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Theorem mulogsum 25221

Description: Asymptotic formula for Σ𝑛 ≤ 𝑥, (μ(𝑛) / 𝑛)log(𝑥 / 𝑛) = 𝑂(1). Equation 10.2.6 of [Shapiro], p. 406. (Contributed by Mario Carneiro, 14-May-2016.)

**Assertion**
Ref	Expression
mulogsum	⊢ (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ 𝑂(1)

Distinct variable group: 𝑥,𝑛

Proof of Theorem mulogsum Dummy variables 𝑘 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpssre 11843	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
2		ax-1cn 9994	. . . 4 ⊢ 1 ∈ ℂ
3		o1const 14350	. . . 4 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
4	1, 2, 3	mp2an 708	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1)
5		1cnd 10056	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
6		fzfid 12772	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
7		elfznn 12370	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
8	7	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
9		mucl 24867	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ)
10	8, 9	syl 17	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
11	10	zred 11482	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ)
12	11, 8	nndivred 11069	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℝ)
13	7	nnrpd 11870	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ⁺)
14		rpdivcl 11856	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑥 / 𝑛) ∈ ℝ⁺)
15	13, 14	sylan2 491	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
16	15	relogcld 24369	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
17	12, 16	remulcld 10070	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℝ)
18	17	recnd 10068	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ)
19	6, 18	fsumcl 14464	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ)
20	19	adantl 482	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ)
21		mulogsumlem 25220	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1)
22		sumex 14418	. . . . . . . 8 ⊢ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ V
23	22	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ V)
24	21	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1))
25	23, 24	o1mptrcl 14353	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ)
26	5, 20	subcld 10392	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ ℂ)
27		1red 10055	. . . . . 6 ⊢ (⊤ → 1 ∈ ℝ)
28		elfznn 12370	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (1...(⌊‘𝑥)) → 𝑘 ∈ ℕ)
29	28	ssriv 3607	. . . . . . . . . . . . . . . 16 ⊢ (1...(⌊‘𝑥)) ⊆ ℕ
30	29	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (1...(⌊‘𝑥)) ⊆ ℕ)
31	30	sselda 3603	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
32	31, 9	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
33	32	zred 11482	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ)
34	33, 31	nndivred 11069	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℝ)
35	34	recnd 10068	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℂ)
36		fzfid 12772	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
37		elfznn 12370	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
38	37	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
39	38	nnrpd 11870	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℝ⁺)
40	39	rpcnne0d 11881	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
41		reccl 10692	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℂ ∧ 𝑚 ≠ 0) → (1 / 𝑚) ∈ ℂ)
42	40, 41	syl 17	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (1 / 𝑚) ∈ ℂ)
43	36, 42	fsumcl 14464	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ∈ ℂ)
44		simpl 473	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → 𝑥 ∈ ℝ⁺)
45	44, 13, 14	syl2an 494	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
46	45	relogcld 24369	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
47	46	recnd 10068	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℂ)
48	35, 43, 47	subdid 10486	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) = ((((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) − (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
49	48	sumeq2dv 14433	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) − (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
50		fzfid 12772	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (1...(⌊‘𝑥)) ∈ Fin)
51	35, 43	mulcld 10060	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) ∈ ℂ)
52	18	adantlr 751	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ)
53	50, 51, 52	fsumsub 14520	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))((((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) − (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
54		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑛 · 𝑚) → (1 / 𝑘) = (1 / (𝑛 · 𝑚)))
55	54	oveq2d 6666	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑛 · 𝑚) → ((μ‘𝑛) · (1 / 𝑘)) = ((μ‘𝑛) · (1 / (𝑛 · 𝑚))))
56		rpre 11839	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
57	56	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → 𝑥 ∈ ℝ)
58		ssrab2 3687	. . . . . . . . . . . . . . 15 ⊢ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ⊆ ℕ
59		simprr 796	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})
60	58, 59	sseldi 3601	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → 𝑛 ∈ ℕ)
61	60, 9	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → (μ‘𝑛) ∈ ℤ)
62	61	zcnd 11483	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → (μ‘𝑛) ∈ ℂ)
