Theorem logdivbnd 26904

Description: A bound on a sum of logs, used in pntlemk 26954. This is not as precise as logdivsum 26881 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 12227	. . . 4 ⊢ 2 ∈ ℝ
2		fzfid 13878	. . . . 5 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ∈ Fin)
3		elfzuz 13437	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ≥‘1))
4	3	adantl 482	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ (ℤ≥‘1))
5		nnuz 12806	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
6	4, 5	eleqtrrdi 2849	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
7	6	nnrpd 12955	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℝ+)
8	7	relogcld 25978	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (log‘𝑛) ∈ ℝ)
9	8, 6	nndivred 12207	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((log‘𝑛) / 𝑛) ∈ ℝ)
10	2, 9	fsumrecl 15619	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ∈ ℝ)
11		remulcl 11136	. . . 4 ⊢ ((2 ∈ ℝ ∧ Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ∈ ℝ) → (2 · Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) ∈ ℝ)
12	1, 10, 11	sylancr 587	. . 3 ⊢ (𝑁 ∈ ℕ → (2 · Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) ∈ ℝ)
13		elfznn 13470	. . . . . . 7 ⊢ (𝑖 ∈ (1...𝑁) → 𝑖 ∈ ℕ)
14	13	adantl 482	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → 𝑖 ∈ ℕ)
15	14	nnrecred 12204	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → (1 / 𝑖) ∈ ℝ)
16	2, 15	fsumrecl 15619	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...𝑁)(1 / 𝑖) ∈ ℝ)
17	16	resqcld 14030	. . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) ∈ ℝ)
18		nnrp 12926	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+)
19	18	relogcld 25978	. . . . 5 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ∈ ℝ)
20		peano2re 11328	. . . . 5 ⊢ ((log‘𝑁) ∈ ℝ → ((log‘𝑁) + 1) ∈ ℝ)
21	19, 20	syl 17	. . . 4 ⊢ (𝑁 ∈ ℕ → ((log‘𝑁) + 1) ∈ ℝ)
22	21	resqcld 14030	. . 3 ⊢ (𝑁 ∈ ℕ → (((log‘𝑁) + 1)↑2) ∈ ℝ)
23	10	recnd 11183	. . . . 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 13878	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1...𝑛) ∈ Fin)
26		elfznn 13470	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...𝑛) → 𝑖 ∈ ℕ)
27	26	adantl 482	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → 𝑖 ∈ ℕ)
28	27	nnrecred 12204	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → (1 / 𝑖) ∈ ℝ)
29	25, 28	fsumrecl 15619	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) ∈ ℝ)
30	29, 6	nndivred 12207	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) ∈ ℝ)
31	2, 30	fsumrecl 15619	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) ∈ ℝ)
32		fzfid 13878	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1...(𝑛 − 1)) ∈ Fin)
33		elfznn 13470	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(𝑛 − 1)) → 𝑖 ∈ ℕ)
34	33	adantl 482	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...(𝑛 − 1))) → 𝑖 ∈ ℕ)
35	34	nnrecred 12204	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...(𝑛 − 1))) → (1 / 𝑖) ∈ ℝ)
36	32, 35	fsumrecl 15619	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) ∈ ℝ)
37	36, 6	nndivred 12207	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛) ∈ ℝ)
38	2, 37	fsumrecl 15619	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛) ∈ ℝ)
39	6	nncnd 12169	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℂ)
40		ax-1cn 11109	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
41		npcan 11410	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
42	39, 40, 41	sylancl 586	. . . . . . . . . . . . . 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 7373	. . . . . . . . . . . 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 26357	. . . . . . . . . . . . 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 2839	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) − (log‘𝑛)) ∈ (0[,]γ))
49		0re 11157	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
50		emre 26355	. . . . . . . . . . . . 13 ⊢ γ ∈ ℝ
51	49, 50	elicc2i 13330	. . . . . . . . . . . 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 1146	. . . . . . . . . . 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 11745	. . . . . . . . . 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 231	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (log‘𝑛) ≤ Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖))
