pntrsumo1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pntrsumo1		Structured version Visualization version GIF version

Theorem pntrsumo1 27782

Description: A bound on a sum over 𝑅. Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 25-May-2016.)

**Hypothesis**
Ref	Expression
pntrval.r	⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))

**Assertion**
Ref	Expression
pntrsumo1	⊢ (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1)

Distinct variable groups: 𝑛,𝑎,𝑥 𝑅,𝑛,𝑥 Allowed substitution hint: 𝑅(𝑎)

Proof of Theorem pntrsumo1 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1re 11225	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
2		elicopnf 13490	. . . . . . . . . . 11 ⊢ (1 ∈ ℝ → (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)))
3	1, 2	ax-mp 5	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥))
4	3	simplbi 502	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ)
5		0red 11228	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → 0 ∈ ℝ)
6		1red 11226	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ∈ ℝ)
7		0lt1 11753	. . . . . . . . . . 11 ⊢ 0 < 1
8	7	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → 0 < 1)
9	3	simprbi 503	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ≤ 𝑥)
10	5, 6, 4, 8, 9	ltletrd 11387	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → 0 < 𝑥)
11	4, 10	elrpd 13075	. . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ⁺)
12	11	ssriv 3942	. . . . . . 7 ⊢ (1[,)+∞) ⊆ ℝ⁺
13	12	a1i 11	. . . . . 6 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ⁺)
14		rpssre 13042	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
15	13, 14	sstrdi 3950	. . . . 5 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ)
16	15	resmptd 6044	. . . 4 ⊢ (⊤ → ((𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ↾ (1[,)+∞)) = (𝑥 ∈ (1[,)+∞) ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
17		chpcl 27341	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
18	4, 17	syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘𝑥) ∈ ℝ)
19		peano2re 11400	. . . . . . . . . . 11 ⊢ ((ψ‘𝑥) ∈ ℝ → ((ψ‘𝑥) + 1) ∈ ℝ)
20	18, 19	syl 18	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘𝑥) + 1) ∈ ℝ)
21	11	rprege0d 13085	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
22		flge0nn0 13873	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
23	21, 22	syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℕ₀)
24		nn0p1nn 12560	. . . . . . . . . . 11 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
25	23, 24	syl 18	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ∈ ℕ)
26	20, 25	nndivred 12307	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℝ)
27	26	recnd 11254	. . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℂ)
28		ax-1cn 11175	. . . . . . . 8 ⊢ 1 ∈ ℂ
29		subcl 11473	. . . . . . . 8 ⊢ (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℂ ∧ 1 ∈ ℂ) → ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) ∈ ℂ)
30	27, 28, 29	sylancl 598	. . . . . . 7 ⊢ (𝑥 ∈ (1[,)+∞) → ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) ∈ ℂ)
31		fzfid 14029	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (1...(⌊‘𝑥)) ∈ Fin)
32		elfznn 13600	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
33	32	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
34		nnrp 13046	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
35		pntrval.r	. . . . . . . . . . . . . . 15 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
36	35	pntrf 27780	. . . . . . . . . . . . . 14 ⊢ 𝑅:ℝ⁺⟶ℝ
37	36	ffvelcdmi 7082	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℝ⁺ → (𝑅‘𝑛) ∈ ℝ)
38	34, 37	syl 18	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑅‘𝑛) ∈ ℝ)
39		peano2nn 12262	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
40		nnmulcl 12274	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ ℕ) → (𝑛 · (𝑛 + 1)) ∈ ℕ)
41	39, 40	mpdan 700	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 · (𝑛 + 1)) ∈ ℕ)
42	38, 41	nndivred 12307	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℝ)
43	33, 42	syl 18	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℝ)
44	31, 43	fsumrecl 15810	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℝ)
45	44	recnd 11254	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
46	4, 45	syl 18	. . . . . . 7 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
47		oveq2 7427	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (1 / 𝑚) = (1 / 𝑛))
48		fvoveq1 7442	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1)))
49		oveq1 7426	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
50	48, 49	oveq12d 7437	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘(𝑛 − 1)) − (𝑛 − 1)))
51	47, 50	jca 521	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((1 / 𝑚) = (1 / 𝑛) ∧ ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘(𝑛 − 1)) − (𝑛 − 1))))
52		oveq2 7427	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1)))
53		fvoveq1 7442	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1)))
54		oveq1 7426	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 − 1) = ((𝑛 + 1) − 1))
