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Theorem mulogsumlem 27575

Description: Lemma for mulogsum 27576. (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 14014	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
2		elfznn 13593	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
3	2	adantl 481	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
4		mucl 27184	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
6	5	zred 12722	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ)
7	6, 3	nndivred 12320	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℝ)
8	7	recnd 11289	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℂ)
9	1, 8	fsumcl 15769	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ)
10	9	adantl 481	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ)
11		emre 27049	. . . . . 6 ⊢ γ ∈ ℝ
12	11	recni 11275	. . . . 5 ⊢ γ ∈ ℂ
13	12	a1i 11	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → γ ∈ ℂ)
14		mudivsum 27574	. . . . 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 13042	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
17		o1const 15656	. . . . . 6 ⊢ ((ℝ⁺ ⊆ ℝ ∧ γ ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ γ) ∈ 𝑂(1))
18	16, 12, 17	mp2an 692	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ ↦ γ) ∈ 𝑂(1)
19	18	a1i 11	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ γ) ∈ 𝑂(1))
20	10, 13, 15, 19	o1mul2 15661	. . 3 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ)) ∈ 𝑂(1))
21		fzfid 14014	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
22		elfznn 13593	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
23	22	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
24	23	nnrecred 12317	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (1 / 𝑚) ∈ ℝ)
25	21, 24	fsumrecl 15770	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ∈ ℝ)
26	2	nnrpd 13075	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ⁺)
27		rpdivcl 13060	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑥 / 𝑛) ∈ ℝ⁺)
28	26, 27	sylan2 593	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
29	28	relogcld 26665	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
30	25, 29	resubcld 11691	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ∈ ℝ)
31	7, 30	remulcld 11291	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ)
32	1, 31	fsumrecl 15770	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ)
33	32	recnd 11289	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ)
34	33	adantl 481	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ)
35		mulcl 11239	. . . . . 6 ⊢ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ ∧ γ ∈ ℂ) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) ∈ ℂ)
36	9, 12, 35	sylancl 586	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) ∈ ℂ)
37	36	adantl 481	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) ∈ ℂ)
38		nnrecre 12308	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℝ)
39	38	recnd 11289	. . . . . . . . . . . 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 15769	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ∈ ℂ)
42	29	recnd 11289	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℂ)
43	41, 42	subcld 11620	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ∈ ℂ)
44	8, 43	mulcld 11281	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ)
45		mulcl 11239	. . . . . . . . 9 ⊢ ((((μ‘𝑛) / 𝑛) ∈ ℂ ∧ γ ∈ ℂ) → (((μ‘𝑛) / 𝑛) · γ) ∈ ℂ)
46	8, 12, 45	sylancl 586	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · γ) ∈ ℂ)
47	1, 44, 46	fsumsub 15824	. . . . . . 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 11651	. . . . . . . . . 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 7447	. . . . . . . . 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 11719	. . . . . . . . 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 2779	. . . . . . . 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 15738	. . . . . . 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 15821	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · γ))
56	55	oveq2d 7447	. . . . . . 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 2787	. . . . . 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 5245	. . . . 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 11280	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛)) + γ) ∈ ℂ)
61	41, 60	subcld 11620	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)) ∈ ℂ)
62	8, 61	mulcld 11281	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ∈ ℂ)
63	1, 62	fsumcl 15769	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ∈ ℂ)
64	63	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ∈ ℂ)
65		1red 11262	. . . . . 6 ⊢ (⊤ → 1 ∈ ℝ)
66	63	abscld 15475	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ∈ ℝ)
67	62	abscld 15475	. . . . . . . . 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 15770	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ∈ ℝ)
69		1red 11262	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 1 ∈ ℝ)
70	1, 62	fsumabs 15837	. . . . . . . 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 13050	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
72		flge0nn0 13860	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
73	71, 72	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℕ₀)
74	73	nn0red 12588	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℝ)
75		rerpdivcl 13065	. . . . . . . . . 10 ⊢ (((⌊‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) / 𝑥) ∈ ℝ)
76	74, 75	mpancom 688	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) / 𝑥) ∈ ℝ)
77		rpreccl 13061	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
78	77	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑥) ∈ ℝ⁺)
79	78	rpred 13077	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑥) ∈ ℝ)
80	8	abscld 15475	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) / 𝑛)) ∈ ℝ)
81	3	nnrecred 12317	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
82	61	abscld 15475	. . . . . . . . . . . . 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 13060	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (𝑛 / 𝑥) ∈ ℝ⁺)
85	26, 83, 84	syl2anr 597	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 / 𝑥) ∈ ℝ⁺)
86	85	rpred 13077	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 / 𝑥) ∈ ℝ)
87	8	absge0d 15483	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((μ‘𝑛) / 𝑛)))