56	8, 36, 7, 55	lediv1dd 13015	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((log‘𝑛) / 𝑛) ≤ (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛))
57	27	nnrpd 12955	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → 𝑖 ∈ ℝ+)
58	57	rpreccld 12967	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → (1 / 𝑖) ∈ ℝ+)
59	58	rpge0d 12961	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → 0 ≤ (1 / 𝑖))
60		elfzelz 13441	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
61	60	adantl 482	. . . . . . . . . . . . 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 12168	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℝ)
65	64	lem1d 12088	. . . . . . . . . . . 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 1343	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ (ℤ≥‘(𝑛 − 1)))
68		fzss2 13481	. . . . . . . . . . 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 15681	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) ≤ Σ𝑖 ∈ (1...𝑛)(1 / 𝑖))
71	6	nngt0d 12202	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 0 < 𝑛)
72		lediv1 12020	. . . . . . . . . 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 1374	. . . . . . . . 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 231	. . . . . . . 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 11312	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((log‘𝑛) / 𝑛) ≤ (Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛))
76	2, 9, 30, 75	fsumle 15684	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ≤ Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛))
77	2, 9, 37, 56	fsumle 15684	. . . . . 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 11774	. . . . 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 7364	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
80	79	oveq2d 7373	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (1...(𝑚 − 1)) = (1...(𝑛 − 1)))
81	80	sumeq1d 15586	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖))
82	81, 81	jca 512	. . . . . . . 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 7364	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 − 1) = ((𝑛 + 1) − 1))
84	83	oveq2d 7373	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (1...(𝑚 − 1)) = (1...((𝑛 + 1) − 1)))
85	84	sumeq1d 15586	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖))
86	85, 85	jca 512	. . . . . . . 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 7364	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
88		1m1e0 12225	. . . . . . . . . . . . . 14 ⊢ (1 − 1) = 0
89	87, 88	eqtrdi 2792	. . . . . . . . . . . . 13 ⊢ (𝑚 = 1 → (𝑚 − 1) = 0)
90	89	oveq2d 7373	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (1...(𝑚 − 1)) = (1...0))
91		fz10 13462	. . . . . . . . . . . 12 ⊢ (1...0) = ∅
92	90, 91	eqtrdi 2792	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (1...(𝑚 − 1)) = ∅)
93	92	sumeq1d 15586	. . . . . . . . . 10 ⊢ (𝑚 = 1 → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ ∅ (1 / 𝑖))
94		sum0 15606	. . . . . . . . . 10 ⊢ Σ𝑖 ∈ ∅ (1 / 𝑖) = 0
95	93, 94	eqtrdi 2792	. . . . . . . . 9 ⊢ (𝑚 = 1 → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = 0)
96	95, 95	jca 512	. . . . . . . 8 ⊢ (𝑚 = 1 → (Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = 0 ∧ Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = 0))
97		oveq1 7364	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑁 + 1) → (𝑚 − 1) = ((𝑁 + 1) − 1))
98	97	oveq2d 7373	. . . . . . . . . 10 ⊢ (𝑚 = (𝑁 + 1) → (1...(𝑚 − 1)) = (1...((𝑁 + 1) − 1)))
99	98	sumeq1d 15586	. . . . . . . . 9 ⊢ (𝑚 = (𝑁 + 1) → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖))
100	99, 99	jca 512	. . . . . . . 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 12165	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
102	101, 5	eleqtrdi 2848	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ (ℤ≥‘1))
103		fzfid 13878	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → (1...(𝑚 − 1)) ∈ Fin)
104		elfznn 13470	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (1...(𝑚 − 1)) → 𝑖 ∈ ℕ)
105	104	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) ∧ 𝑖 ∈ (1...(𝑚 − 1))) → 𝑖 ∈ ℕ)
