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Theorem mulogsumlem 25689

Description: Lemma for mulogsum 25690. (Contributed by Mario Carneiro, 14-May-2016.)

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

Distinct variable group: 𝑚,𝑛,𝑥

**Proof of Theorem mulogsumlem**
Step	Hyp	Ref	Expression
1		fzfid 13096	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
2		elfznn 12692	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
3	2	adantl 475	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
4		mucl 25330	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
6	5	zred 11839	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ)
7	6, 3	nndivred 11434	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℝ)
8	7	recnd 10407	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℂ)
9	1, 8	fsumcl 14880	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ)
10	9	adantl 475	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ)
11		emre 25195	. . . . . 6 ⊢ γ ∈ ℝ
12	11	recni 10393	. . . . 5 ⊢ γ ∈ ℂ
13	12	a1i 11	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → γ ∈ ℂ)
14		mudivsum 25688	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ 𝑂(1)
15	14	a1i 11	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ 𝑂(1))
16		rpssre 12149	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
17		o1const 14767	. . . . . 6 ⊢ ((ℝ⁺ ⊆ ℝ ∧ γ ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ γ) ∈ 𝑂(1))
18	16, 12, 17	mp2an 682	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ ↦ γ) ∈ 𝑂(1)
19	18	a1i 11	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ γ) ∈ 𝑂(1))
20	10, 13, 15, 19	o1mul2 14772	. . 3 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ)) ∈ 𝑂(1))
21		fzfid 13096	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
22		elfznn 12692	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
23	22	adantl 475	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
24	23	nnrecred 11431	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (1 / 𝑚) ∈ ℝ)
25	21, 24	fsumrecl 14881	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ∈ ℝ)
26	2	nnrpd 12184	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ⁺)
27		rpdivcl 12169	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑥 / 𝑛) ∈ ℝ⁺)
28	26, 27	sylan2 586	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
29	28	relogcld 24817	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
30	25, 29	resubcld 10806	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ∈ ℝ)
31	7, 30	remulcld 10409	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ)
32	1, 31	fsumrecl 14881	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ)
33	32	recnd 10407	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ)
34	33	adantl 475	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ)
35		mulcl 10358	. . . . . 6 ⊢ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ ∧ γ ∈ ℂ) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) ∈ ℂ)
36	9, 12, 35	sylancl 580	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) ∈ ℂ)
37	36	adantl 475	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) ∈ ℂ)
38		nnrecre 11422	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℝ)
39	38	recnd 10407	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℂ)
40	23, 39	syl 17	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (1 / 𝑚) ∈ ℂ)
41	21, 40	fsumcl 14880	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ∈ ℂ)
42	29	recnd 10407	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℂ)
43	41, 42	subcld 10736	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ∈ ℂ)
44	8, 43	mulcld 10399	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ)
45		mulcl 10358	. . . . . . . . 9 ⊢ ((((μ‘𝑛) / 𝑛) ∈ ℂ ∧ γ ∈ ℂ) → (((μ‘𝑛) / 𝑛) · γ) ∈ ℂ)
46	8, 12, 45	sylancl 580	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · γ) ∈ ℂ)
47	1, 44, 46	fsumsub 14933	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − (((μ‘𝑛) / 𝑛) · γ)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · γ)))
48	12	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → γ ∈ ℂ)
49	41, 42, 48	subsub4d 10767	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) − γ) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))
50	49	oveq2d 6940	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) − γ)) = (((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))))
51	8, 43, 48	subdid 10834	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) − γ)) = ((((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − (((μ‘𝑛) / 𝑛) · γ)))
52	50, 51	eqtr3d 2816	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) = ((((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − (((μ‘𝑛) / 𝑛) · γ)))
53	52	sumeq2dv 14850	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − (((μ‘𝑛) / 𝑛) · γ)))
54	12	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → γ ∈ ℂ)
55	1, 54, 8	fsummulc1 14930	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · γ))
56	55	oveq2d 6940	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · γ)))
57	47, 53, 56	3eqtr4d 2824	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ)))
58	57	mpteq2ia 4977	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ)))
59	16	a1i 11	. . . . . 6 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
60	42, 48	addcld 10398	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛)) + γ) ∈ ℂ)
61	41, 60	subcld 10736	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)) ∈ ℂ)
62	8, 61	mulcld 10399	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ∈ ℂ)
63	1, 62	fsumcl 14880	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ∈ ℂ)
64	63	adantl 475	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ∈ ℂ)
65		1red 10379	. . . . . 6 ⊢ (⊤ → 1 ∈ ℝ)
66	63	abscld 14590	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ∈ ℝ)
67	62	abscld 14590	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ∈ ℝ)
68	1, 67	fsumrecl 14881	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ∈ ℝ)
69		1red 10379	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 1 ∈ ℝ)
70	1, 62	fsumabs 14946	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))))
71		rprege0 12159	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
72		flge0nn0 12945	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
73	71, 72	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℕ₀)
74	73	nn0red 11708	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℝ)
75		rerpdivcl 12174	. . . . . . . . . 10 ⊢ (((⌊‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) / 𝑥) ∈ ℝ)
76	74, 75	mpancom 678	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) / 𝑥) ∈ ℝ)
77		rpreccl 12170	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
78	77	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑥) ∈ ℝ⁺)
79	78	rpred 12186	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑥) ∈ ℝ)
80	8	abscld 14590	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) / 𝑛)) ∈ ℝ)
81	3	nnrecred 11431	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
82	61	abscld 14590	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ∈ ℝ)
83		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ⁺)
84		rpdivcl 12169	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (𝑛 / 𝑥) ∈ ℝ⁺)