88	61	absge0d 15483	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))))
89	6	recnd 11289	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℂ)
90	3	nncnd 12282	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
91	3	nnne0d 12316	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
92	89, 90, 91	absdivd 15494	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) / 𝑛)) = ((abs‘(μ‘𝑛)) / (abs‘𝑛)))
93	3	nnrpd 13075	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
94		rprege0 13050	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ∈ ℝ ∧ 0 ≤ 𝑛))
95	93, 94	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℝ ∧ 0 ≤ 𝑛))
96		absid 15335	. . . . . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(μ‘𝑛)) / (abs‘𝑛)) = ((abs‘(μ‘𝑛)) / 𝑛))
99	92, 98	eqtrd 2777	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) / 𝑛)) = ((abs‘(μ‘𝑛)) / 𝑛))
100	89	abscld 15475	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑛)) ∈ ℝ)
101		1red 11262	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
102		mule1 27191	. . . . . . . . . . . . . . . 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 13135	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(μ‘𝑛)) / 𝑛) ≤ (1 / 𝑛))
105	99, 104	eqbrtrd 5165	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) / 𝑛)) ≤ (1 / 𝑛))
106		harmonicbnd4 27054	. . . . . . . . . . . . . . 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 13053	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
109	108	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
110		rpcnne0 13053	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
111	93, 110	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
112		recdiv 11973	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (1 / (𝑥 / 𝑛)) = (𝑛 / 𝑥))
113	109, 111, 112	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑥 / 𝑛)) = (𝑛 / 𝑥))
114	107, 113	breqtrd 5169	. . . . . . . . . . . . 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 12210	. . . . . . . . . . . 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 15493	. . . . . . . . . . . 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 11256	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
118		dmdcan 11977	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℂ ∧ 𝑛 ≠ 0) ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ 1 ∈ ℂ) → ((𝑛 / 𝑥) · (1 / 𝑛)) = (1 / 𝑥))
119	111, 109, 117, 118	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 / 𝑥) · (1 / 𝑛)) = (1 / 𝑥))
120	85	rpcnd 13079	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 / 𝑥) ∈ ℂ)
121	81	recnd 11289	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
122	120, 121	mulcomd 11282	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 / 𝑥) · (1 / 𝑛)) = ((1 / 𝑛) · (𝑛 / 𝑥)))
123	119, 122	eqtr3d 2779	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑥) = ((1 / 𝑛) · (𝑛 / 𝑥)))
124	115, 116, 123	3brtr4d 5175	. . . . . . . . . . 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 15835	. . . . . . . . . 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 14385	. . . . . . . . . . . . 13 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
127	73, 126	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
128	127	oveq1d 7446	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)) = ((⌊‘𝑥) · (1 / 𝑥)))
129	77	rpcnd 13079	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℂ)
130		fsumconst 15826	. . . . . . . . . . . 12 ⊢ (((1...(⌊‘𝑥)) ∈ Fin ∧ (1 / 𝑥) ∈ ℂ) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)))
131	1, 129, 130	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)))
132	73	nn0cnd 12589	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℂ)
133		rpcn 13045	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
134		rpne0 13051	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
135	132, 133, 134	divrecd 12046	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) / 𝑥) = ((⌊‘𝑥) · (1 / 𝑥)))
136	128, 131, 135	3eqtr4d 2787	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((⌊‘𝑥) / 𝑥))
137	125, 136	breqtrd 5169	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ ((⌊‘𝑥) / 𝑥))
138		rpre 13043	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
139		flle 13839	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
140	138, 139	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ≤ 𝑥)
141	133	mulridd 11278	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 · 1) = 𝑥)
142	140, 141	breqtrrd 5171	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ≤ (𝑥 · 1))
143		reflcl 13836	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
144	138, 143	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℝ)
145		rpregt0 13049	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
146		ledivmul 12144	. . . . . . . . . . 11 ⊢ (((⌊‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (((⌊‘𝑥) / 𝑥) ≤ 1 ↔ (⌊‘𝑥) ≤ (𝑥 · 1)))
147	144, 69, 145, 146	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (((⌊‘𝑥) / 𝑥) ≤ 1 ↔ (⌊‘𝑥) ≤ (𝑥 · 1)))
148	142, 147	mpbird 257	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) / 𝑥) ≤ 1)
149	68, 76, 69, 137, 148	letrd 11418	. . . . . . . 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 11418	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ 1)
151	150	ad2antrl 728	. . . . . 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 15572	. . . . 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 2846	. . . 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 15666	. . 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 257	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1))
156	155	mptru 1547	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 206 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 ℕ₀cn0 12526 ℤcz 12613 ℝ⁺crp 13034 ...cfz 13547 ⌊cfl 13830 ♯chash 14369 abscabs 15273 𝑂(1)co1 15522 Σcsu 15722 logclog 26596 γcem 27035 μcmu 27138

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 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 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-o1 15526 df-lo1 15527 df-sum 15723 df-ef 16103 df-e 16104 df-sin 16105 df-cos 16106 df-tan 16107 df-pi 16108 df-dvds 16291 df-gcd 16532 df-prm 16709 df-pc 16875 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-ulm 26420 df-log 26598 df-atan 26910 df-em 27036 df-mu 27144

This theorem is referenced by: mulogsum 27576

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