106	105	nnrecred 12204	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) ∧ 𝑖 ∈ (1...(𝑚 − 1))) → (1 / 𝑖) ∈ ℝ)
107	103, 106	fsumrecl 15619	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) ∈ ℝ)
108	107	recnd 11183	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑚 ∈ (1...(𝑁 + 1))) → Σ𝑖 ∈ (1...(𝑚 − 1))(1 / 𝑖) ∈ ℂ)
109	82, 86, 96, 100, 102, 108, 108	fsumparts 15691	. . . . . . 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 12520	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
111		fzval3 13641	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (1...𝑁) = (1..^(𝑁 + 1)))
112	110, 111	syl 17	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (1...𝑁) = (1..^(𝑁 + 1)))
113	112	eqcomd 2742	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (1..^(𝑁 + 1)) = (1...𝑁))
114	36	recnd 11183	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) ∈ ℂ)
115	6	nnrecred 12204	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1 / 𝑛) ∈ ℝ)
116	115	recnd 11183	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1 / 𝑛) ∈ ℂ)
117		pncan 11407	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
118	39, 40, 117	sylancl 586	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → ((𝑛 + 1) − 1) = 𝑛)
119	118	oveq2d 7373	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (1...((𝑛 + 1) − 1)) = (1...𝑛))
120	119	sumeq1d 15586	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...𝑛)(1 / 𝑖))
121	28	recnd 11183	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑖 ∈ (1...𝑛)) → (1 / 𝑖) ∈ ℂ)
122		oveq2 7365	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑛 → (1 / 𝑖) = (1 / 𝑛))
123	4, 121, 122	fsumm1 15636	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) = (Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) + (1 / 𝑛)))
124	120, 123	eqtrd 2776	. . . . . . . . . . 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 11568	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...((𝑛 + 1) − 1))(1 / 𝑖) − Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖)) = (1 / 𝑛))
126	125	oveq2d 7373	. . . . . . . . 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 12203	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ≠ 0)
128	114, 39, 127	divrecd 11934	. . . . . . . . 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 2779	. . . . . . . 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 15592	. . . . . . 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 12161	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
132		pncan 11407	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
133	131, 40, 132	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) − 1) = 𝑁)
134	133	oveq2d 7373	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (1...((𝑁 + 1) − 1)) = (1...𝑁))
135	134	sumeq1d 15586	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...((𝑁 + 1) − 1))(1 / 𝑖) = Σ𝑖 ∈ (1...𝑁)(1 / 𝑖))
136	135, 135	oveq12d 7375	. . . . . . . . . . 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 11183	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...𝑁)(1 / 𝑖) ∈ ℂ)
138	137	sqvald 14048	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) = (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖) · Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)))
139	136, 138	eqtr4d 2779	. . . . . . . . . 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 11147	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
141	140	mul01i 11345	. . . . . . . . . . 11 ⊢ (0 · 0) = 0
142	141	a1i 11	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (0 · 0) = 0)
143	139, 142	oveq12d 7375	. . . . . . . . 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 14049	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) ∈ ℂ)
145	144	subid1d 11501	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) − 0) = (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2))
146	143, 145	eqtrd 2776	. . . . . . . 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 7375	. . . . . . . . . 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 11183	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) ∈ ℂ)
149	148, 39, 127	divrec2d 11935	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) = ((1 / 𝑛) · Σ𝑖 ∈ (1...𝑛)(1 / 𝑖)))