55	53, 54	oveq12d 7437	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)))
56	52, 55	jca 521	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → ((1 / 𝑚) = (1 / (𝑛 + 1)) ∧ ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1))))
57		oveq2 7427	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (1 / 𝑚) = (1 / 1))
58		1div1e1 11922	. . . . . . . . . . . 12 ⊢ (1 / 1) = 1
59	57, 58	eqtrdi 2816	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (1 / 𝑚) = 1)
60		oveq1 7426	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
61		1m1e0 12330	. . . . . . . . . . . . . . . 16 ⊢ (1 − 1) = 0
62	60, 61	eqtrdi 2816	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 1 → (𝑚 − 1) = 0)
63	62	fveq2d 6889	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = (ψ‘0))
64		2pos 12362	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
65		0re 11227	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
66		chpeq0 27425	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ ℝ → ((ψ‘0) = 0 ↔ 0 < 2))
67	65, 66	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((ψ‘0) = 0 ↔ 0 < 2)
68	64, 67	mpbir 234	. . . . . . . . . . . . . 14 ⊢ (ψ‘0) = 0
69	63, 68	eqtrdi 2816	. . . . . . . . . . . . 13 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = 0)
70	69, 62	oveq12d 7437	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = (0 − 0))
71		0m0e0 12376	. . . . . . . . . . . 12 ⊢ (0 − 0) = 0
72	70, 71	eqtrdi 2816	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = 0)
73	59, 72	jca 521	. . . . . . . . . 10 ⊢ (𝑚 = 1 → ((1 / 𝑚) = 1 ∧ ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = 0))
74		oveq2 7427	. . . . . . . . . . 11 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (1 / 𝑚) = (1 / ((⌊‘𝑥) + 1)))
75		fvoveq1 7442	. . . . . . . . . . . 12 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1)))
76		oveq1 7426	. . . . . . . . . . . 12 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (𝑚 − 1) = (((⌊‘𝑥) + 1) − 1))
77	75, 76	oveq12d 7437	. . . . . . . . . . 11 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1)))
78	74, 77	jca 521	. . . . . . . . . 10 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → ((1 / 𝑚) = (1 / ((⌊‘𝑥) + 1)) ∧ ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))))
79		nnuz 12919	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
80	25, 79	eleqtrdi 2875	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
81		elfznn 13600	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
82	81	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
83	82	nnrecred 12304	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (1 / 𝑚) ∈ ℝ)
84	83	recnd 11254	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (1 / 𝑚) ∈ ℂ)
85	82	nnred 12265	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ)
86		peano2rem 11542	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℝ → (𝑚 − 1) ∈ ℝ)
87	85, 86	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑚 − 1) ∈ ℝ)
88		chpcl 27341	. . . . . . . . . . . . 13 ⊢ ((𝑚 − 1) ∈ ℝ → (ψ‘(𝑚 − 1)) ∈ ℝ)
89	87, 88	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℝ)
90	89, 87	resubcld 11659	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) ∈ ℝ)
91	90	recnd 11254	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) ∈ ℂ)
92	51, 56, 73, 78, 80, 84, 91	fsumparts 15883	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))((1 / 𝑛) · (((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) − ((ψ‘(𝑛 − 1)) − (𝑛 − 1)))) = ((((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) − (1 · 0)) − Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((1 / (𝑛 + 1)) − (1 / 𝑛)) · ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)))))
93	4	flcld 13851	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℤ)
94		fzval3 13782	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ ℤ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
95	93, 94	syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
96	95	eqcomd 2771	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → (1..^((⌊‘𝑥) + 1)) = (1...(⌊‘𝑥)))
97	32	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
98	97	nncnd 12266	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
99		pncan 11480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
100	98, 28, 99	sylancl 598	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = 𝑛)
101	97	nnred 12265	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
102	100, 101	eqeltrd 2865	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) ∈ ℝ)
103		chpcl 27341	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 + 1) − 1) ∈ ℝ → (ψ‘((𝑛 + 1) − 1)) ∈ ℝ)
104	102, 103	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) ∈ ℝ)
105	104	recnd 11254	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) ∈ ℂ)
106	102	recnd 11254	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) ∈ ℂ)
107		peano2rem 11542	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
108	101, 107	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
109		chpcl 27341	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 − 1) ∈ ℝ → (ψ‘(𝑛 − 1)) ∈ ℝ)
110	108, 109	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℝ)