85	26, 83, 84	syl2anr 590	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 / 𝑥) ∈ ℝ⁺)
86	85	rpred 12186	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 / 𝑥) ∈ ℝ)
87	8	absge0d 14598	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((μ‘𝑛) / 𝑛)))
88	61	absge0d 14598	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))))
89	6	recnd 10407	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℂ)
90	3	nncnd 11397	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
91	3	nnne0d 11430	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
92	89, 90, 91	absdivd 14609	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) / 𝑛)) = ((abs‘(μ‘𝑛)) / (abs‘𝑛)))
93	3	nnrpd 12184	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
94		rprege0 12159	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ∈ ℝ ∧ 0 ≤ 𝑛))
95	93, 94	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℝ ∧ 0 ≤ 𝑛))
96		absid 14450	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℝ ∧ 0 ≤ 𝑛) → (abs‘𝑛) = 𝑛)
97	95, 96	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝑛) = 𝑛)
98	97	oveq2d 6940	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(μ‘𝑛)) / (abs‘𝑛)) = ((abs‘(μ‘𝑛)) / 𝑛))
99	92, 98	eqtrd 2814	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) / 𝑛)) = ((abs‘(μ‘𝑛)) / 𝑛))
100	89	abscld 14590	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑛)) ∈ ℝ)
101		1red 10379	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
102		mule1 25337	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (abs‘(μ‘𝑛)) ≤ 1)
103	3, 102	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑛)) ≤ 1)
104	100, 101, 93, 103	lediv1dd 12244	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(μ‘𝑛)) / 𝑛) ≤ (1 / 𝑛))
105	99, 104	eqbrtrd 4910	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) / 𝑛)) ≤ (1 / 𝑛))
106		harmonicbnd4 25200	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 / 𝑛) ∈ ℝ⁺ → (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ≤ (1 / (𝑥 / 𝑛)))
107	28, 106	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ≤ (1 / (𝑥 / 𝑛)))
108		rpcnne0 12162	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
109	108	adantr 474	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
110		rpcnne0 12162	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
111	93, 110	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
112		recdiv 11084	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (1 / (𝑥 / 𝑛)) = (𝑛 / 𝑥))
113	109, 111, 112	syl2anc 579	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑥 / 𝑛)) = (𝑛 / 𝑥))
114	107, 113	breqtrd 4914	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ≤ (𝑛 / 𝑥))
115	80, 81, 82, 86, 87, 88, 105, 114	lemul12ad 11323	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘((μ‘𝑛) / 𝑛)) · (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ ((1 / 𝑛) · (𝑛 / 𝑥)))
116	8, 61	absmuld 14608	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) = ((abs‘((μ‘𝑛) / 𝑛)) · (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))))
117		1cnd 10373	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
118		dmdcan 11088	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℂ ∧ 𝑛 ≠ 0) ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ 1 ∈ ℂ) → ((𝑛 / 𝑥) · (1 / 𝑛)) = (1 / 𝑥))
119	111, 109, 117, 118	syl3anc 1439	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 / 𝑥) · (1 / 𝑛)) = (1 / 𝑥))
120	85	rpcnd 12188	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 / 𝑥) ∈ ℂ)
121	81	recnd 10407	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
122	120, 121	mulcomd 10400	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 / 𝑥) · (1 / 𝑛)) = ((1 / 𝑛) · (𝑛 / 𝑥)))
123	119, 122	eqtr3d 2816	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑥) = ((1 / 𝑛) · (𝑛 / 𝑥)))
124	115, 116, 123	3brtr4d 4920	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ (1 / 𝑥))
125	1, 67, 79, 124	fsumle 14944	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑥))
126		hashfz1 13457	. . . . . . . . . . . . 13 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
127	73, 126	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
128	127	oveq1d 6939	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)) = ((⌊‘𝑥) · (1 / 𝑥)))
129	77	rpcnd 12188	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℂ)
130		fsumconst 14935	. . . . . . . . . . . 12 ⊢ (((1...(⌊‘𝑥)) ∈ Fin ∧ (1 / 𝑥) ∈ ℂ) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)))
131	1, 129, 130	syl2anc 579	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)))
132	73	nn0cnd 11709	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℂ)
133		rpcn 12154	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
134		rpne0 12160	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
135	132, 133, 134	divrecd 11157	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) / 𝑥) = ((⌊‘𝑥) · (1 / 𝑥)))
136	128, 131, 135	3eqtr4d 2824	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((⌊‘𝑥) / 𝑥))
137	125, 136	breqtrd 4914	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ ((⌊‘𝑥) / 𝑥))
138		rpre 12150	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
139		flle 12924	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
140	138, 139	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ≤ 𝑥)
141	133	mulid1d 10396	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 · 1) = 𝑥)
142	140, 141	breqtrrd 4916	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ≤ (𝑥 · 1))
143		reflcl 12921	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
144	138, 143	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℝ)
145		rpregt0 12158	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
146		ledivmul 11256	. . . . . . . . . . 11 ⊢ (((⌊‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (((⌊‘𝑥) / 𝑥) ≤ 1 ↔ (⌊‘𝑥) ≤ (𝑥 · 1)))
147	144, 69, 145, 146	syl3anc 1439	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (((⌊‘𝑥) / 𝑥) ≤ 1 ↔ (⌊‘𝑥) ≤ (𝑥 · 1)))
148	142, 147	mpbird 249	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) / 𝑥) ≤ 1)
149	68, 76, 69, 137, 148	letrd 10535	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ 1)
150	66, 68, 69, 70, 149	letrd 10535	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ 1)
151	150	ad2antrl 718	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ 1)
152	59, 64, 65, 65, 151	elo1d 14684	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ∈ 𝑂(1))
153	58, 152	syl5eqelr 2864	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ))) ∈ 𝑂(1))
154	34, 37, 153	o1dif 14777	. . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1) ↔ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ)) ∈ 𝑂(1)))
155	20, 154	mpbird 249	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1))
156	155	mptru 1609	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 198 ∧ wa 386 = wceq 1601 ⊤wtru 1602 ∈ wcel 2107 ≠ wne 2969 ⊆ wss 3792 class class class wbr 4888 ↦ cmpt 4967 ‘cfv 6137 (class class class)co 6924 Fincfn 8243 ℂcc 10272 ℝcr 10273 0cc0 10274 1c1 10275 + caddc 10277 · cmul 10279 < clt 10413 ≤ cle 10414 − cmin 10608 / cdiv 11035 ℕcn 11379 ℕ₀cn0 11647 ℤcz 11733 ℝ⁺crp 12142 ...cfz 12648 ⌊cfl 12915 ♯chash 13441 abscabs 14387 𝑂(1)co1 14634 Σcsu 14833 logclog 24749 γcem 25181 μcmu 25284

