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Theorem mulogsumlem 26088

Description: Lemma for mulogsum 26089. (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 13326	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
2		elfznn 12921	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
3	2	adantl 484	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
4		mucl 25699	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
6	5	zred 12069	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ)
7	6, 3	nndivred 11673	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℝ)
8	7	recnd 10650	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℂ)
9	1, 8	fsumcl 15070	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ)
10	9	adantl 484	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ)
11		emre 25564	. . . . . 6 ⊢ γ ∈ ℝ
12	11	recni 10636	. . . . 5 ⊢ γ ∈ ℂ
13	12	a1i 11	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → γ ∈ ℂ)
14		mudivsum 26087	. . . . 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 12378	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
17		o1const 14956	. . . . . 6 ⊢ ((ℝ⁺ ⊆ ℝ ∧ γ ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ γ) ∈ 𝑂(1))
18	16, 12, 17	mp2an 690	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ ↦ γ) ∈ 𝑂(1)
19	18	a1i 11	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ γ) ∈ 𝑂(1))
20	10, 13, 15, 19	o1mul2 14961	. . 3 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ)) ∈ 𝑂(1))
21		fzfid 13326	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
22		elfznn 12921	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
23	22	adantl 484	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
24	23	nnrecred 11670	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (1 / 𝑚) ∈ ℝ)
25	21, 24	fsumrecl 15071	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ∈ ℝ)
26	2	nnrpd 12411	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ⁺)
27		rpdivcl 12396	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑥 / 𝑛) ∈ ℝ⁺)
28	26, 27	sylan2 594	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
29	28	relogcld 25187	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
30	25, 29	resubcld 11049	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ∈ ℝ)
31	7, 30	remulcld 10652	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ)
32	1, 31	fsumrecl 15071	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ)
33	32	recnd 10650	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ)
34	33	adantl 484	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ)
35		mulcl 10602	. . . . . 6 ⊢ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ ∧ γ ∈ ℂ) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) ∈ ℂ)
36	9, 12, 35	sylancl 588	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) ∈ ℂ)
37	36	adantl 484	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) ∈ ℂ)
38		nnrecre 11661	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℝ)
39	38	recnd 10650	. . . . . . . . . . . 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 15070	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ∈ ℂ)
42	29	recnd 10650	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℂ)
43	41, 42	subcld 10978	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ∈ ℂ)
44	8, 43	mulcld 10642	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ)
45		mulcl 10602	. . . . . . . . 9 ⊢ ((((μ‘𝑛) / 𝑛) ∈ ℂ ∧ γ ∈ ℂ) → (((μ‘𝑛) / 𝑛) · γ) ∈ ℂ)
46	8, 12, 45	sylancl 588	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · γ) ∈ ℂ)
47	1, 44, 46	fsumsub 15123	. . . . . . 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 11009	. . . . . . . . . 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 7153	. . . . . . . . 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 11077	. . . . . . . . 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 2857	. . . . . . . 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 15040	. . . . . . 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 15120	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · γ))
56	55	oveq2d 7153	. . . . . . 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 2865	. . . . . 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 5138	. . . . 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 10641	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛)) + γ) ∈ ℂ)
61	41, 60	subcld 10978	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)) ∈ ℂ)
62	8, 61	mulcld 10642	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ∈ ℂ)
63	1, 62	fsumcl 15070	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ∈ ℂ)
64	63	adantl 484	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ∈ ℂ)
65		1red 10623	. . . . . 6 ⊢ (⊤ → 1 ∈ ℝ)
66	63	abscld 14776	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ∈ ℝ)
67	62	abscld 14776	. . . . . . . . 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 15071	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ∈ ℝ)
69		1red 10623	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 1 ∈ ℝ)
70	1, 62	fsumabs 15136	. . . . . . . 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 12386	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
72		flge0nn0 13175	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
73	71, 72	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℕ₀)
74	73	nn0red 11938	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℝ)
75		rerpdivcl 12401	. . . . . . . . . 10 ⊢ (((⌊‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) / 𝑥) ∈ ℝ)
76	74, 75	mpancom 686	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) / 𝑥) ∈ ℝ)
77		rpreccl 12397	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
78	77	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑥) ∈ ℝ⁺)
79	78	rpred 12413	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑥) ∈ ℝ)
80	8	abscld 14776	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) / 𝑛)) ∈ ℝ)
81	3	nnrecred 11670	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
82	61	abscld 14776	. . . . . . . . . . . . 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 12396	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (𝑛 / 𝑥) ∈ ℝ⁺)
85	26, 83, 84	syl2anr 598	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 / 𝑥) ∈ ℝ⁺)
86	85	rpred 12413	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 / 𝑥) ∈ ℝ)