150	147, 149	eqtr4d 2779	. . . . . . . . 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 15592	. . . . . . . 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 7375	. . . . . . 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 2785	. . . . . 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 11183	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...𝑛)(1 / 𝑖) / 𝑛) ∈ ℂ)
155	38	recnd 11183	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)(Σ𝑖 ∈ (1...(𝑛 − 1))(1 / 𝑖) / 𝑛) ∈ ℂ)
156	144, 154, 155	subaddd 11530	. . . . . 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 231	. . . . 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 5131	. . . 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 5127	. . 3 ⊢ (𝑁 ∈ ℕ → (2 · Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛)) ≤ (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2))
160		flid 13713	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (⌊‘𝑁) = 𝑁)
161	110, 160	syl 17	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (⌊‘𝑁) = 𝑁)
162	161	oveq2d 7373	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (1...(⌊‘𝑁)) = (1...𝑁))
163	162	sumeq1d 15586	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...(⌊‘𝑁))(1 / 𝑖) = Σ𝑖 ∈ (1...𝑁)(1 / 𝑖))
164		nnre 12160	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
165		nnge1 12181	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
166		harmonicubnd 26359	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ 1 ≤ 𝑁) → Σ𝑖 ∈ (1...(⌊‘𝑁))(1 / 𝑖) ≤ ((log‘𝑁) + 1))
167	164, 165, 166	syl2anc 584	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...(⌊‘𝑁))(1 / 𝑖) ≤ ((log‘𝑁) + 1))
168	163, 167	eqbrtrrd 5129	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑖 ∈ (1...𝑁)(1 / 𝑖) ≤ ((log‘𝑁) + 1))
169	14	nnrpd 12955	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → 𝑖 ∈ ℝ+)
170	169	rpreccld 12967	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → (1 / 𝑖) ∈ ℝ+)
171	170	rpge0d 12961	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...𝑁)) → 0 ≤ (1 / 𝑖))
172	2, 15, 171	fsumge0 15680	. . . . 5 ⊢ (𝑁 ∈ ℕ → 0 ≤ Σ𝑖 ∈ (1...𝑁)(1 / 𝑖))
173	49	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℝ)
174		log1 25941	. . . . . . 7 ⊢ (log‘1) = 0
175		1rp 12919	. . . . . . . . 9 ⊢ 1 ∈ ℝ+
176		logleb 25958	. . . . . . . . 9 ⊢ ((1 ∈ ℝ+ ∧ 𝑁 ∈ ℝ+) → (1 ≤ 𝑁 ↔ (log‘1) ≤ (log‘𝑁)))
177	175, 18, 176	sylancr 587	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (1 ≤ 𝑁 ↔ (log‘1) ≤ (log‘𝑁)))
178	165, 177	mpbid 231	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (log‘1) ≤ (log‘𝑁))
179	174, 178	eqbrtrrid 5141	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 ≤ (log‘𝑁))
180	19	lep1d 12086	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (log‘𝑁) ≤ ((log‘𝑁) + 1))
181	173, 19, 21, 179, 180	letrd 11312	. . . . 5 ⊢ (𝑁 ∈ ℕ → 0 ≤ ((log‘𝑁) + 1))
182	16, 21, 172, 181	le2sqd 14160	. . . 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 231	. . 3 ⊢ (𝑁 ∈ ℕ → (Σ𝑖 ∈ (1...𝑁)(1 / 𝑖)↑2) ≤ (((log‘𝑁) + 1)↑2))
184	12, 17, 22, 159, 183	letrd 11312	. 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 12256	. . . 4 ⊢ 0 < 2
187	186	a1i 11	. . 3 ⊢ (𝑁 ∈ ℕ → 0 < 2)
188		lemuldiv2 12036	. . 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 1374	. 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 231	1 ⊢ (𝑁 ∈ ℕ → Σ𝑛 ∈ (1...𝑁)((log‘𝑛) / 𝑛) ≤ ((((log‘𝑁) + 1)↑2) / 2))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ⊆ wss 3910  ∅c0 4282   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  ℂcc 11049  ℝcr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056   < clt 11189   ≤ cle 11190   − cmin 11385   / cdiv 11812  ℕcn 12153  2c2 12208  ℕ₀cn0 12413  ℤcz 12499  ℤ_≥cuz 12763  ℝ⁺crp 12915  [,]cicc 13267  ...cfz 13424  ..^cfzo 13567  ⌊cfl 13695  ↑cexp 13967  Σcsu 15570  logclog 25910  γcem 26341

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-xnn0 12486  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-ef 15950  df-e 15951  df-sin 15952  df-cos 15953  df-tan 15954  df-pi 15955  df-dvds 16137  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-haus 22666  df-cmp 22738  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-xms 23673  df-ms 23674  df-tms 23675  df-cncf 24241  df-limc 25230  df-dv 25231  df-ulm 25736  df-log 25912  df-atan 26217  df-em 26342

This theorem is referenced by:  pntlemk  26954