Theorem logdivbnd 27916

Description: A bound on a sum of logs, used in pntlemk 27966. This is not as precise as logdivsum 27893 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 12447	. . . 4 ⊢ 2 ∈ ℝ
2		fzfid 14146	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ∈ Fin)
3		elfzuz 13682	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ≥‘1))
4	3	adantl 487	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ (ℤ≥‘1))
5		nnuz 13034	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
6	4, 5	eleqtrrdi 2875	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
7	6	nnrpd 13192	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℝ+)
8	7	relogcld 26984	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (log‘𝑛) ∈ ℝ)
9	8, 6	nndivred 12422	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((log‘𝑛) / 𝑛) ∈ ℝ)
10	2, 9	fsumrecl 15932	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ∈ ℝ)
11		remulcl 11315	. . . 4 ⊢ ((2 ∈ ℝ ∧ Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ∈ ℝ) → (2 · Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) ∈ ℝ)
12	1, 10, 11	sylancr 599	. . 3 ⊢ (𝑁 ∈ ℕ → (2 · Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) ∈ ℝ)
13		elfznn 13717	. . . . . . 7 ⊢ (𝑖 ∈ (1...𝑁) → 𝑖 ∈ ℕ)
14	13	adantl 487	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → 𝑖 ∈ ℕ)
15	14	nnrecred 12419	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → (1 / 𝑖) ∈ ℝ)
16	2, 15	fsumrecl 15932	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...𝑁)(1 / 𝑖) ∈ ℝ)
17	16	resqcld 14298	. . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) ∈ ℝ)
18		nnrp 13162	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+)
19	18	relogcld 26984	. . . . 5 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ∈ ℝ)
20		peano2re 11513	. . . . 5 ⊢ ((log‘𝑁) ∈ ℝ → ((log‘𝑁) + 1) ∈ ℝ)
21	19, 20	syl 18	. . . 4 ⊢ (𝑁 ∈ ℕ → ((log‘𝑁) + 1) ∈ ℝ)
22	21	resqcld 14298	. . 3 ⊢ (𝑁 ∈ ℕ → (((log‘𝑁) + 1)↑2) ∈ ℝ)
23	10	recnd 11367	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ∈ ℂ)
24	23	2timesd 12619	. . . 4 ⊢ (𝑁 ∈ ℕ → (2 · Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) = (Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) + Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)))
25		fzfid 14146	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1...𝑛) ∈ Fin)
26		elfznn 13717	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...𝑛) → 𝑖 ∈ ℕ)
27	26	adantl 487	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → 𝑖 ∈ ℕ)
28	27	nnrecred 12419	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → (1 / 𝑖) ∈ ℝ)
29	25, 28	fsumrecl 15932	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) ∈ ℝ)
30	29, 6	nndivred 12422	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) ∈ ℝ)
31	2, 30	fsumrecl 15932	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) ∈ ℝ)
32		fzfid 14146	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1...(𝑛 − 1)) ∈ Fin)
33		elfznn 13717	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑛 − 1)) → 𝑖 ∈ ℕ)
34	33	adantl 487	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...(𝑛 − 1))) → 𝑖 ∈ ℕ)
35	34	nnrecred 12419	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...(𝑛 − 1))) → (1 / 𝑖) ∈ ℝ)
36	32, 35	fsumrecl 15932	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) ∈ ℝ)
37	36, 6	nndivred 12422	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛) ∈ ℝ)
38	2, 37	fsumrecl 15932	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛) ∈ ℝ)
39	6	nncnd 12381	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℂ)
40		ax-1cn 11288	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
41		npcan 11596	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
42	39, 40, 41	sylancl 598	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((𝑛 − 1) + 1) = 𝑛)
43	42	fveq2d 6915	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (log‘((𝑛 − 1) + 1)) = (log‘𝑛))
44	43	oveq2d 7462	. . . . . . . . . . . 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 12677	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
46		harmonicbnd3 27368	. . . . . . . . . . . . 13 ⊢ ((𝑛 − 1) ∈ ℕ0 → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘((𝑛 − 1) + 1))) ∈ (0[,]γ))
47	6, 45, 46	3syl 19	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘((𝑛 − 1) + 1))) ∈ (0[,]γ))
48	44, 47	eqeltrrd 2865	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘𝑛)) ∈ (0[,]γ))
49		0re 11340	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
50		emre 27366	. . . . . . . . . . . . 13 ⊢ γ ∈ ℝ
51	49, 50	elicc2i 13573	. . . . . . . . . . . 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 1164	. . . . . . . . . . 11 ⊢ ((Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘𝑛)) ∈ (0[,]γ) → 0 ≤ (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘𝑛)))
53	48, 52	syl 18	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 0 ≤ (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘𝑛)))
54	36, 8	subge0d 11936	. . . . . . . . . 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 235	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (log‘𝑛) ≤ Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖))