87	8	absge0d 14784	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((μ‘𝑛) / 𝑛)))
88	61	absge0d 14784	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))))
89	6	recnd 10650	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℂ)
90	3	nncnd 11635	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
91	3	nnne0d 11669	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
92	89, 90, 91	absdivd 14795	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) / 𝑛)) = ((abs‘(μ‘𝑛)) / (abs‘𝑛)))
93	3	nnrpd 12411	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
94		rprege0 12386	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ∈ ℝ ∧ 0 ≤ 𝑛))
95	93, 94	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℝ ∧ 0 ≤ 𝑛))
96		absid 14636	. . . . . . . . . . . . . . . . 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 7153	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(μ‘𝑛)) / (abs‘𝑛)) = ((abs‘(μ‘𝑛)) / 𝑛))
99	92, 98	eqtrd 2855	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) / 𝑛)) = ((abs‘(μ‘𝑛)) / 𝑛))
100	89	abscld 14776	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑛)) ∈ ℝ)
101		1red 10623	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
102		mule1 25706	. . . . . . . . . . . . . . . 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 12471	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(μ‘𝑛)) / 𝑛) ≤ (1 / 𝑛))
105	99, 104	eqbrtrd 5069	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) / 𝑛)) ≤ (1 / 𝑛))
106		harmonicbnd4 25569	. . . . . . . . . . . . . . 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 12389	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
109	108	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
110		rpcnne0 12389	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
111	93, 110	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
112		recdiv 11327	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (1 / (𝑥 / 𝑛)) = (𝑛 / 𝑥))
113	109, 111, 112	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑥 / 𝑛)) = (𝑛 / 𝑥))
114	107, 113	breqtrd 5073	. . . . . . . . . . . . 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 11563	. . . . . . . . . . . 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 14794	. . . . . . . . . . . 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 10617	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
118		dmdcan 11331	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℂ ∧ 𝑛 ≠ 0) ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ 1 ∈ ℂ) → ((𝑛 / 𝑥) · (1 / 𝑛)) = (1 / 𝑥))
119	111, 109, 117, 118	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 / 𝑥) · (1 / 𝑛)) = (1 / 𝑥))
120	85	rpcnd 12415	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 / 𝑥) ∈ ℂ)
121	81	recnd 10650	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
122	120, 121	mulcomd 10643	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 / 𝑥) · (1 / 𝑛)) = ((1 / 𝑛) · (𝑛 / 𝑥)))
123	119, 122	eqtr3d 2857	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑥) = ((1 / 𝑛) · (𝑛 / 𝑥)))
124	115, 116, 123	3brtr4d 5079	. . . . . . . . . . 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 15134	. . . . . . . . . 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 13691	. . . . . . . . . . . . 13 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
127	73, 126	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
128	127	oveq1d 7152	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)) = ((⌊‘𝑥) · (1 / 𝑥)))
129	77	rpcnd 12415	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℂ)
130		fsumconst 15125	. . . . . . . . . . . 12 ⊢ (((1...(⌊‘𝑥)) ∈ Fin ∧ (1 / 𝑥) ∈ ℂ) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)))
131	1, 129, 130	syl2anc 586	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)))
132	73	nn0cnd 11939	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℂ)
133		rpcn 12381	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
134		rpne0 12387	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
135	132, 133, 134	divrecd 11400	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) / 𝑥) = ((⌊‘𝑥) · (1 / 𝑥)))
136	128, 131, 135	3eqtr4d 2865	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((⌊‘𝑥) / 𝑥))
137	125, 136	breqtrd 5073	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ ((⌊‘𝑥) / 𝑥))
138		rpre 12379	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
139		flle 13154	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
140	138, 139	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ≤ 𝑥)
141	133	mulid1d 10639	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 · 1) = 𝑥)
142	140, 141	breqtrrd 5075	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ≤ (𝑥 · 1))
143		reflcl 13151	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
144	138, 143	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℝ)
145		rpregt0 12385	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
146		ledivmul 11497	. . . . . . . . . . 11 ⊢ (((⌊‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (((⌊‘𝑥) / 𝑥) ≤ 1 ↔ (⌊‘𝑥) ≤ (𝑥 · 1)))
147	144, 69, 145, 146	syl3anc 1367	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (((⌊‘𝑥) / 𝑥) ≤ 1 ↔ (⌊‘𝑥) ≤ (𝑥 · 1)))
148	142, 147	mpbird 259	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) / 𝑥) ≤ 1)
149	68, 76, 69, 137, 148	letrd 10778	. . . . . . . 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 10778	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ 1)
151	150	ad2antrl 726	. . . . . 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 14873	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ∈ 𝑂(1))
153	58, 152	eqeltrrid 2916	. . . 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 14966	. . 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 259	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1))
156	155	mptru 1544	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 208 ∧ wa 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ≠ wne 3011 ⊆ wss 3919 class class class wbr 5047 ↦ cmpt 5127 ‘cfv 6336 (class class class)co 7137 Fincfn 8490 ℂcc 10516 ℝcr 10517 0cc0 10518 1c1 10519 + caddc 10521 · cmul 10523 < clt 10656 ≤ cle 10657 − cmin 10851 / cdiv 11278 ℕcn 11619 ℕ₀cn0 11879 ℤcz 11963 ℝ⁺crp 12371 ...cfz 12877 ⌊cfl 13145 ♯chash 13675 abscabs 14573 𝑂(1)co1 14823 Σcsu 15022 logclog 25119 γcem 25550 μcmu 25653