63	28	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℕ)
64	63	nnrecred 11066	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (1 / 𝑘) ∈ ℝ)
65	64	recnd 10068	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (1 / 𝑘) ∈ ℂ)
66	65	adantrr 753	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → (1 / 𝑘) ∈ ℂ)
67	62, 66	mulcld 10060	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → ((μ‘𝑛) · (1 / 𝑘)) ∈ ℂ)
68	55, 57, 67	dvdsflsumcom 24914	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((μ‘𝑛) · (1 / 𝑘)) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · (1 / (𝑛 · 𝑚))))
69		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → (1 / 𝑘) = (1 / 1))
70		1div1e1 10717	. . . . . . . . . . . 12 ⊢ (1 / 1) = 1
71	69, 70	syl6eq 2672	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → (1 / 𝑘) = 1)
72		flge1nn 12622	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
73	56, 72	sylan 488	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
74		nnuz 11723	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ_≥‘1)
75	73, 74	syl6eleq 2711	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ (ℤ_≥‘1))
76		eluzfz1 12348	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ (ℤ_≥‘1) → 1 ∈ (1...(⌊‘𝑥)))
77	75, 76	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → 1 ∈ (1...(⌊‘𝑥)))
78	71, 50, 30, 77, 65	musumsum 24918	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((μ‘𝑛) · (1 / 𝑘)) = 1)
79	32	zcnd 11483	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℂ)
80	79	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (μ‘𝑛) ∈ ℂ)
81	31	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑛 ∈ ℕ)
82	81	nnrpd 11870	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑛 ∈ ℝ⁺)
83	82	rpcnne0d 11881	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
84		divdiv1 10736	. . . . . . . . . . . . . . 15 ⊢ (((μ‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0) ∧ (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0)) → (((μ‘𝑛) / 𝑛) / 𝑚) = ((μ‘𝑛) / (𝑛 · 𝑚)))
85	80, 83, 40, 84	syl3anc 1326	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (((μ‘𝑛) / 𝑛) / 𝑚) = ((μ‘𝑛) / (𝑛 · 𝑚)))
86	35	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((μ‘𝑛) / 𝑛) ∈ ℂ)
87	38	nncnd 11036	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℂ)
88	38	nnne0d 11065	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ≠ 0)
89	86, 87, 88	divrecd 10804	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (((μ‘𝑛) / 𝑛) / 𝑚) = (((μ‘𝑛) / 𝑛) · (1 / 𝑚)))
90		nnmulcl 11043	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (𝑛 · 𝑚) ∈ ℕ)
91	31, 37, 90	syl2an 494	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑛 · 𝑚) ∈ ℕ)
92	91	nncnd 11036	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑛 · 𝑚) ∈ ℂ)
93	91	nnne0d 11065	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑛 · 𝑚) ≠ 0)
94	80, 92, 93	divrecd 10804	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((μ‘𝑛) / (𝑛 · 𝑚)) = ((μ‘𝑛) · (1 / (𝑛 · 𝑚))))
95	85, 89, 94	3eqtr3rd 2665	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((μ‘𝑛) · (1 / (𝑛 · 𝑚))) = (((μ‘𝑛) / 𝑛) · (1 / 𝑚)))
96	95	sumeq2dv 14433	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · (1 / (𝑛 · 𝑚))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(((μ‘𝑛) / 𝑛) · (1 / 𝑚)))
97	36, 35, 42	fsummulc2 14516	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(((μ‘𝑛) / 𝑛) · (1 / 𝑚)))
98	96, 97	eqtr4d 2659	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · (1 / (𝑛 · 𝑚))) = (((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)))
99	98	sumeq2dv 14433	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · (1 / (𝑛 · 𝑚))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)))
100	68, 78, 99	3eqtr3rd 2665	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) = 1)
101	100	oveq1d 6665	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = (1 − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
102	49, 53, 101	3eqtrd 2660	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) = (1 − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
103	102	adantl 482	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) = (1 − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
104	25, 26, 27, 103	o1eq 14301	. . . . 5 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1) ↔ (𝑥 ∈ ℝ⁺ ↦ (1 − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1)))
105	21, 104	mpbii 223	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (1 − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1))
106	5, 20, 105	o1dif 14360	. . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1) ↔ (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ 𝑂(1)))
107	4, 106	mpbii 223	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ 𝑂(1))
108	107	trud 1493	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 384 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 ≠ wne 2794 {crab 2916 Vcvv 3200 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 · cmul 9941 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 ℤcz 11377 ℤ_≥cuz 11687 ℝ⁺crp 11832 ...cfz 12326 ⌊cfl 12591 𝑂(1)co1 14217 Σcsu 14416 ∥ cdvds 14983 logclog 24301 μcmu 24821

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-o1 14221 df-lo1 14222 df-sum 14417 df-ef 14798 df-e 14799 df-sin 14800 df-cos 14801 df-pi 14803 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 df-em 24719 df-mu 24827

This theorem is referenced by: mulog2sumlem3 25225 selberglem1 25234

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