56	8, 36, 7, 55	lediv1dd 13252	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((log‘𝑛) / 𝑛) ≤ (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛))
57	27	nnrpd 13192	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → 𝑖 ∈ ℝ+)
58	57	rpreccld 13204	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → (1 / 𝑖) ∈ ℝ+)
59	58	rpge0d 13198	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → 0 ≤ (1 / 𝑖))
60		elfzelz 13686	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
61	60	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℤ)
62		peano2zm 12769	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℤ → (𝑛 − 1) ∈ ℤ)
63	61, 62	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ ℤ)
64	6	nnred 12380	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℝ)
65	64	lem1d 12280	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑛)
66		eluz2 13001	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘(𝑛 − 1)) ↔ ((𝑛 − 1) ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ (𝑛 − 1) ≤ 𝑛))
67	63, 61, 65, 66	syl3anbrc 1362	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ (ℤ≥‘(𝑛 − 1)))
68		fzss2 13728	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘(𝑛 − 1)) → (1...(𝑛 − 1)) ⊆ (1...𝑛))
69	67, 68	syl 18	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1...(𝑛 − 1)) ⊆ (1...𝑛))
70	25, 28, 59, 69	fsumless 15995	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) ≤ Σ𝑖 ∈ (1...𝑛)(1 / 𝑖))
71	6	nngt0d 12417	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 0 < 𝑛)
72		lediv1 12212	. . . . . . . . . 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 1402	. . . . . . . . 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 235	. . . . . . . 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 11497	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((log‘𝑛) / 𝑛) ≤ (Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛))
76	2, 9, 30, 75	fsumle 15998	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ≤ Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛))
77	2, 9, 37, 56	fsumle 15998	. . . . . 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 11965	. . . . 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 7453	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
80	79	oveq2d 7462	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (1...(𝑚 − 1)) = (1...(𝑛 − 1)))
81	80	sumeq1d 15899	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖))
82	81, 81	jca 521	. . . . . . . 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 7453	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 − 1) = ((𝑛 + 1) − 1))
84	83	oveq2d 7462	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (1...(𝑚 − 1)) = (1...((𝑛 + 1) − 1)))
85	84	sumeq1d 15899	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖))
86	85, 85	jca 521	. . . . . . . 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 7453	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
88		1m1e0 12445	. . . . . . . . . . . . . 14 ⊢ (1 − 1) = 0
89	87, 88	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ (𝑚 = 1 → (𝑚 − 1) = 0)
90	89	oveq2d 7462	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (1...(𝑚 − 1)) = (1...0))
91		fz10 13708	. . . . . . . . . . . 12 ⊢ (1...0) = ∅
92	90, 91	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (1...(𝑚 − 1)) = ∅)
93	92	sumeq1d 15899	. . . . . . . . . 10 ⊢ (𝑚 = 1 → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ ∅ (1 / 𝑖))
94		sum0 15919	. . . . . . . . . 10 ⊢ Σ𝑖 ∈ ∅ (1 / 𝑖) = 0
95	93, 94	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑚 = 1 → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = 0)
96	95, 95	jca 521	. . . . . . . 8 ⊢ (𝑚 = 1 → (Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = 0 ∧ Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = 0))
97		oveq1 7453	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑁 + 1) → (𝑚 − 1) = ((𝑁 + 1) − 1))
98	97	oveq2d 7462	. . . . . . . . . 10 ⊢ (𝑚 = (𝑁 + 1) → (1...(𝑚 − 1)) = (1...((𝑁 + 1) − 1)))
99	98	sumeq1d 15899	. . . . . . . . 9 ⊢ (𝑚 = (𝑁 + 1) → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖))
100	99, 99	jca 521	. . . . . . . 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 12377	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
102	101, 5	eleqtrdi 2874	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ (ℤ≥‘1))
103		fzfid 14146	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → (1...(𝑚 − 1)) ∈ Fin)
104		elfznn 13717	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (1...(𝑚 − 1)) → 𝑖 ∈ ℕ)
105	104	adantl 487	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) ∧ 𝑖 ∈ (1...(𝑚 − 1))) → 𝑖 ∈ ℕ)
106	105	nnrecred 12419	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) ∧ 𝑖 ∈ (1...(𝑚 − 1))) → (1 / 𝑖) ∈ ℝ)
107	103, 106	fsumrecl 15932	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) ∈ ℝ)
108	107	recnd 11367	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) ∈ ℂ)