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 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354

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 4674 df-int 4713 df-iun 4757 df-iin 4758 df-disj 4857 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-fi 8607 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-xnn0 11720 df-z 11734 df-dec 11851 df-uz 11998 df-q 12101 df-rp 12143 df-xneg 12262 df-xadd 12263 df-xmul 12264 df-ioo 12496 df-ioc 12497 df-ico 12498 df-icc 12499 df-fz 12649 df-fzo 12790 df-fl 12917 df-mod 12993 df-seq 13125 df-exp 13184 df-fac 13385 df-bc 13414 df-hash 13442 df-shft 14220 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 df-limsup 14619 df-clim 14636 df-rlim 14637 df-o1 14638 df-lo1 14639 df-sum 14834 df-ef 15209 df-e 15210 df-sin 15211 df-cos 15212 df-pi 15214 df-dvds 15397 df-gcd 15633 df-prm 15801 df-pc 15957 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-starv 16364 df-sca 16365 df-vsca 16366 df-ip 16367 df-tset 16368 df-ple 16369 df-ds 16371 df-unif 16372 df-hom 16373 df-cco 16374 df-rest 16480 df-topn 16481 df-0g 16499 df-gsum 16500 df-topgen 16501 df-pt 16502 df-prds 16505 df-xrs 16559 df-qtop 16564 df-imas 16565 df-xps 16567 df-mre 16643 df-mrc 16644 df-acs 16646 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-submnd 17733 df-mulg 17939 df-cntz 18144 df-cmn 18592 df-psmet 20145 df-xmet 20146 df-met 20147 df-bl 20148 df-mopn 20149 df-fbas 20150 df-fg 20151 df-cnfld 20154 df-top 21117 df-topon 21134 df-topsp 21156 df-bases 21169 df-cld 21242 df-ntr 21243 df-cls 21244 df-nei 21321 df-lp 21359 df-perf 21360 df-cn 21450 df-cnp 21451 df-haus 21538 df-tx 21785 df-hmeo 21978 df-fil 22069 df-fm 22161 df-flim 22162 df-flf 22163 df-xms 22544 df-ms 22545 df-tms 22546 df-cncf 23100 df-limc 24078 df-dv 24079 df-log 24751 df-em 25182 df-mu 25290

This theorem is referenced by: mulogsum 25690

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