . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) ≤ (((log‘𝑁) + 1)↑2))
184	12, 17, 22, 159, 183	letrd 11302	. 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 12260	. . . 4 ⊢ 0 < 2
187	186	a1i 11	. . 3 ⊢ (𝑁 ∈ ℕ → 0 < 2)
188		lemuldiv2 12035	. . 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 1377	. 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 232	1 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ≤ ((((log‘𝑁) + 1)↑2) / 2))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903  ∅c0 4287   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  ℂcc 11036  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   · cmul 11043   < clt 11178   ≤ cle 11179   − cmin 11376   / cdiv 11806  ℕcn 12157  2c2 12212  ℕ₀cn0 12413  ℤcz 12500  ℤ_≥cuz 12763  ℝ⁺crp 12917  [,]cicc 13276  ...cfz 13435  ..^cfzo 13582  ⌊cfl 13722  ↑cexp 13996  Σcsu 15621  logclog 26531  γcem 26970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116  ax-addf 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-of 7632  df-om 7819  df-1st 7943  df-2nd 7944  df-supp 8113  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-er 8645  df-map 8777  df-pm 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9277  df-fi 9326  df-sup 9357  df-inf 9358  df-oi 9427  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-xnn0 12487  df-z 12501  df-dec 12620  df-uz 12764  df-q 12874  df-rp 12918  df-xneg 13038  df-xadd 13039  df-xmul 13040  df-ioo 13277  df-ioc 13278  df-ico 13279  df-icc 13280  df-fz 13436  df-fzo 13583  df-fl 13724  df-mod 13802  df-seq 13937  df-exp 13997  df-fac 14209  df-bc 14238  df-hash 14266  df-shft 15002  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171  df-limsup 15406  df-clim 15423  df-rlim 15424  df-sum 15622  df-ef 16002  df-e 16003  df-sin 16004  df-cos 16005  df-tan 16006  df-pi 16007  df-dvds 16192  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-mulr 17203  df-starv 17204  df-sca 17205  df-vsca 17206  df-ip 17207  df-tset 17208  df-ple 17209  df-ds 17211  df-unif 17212  df-hom 17213  df-cco 17214  df-rest 17354  df-topn 17355  df-0g 17373  df-gsum 17374  df-topgen 17375  df-pt 17376  df-prds 17379  df-xrs 17435  df-qtop 17440  df-imas 17441  df-xps 17443  df-mre 17517  df-mrc 17518  df-acs 17520  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-submnd 18721  df-mulg 19010  df-cntz 19258  df-cmn 19723  df-psmet 21313  df-xmet 21314  df-met 21315  df-bl 21316  df-mopn 21317  df-fbas 21318  df-fg 21319  df-cnfld 21322  df-top 22850  df-topon 22867  df-topsp 22889  df-bases 22902  df-cld 22975  df-ntr 22976  df-cls 22977  df-nei 23054  df-lp 23092  df-perf 23093  df-cn 23183  df-cnp 23184  df-haus 23271  df-cmp 23343  df-tx 23518  df-hmeo 23711  df-fil 23802  df-fm 23894  df-flim 23895  df-flf 23896  df-xms 24276  df-ms 24277  df-tms 24278  df-cncf 24839  df-limc 25835  df-dv 25836  df-ulm 26354  df-log 26533  df-atan 26845  df-em 26971

This theorem is referenced by:  pntlemk  27585