109	82, 86, 96, 100, 102, 108, 108	fsumparts 16005	. . . . . . 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 12744	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
111		fzval3 13899	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (1...𝑁) = (1..^(𝑁 + 1)))
112	110, 111	syl 18	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (1...𝑁) = (1..^(𝑁 + 1)))
113	112	eqcomd 2769	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (1..^(𝑁 + 1)) = (1...𝑁))
114	36	recnd 11367	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) ∈ ℂ)
115	6	nnrecred 12419	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1 / 𝑛) ∈ ℝ)
116	115	recnd 11367	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1 / 𝑛) ∈ ℂ)
117		pncan 11593	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
118	39, 40, 117	sylancl 598	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((𝑛 + 1) − 1) = 𝑛)
119	118	oveq2d 7462	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1...((𝑛 + 1) − 1)) = (1...𝑛))
120	119	sumeq1d 15899	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...𝑛)(1 / 𝑖))
121	28	recnd 11367	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → (1 / 𝑖) ∈ ℂ)
122		oveq2 7454	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑛 → (1 / 𝑖) = (1 / 𝑛))
123	4, 121, 122	fsumm1 15949	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) = (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) + (1 / 𝑛)))
124	120, 123	eqtrd 2798	. . . . . . . . . . 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 11758	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) − Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖)) = (1 / 𝑛))
126	125	oveq2d 7462	. . . . . . . . 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 12418	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ≠ 0)
128	114, 39, 127	divrecd 12126	. . . . . . . . 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 2801	. . . . . . . 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 15905	. . . . . . 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 12373	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
132		pncan 11593	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
133	131, 40, 132	sylancl 598	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) − 1) = 𝑁)
134	133	oveq2d 7462	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (1...((𝑁 + 1) − 1)) = (1...𝑁))
135	134	sumeq1d 15899	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...𝑁)(1 / 𝑖))
136	135, 135	oveq12d 7464	. . . . . . . . . . 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 11367	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...𝑁)(1 / 𝑖) ∈ ℂ)
138	137	sqvald 14316	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) = (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖) · Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)))
139	136, 138	eqtr4d 2801	. . . . . . . . . 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 11328	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
141	140	mul01i 11530	. . . . . . . . . . 11 ⊢ (0 · 0) = 0
142	141	a1i 11	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (0 · 0) = 0)
143	139, 142	oveq12d 7464	. . . . . . . . 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 14317	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) ∈ ℂ)
145	144	subid1d 11688	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) − 0) = (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2))
146	143, 145	eqtrd 2798	. . . . . . . 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 7464	. . . . . . . . . 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 11367	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) ∈ ℂ)
149	148, 39, 127	divrec2d 12127	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) = ((1 / 𝑛) · Σ𝑖 ∈ (1...𝑛)(1 / 𝑖)))
150	147, 149	eqtr4d 2801	. . . . . . . . 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 15905	. . . . . . . 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 7464	. . . . . . 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 2807	. . . . . 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 11367	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) ∈ ℂ)
155	38	recnd 11367	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛) ∈ ℂ)
156	144, 154, 155	subaddd 11717	. . . . . 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 235	. . . . 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 5135	. . . 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 5131	. . 3 ⊢ (𝑁 ∈ ℕ → (2 · Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) ≤ (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2))
160		flid 13978	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (⌊‘𝑁) = 𝑁)
161	110, 160	syl 18	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (⌊‘𝑁) = 𝑁)
162	161	oveq2d 7462	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1...(⌊‘𝑁)) = (1...𝑁))
163	162	sumeq1d 15899	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...(⌊‘𝑁))(1 / 𝑖) = Σ𝑖 ∈ (1...𝑁)(1 / 𝑖))