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-inf2 9085 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 ax-pre-sup 10596 ax-addf 10597 ax-mulf 10598

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rmo 3141 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-int 4858 df-iun 4902 df-iin 4903 df-disj 5013 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-se 5496 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-isom 6345 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-of 7390 df-om 7562 df-1st 7670 df-2nd 7671 df-supp 7812 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-1o 8083 df-2o 8084 df-oadd 8087 df-er 8270 df-map 8389 df-pm 8390 df-ixp 8443 df-en 8491 df-dom 8492 df-sdom 8493 df-fin 8494 df-fsupp 8815 df-fi 8856 df-sup 8887 df-inf 8888 df-oi 8955 df-dju 9311 df-card 9349 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-div 11279 df-nn 11620 df-2 11682 df-3 11683 df-4 11684 df-5 11685 df-6 11686 df-7 11687 df-8 11688 df-9 11689 df-n0 11880 df-xnn0 11950 df-z 11964 df-dec 12081 df-uz 12226 df-q 12331 df-rp 12372 df-xneg 12489 df-xadd 12490 df-xmul 12491 df-ioo 12724 df-ioc 12725 df-ico 12726 df-icc 12727 df-fz 12878 df-fzo 13019 df-fl 13147 df-mod 13223 df-seq 13355 df-exp 13415 df-fac 13619 df-bc 13648 df-hash 13676 df-shft 14406 df-cj 14438 df-re 14439 df-im 14440 df-sqrt 14574 df-abs 14575 df-limsup 14808 df-clim 14825 df-rlim 14826 df-o1 14827 df-lo1 14828 df-sum 15023 df-ef 15401 df-e 15402 df-sin 15403 df-cos 15404 df-tan 15405 df-pi 15406 df-dvds 15588 df-gcd 15822 df-prm 15994 df-pc 16152 df-struct 16463 df-ndx 16464 df-slot 16465 df-base 16467 df-sets 16468 df-ress 16469 df-plusg 16556 df-mulr 16557 df-starv 16558 df-sca 16559 df-vsca 16560 df-ip 16561 df-tset 16562 df-ple 16563 df-ds 16565 df-unif 16566 df-hom 16567 df-cco 16568 df-rest 16674 df-topn 16675 df-0g 16693 df-gsum 16694 df-topgen 16695 df-pt 16696 df-prds 16699 df-xrs 16753 df-qtop 16758 df-imas 16759 df-xps 16761 df-mre 16835 df-mrc 16836 df-acs 16838 df-mgm 17830 df-sgrp 17879 df-mnd 17890 df-submnd 17935 df-mulg 18203 df-cntz 18425 df-cmn 18886 df-psmet 20515 df-xmet 20516 df-met 20517 df-bl 20518 df-mopn 20519 df-fbas 20520 df-fg 20521 df-cnfld 20524 df-top 21480 df-topon 21497 df-topsp 21519 df-bases 21532 df-cld 21605 df-ntr 21606 df-cls 21607 df-nei 21684 df-lp 21722 df-perf 21723 df-cn 21813 df-cnp 21814 df-haus 21901 df-cmp 21973 df-tx 22148 df-hmeo 22341 df-fil 22432 df-fm 22524 df-flim 22525 df-flf 22526 df-xms 22908 df-ms 22909 df-tms 22910 df-cncf 23464 df-limc 24444 df-dv 24445 df-ulm 24946 df-log 25121 df-atan 25426 df-em 25551 df-mu 25659

This theorem is referenced by: mulogsum 26089

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