164		nnre 12372	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
165		nnge1 12396	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
166		harmonicubnd 27370	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ 1 ≤ 𝑁) → Σ𝑖 ∈ (1...(⌊‘𝑁))(1 / 𝑖) ≤ ((log‘𝑁) + 1))
167	164, 165, 166	syl2anc 596	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...(⌊‘𝑁))(1 / 𝑖) ≤ ((log‘𝑁) + 1))
168	163, 167	eqbrtrrd 5133	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...𝑁)(1 / 𝑖) ≤ ((log‘𝑁) + 1))
169	14	nnrpd 13192	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → 𝑖 ∈ ℝ+)
170	169	rpreccld 13204	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → (1 / 𝑖) ∈ ℝ+)
171	170	rpge0d 13198	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → 0 ≤ (1 / 𝑖))
172	2, 15, 171	fsumge0 15994	. . . . 5 ⊢ (𝑁 ∈ ℕ → 0 ≤ Σ𝑖 ∈ (1...𝑁)(1 / 𝑖))
173	49	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℝ)
174		log1 26946	. . . . . . 7 ⊢ (log‘1) = 0
175		1rp 13154	. . . . . . . . 9 ⊢ 1 ∈ ℝ+
176		logleb 26964	. . . . . . . . 9 ⊢ ((1 ∈ ℝ+ ∧ 𝑁 ∈ ℝ+) → (1 ≤ 𝑁 ↔ (log‘1) ≤ (log‘𝑁)))
177	175, 18, 176	sylancr 599	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (1 ≤ 𝑁 ↔ (log‘1) ≤ (log‘𝑁)))
178	165, 177	mpbid 235	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘1) ≤ (log‘𝑁))
179	174, 178	eqbrtrrid 5145	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 ≤ (log‘𝑁))
180	19	lep1d 12278	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ≤ ((log‘𝑁) + 1))
181	173, 19, 21, 179, 180	letrd 11497	. . . . 5 ⊢ (𝑁 ∈ ℕ → 0 ≤ ((log‘𝑁) + 1))
182	16, 21, 172, 181	le2sqd 14431	. . . 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 235	. . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) ≤ (((log‘𝑁) + 1)↑2))
184	12, 17, 22, 159, 183	letrd 11497	. 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 12477	. . . 4 ⊢ 0 < 2
187	186	a1i 11	. . 3 ⊢ (𝑁 ∈ ℕ → 0 < 2)
188		lemuldiv2 12228	. . 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 1402	. 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 235	1 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ≤ ((((log‘𝑁) + 1)↑2) / 2))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900  ∅c0 4281   class class class wbr 5106  ‘cfv 6561  (class class class)co 7446  ℂcc 11228  ℝcr 11229  0cc0 11230  1c1 11231   + caddc 11233   · cmul 11235   < clt 11373   ≤ cle 11374   − cmin 11571   / cdiv 12003  ℕcn 12365  2c2 12427  ℕ₀cn0 12636  ℤcz 12723  ℤ_≥cuz 12995  ℝ⁺crp 13150  [,]cicc 13509  ...cfz 13669  ..^cfzo 13818  ⌊cfl 13960  ↑cexp 14234  Σcsu 15885  logclog 26915  γcem 27352

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-inf2 9672  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307  ax-pre-sup 11308  ax-addf 11309

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-of 7720  df-om 7905  df-1st 8028  df-2nd 8029  df-supp 8208  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-2o 8508  df-oadd 8511  df-er 8748  df-map 8880  df-pm 8881  df-ixp 8957  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-fsupp 9385  df-fi 9433  df-sup 9464  df-inf 9465  df-oi 9534  df-card 10050  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-div 12004  df-nn 12366  df-2 12435  df-3 12436  df-4 12437  df-5 12438  df-6 12439  df-7 12440  df-8 12441  df-9 12442  df-n0 12637  df-xnn0 12710  df-z 12724  df-dec 12845  df-uz 12996  df-q 13106  df-rp 13151  df-xneg 13271  df-xadd 13272  df-xmul 13273  df-ioo 13510  df-ioc 13511  df-ico 13512  df-icc 13513  df-fz 13670  df-fzo 13819  df-fl 13962  df-mod 14040  df-seq 14175  df-exp 14235  df-fac 14448  df-bc 14477  df-hash 14505  df-shft 15252  df-cj 15298  df-re 15299  df-im 15300  df-sqrt 15434  df-abs 15435  df-limsup 15670  df-clim 15687  df-rlim 15688  df-sum 15886  df-ef 16265  df-e 16266  df-sin 16267  df-cos 16268  df-tan 16269  df-pi 16270  df-dvds 16455  df-struct 17357  df-sets 17374  df-slot 17392  df-ndx 17404  df-base 17420  df-ress 17441  df-plusg 17473  df-mulr 17474  df-starv 17475  df-sca 17476  df-vsca 17477  df-ip 17478  df-tset 17479  df-ple 17480  df-ds 17482  df-unif 17483  df-hom 17484  df-cco 17485  df-rest 17625  df-topn 17626  df-0g 17644  df-gsum 17645  df-topgen 17646  df-pt 17647  df-prds 17650  df-xrs 17706  df-qtop 17711  df-imas 17712  df-xps 17714  df-mre 17788  df-mrc 17789  df-acs 17791  df-mgm 18848  df-sgrp 18940  df-mnd 18956  df-submnd 19011  df-mulg 19310  df-cntz 19564  df-cmn 20029  df-psmet 21703  df-xmet 21704  df-met 21705  df-bl 21706  df-mopn 21707  df-fbas 21708  df-fg 21709  df-cnfld 21712  df-top 23245  df-topon 23262  df-topsp 23284  df-bases 23297  df-cld 23370  df-ntr 23371  df-cls 23372  df-nei 23449  df-lp 23487  df-perf 23488  df-cn 23578  df-cnp 23579  df-haus 23666  df-cmp 23738  df-tx 23914  df-hmeo 24107  df-fil 24198  df-fm 24290  df-flim 24291  df-flf 24292  df-xms 24672  df-ms 24673  df-tms 24674  df-cncf 25232  df-limc 26219  df-dv 26220  df-ulm 26737  df-log 26917  df-atan 27228  df-em 27353

This theorem is used by:  pntlemk  27966