111	110	recnd 11254	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℂ)
112		1cnd 11219	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
113	98, 112	subcld 11586	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℂ)
114	105, 106, 111, 113	sub4d 11635	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) − ((ψ‘(𝑛 − 1)) − (𝑛 − 1))) = (((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) − (((𝑛 + 1) − 1) − (𝑛 − 1))))
115		vmacl 27335	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
116	97, 115	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
117	116	recnd 11254	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
118		nnm1nn0 12562	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
119	97, 118	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℕ₀)
120		chpp1 27372	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 1) ∈ ℕ₀ → (ψ‘((𝑛 − 1) + 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))))
121	119, 120	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 − 1) + 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))))
122		npcan 11483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
123	98, 28, 122	sylancl 598	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 − 1) + 1) = 𝑛)
124	123, 100	eqtr4d 2803	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 − 1) + 1) = ((𝑛 + 1) − 1))
125	124	fveq2d 6889	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 − 1) + 1)) = (ψ‘((𝑛 + 1) − 1)))
126	123	fveq2d 6889	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘((𝑛 − 1) + 1)) = (Λ‘𝑛))
127	126	oveq2d 7435	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
128	121, 125, 127	3eqtr3d 2808	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
129	111, 117, 128	mvrladdd 11644	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
130		peano2cn 11399	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℂ → (𝑛 + 1) ∈ ℂ)
131	98, 130	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ∈ ℂ)
132	131, 98, 112	nnncan2d 11621	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − 1) − (𝑛 − 1)) = ((𝑛 + 1) − 𝑛))
133		pncan2 11481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 𝑛) = 1)
134	98, 28, 133	sylancl 598	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 𝑛) = 1)
135	132, 134	eqtrd 2800	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − 1) − (𝑛 − 1)) = 1)
136	129, 135	oveq12d 7437	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) − (((𝑛 + 1) − 1) − (𝑛 − 1))) = ((Λ‘𝑛) − 1))
137	114, 136	eqtrd 2800	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) − ((ψ‘(𝑛 − 1)) − (𝑛 − 1))) = ((Λ‘𝑛) − 1))
138	137	oveq2d 7435	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 / 𝑛) · (((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) − ((ψ‘(𝑛 − 1)) − (𝑛 − 1)))) = ((1 / 𝑛) · ((Λ‘𝑛) − 1)))
139		peano2rem 11542	. . . . . . . . . . . . . 14 ⊢ ((Λ‘𝑛) ∈ ℝ → ((Λ‘𝑛) − 1) ∈ ℝ)
140	116, 139	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) − 1) ∈ ℝ)
141	140	recnd 11254	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) − 1) ∈ ℂ)
142	97	nnne0d 12303	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
143	141, 98, 142	divrec2d 12012	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) − 1) / 𝑛) = ((1 / 𝑛) · ((Λ‘𝑛) − 1)))
144	138, 143	eqtr4d 2803	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 / 𝑛) · (((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) − ((ψ‘(𝑛 − 1)) − (𝑛 − 1)))) = (((Λ‘𝑛) − 1) / 𝑛))
145	96, 144	sumeq12rdv 15783	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))((1 / 𝑛) · (((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) − ((ψ‘(𝑛 − 1)) − (𝑛 − 1)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛))
146	23	nn0cnd 12584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℂ)
147		pncan 11480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((⌊‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
148	146, 28, 147	sylancl 598	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (1[,)+∞) → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
149	148	fveq2d 6889	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘(⌊‘𝑥)))
150		chpfl 27367	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
151	4, 150	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
152	149, 151	eqtrd 2800	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘𝑥))
153	152	oveq1d 7434	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1)) = ((ψ‘𝑥) − (((⌊‘𝑥) + 1) − 1)))
154	18	recnd 11254	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘𝑥) ∈ ℂ)
155	25	nncnd 12266	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ∈ ℂ)
156		1cnd 11219	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ∈ ℂ)
157	154, 155, 156	subsub3d 11616	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘𝑥) − (((⌊‘𝑥) + 1) − 1)) = (((ψ‘𝑥) + 1) − ((⌊‘𝑥) + 1)))
158	153, 157	eqtrd 2800	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1)) = (((ψ‘𝑥) + 1) − ((⌊‘𝑥) + 1)))
159	158	oveq2d 7435	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) = ((1 / ((⌊‘𝑥) + 1)) · (((ψ‘𝑥) + 1) − ((⌊‘𝑥) + 1))))
160	25	nnrecred 12304	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → (1 / ((⌊‘𝑥) + 1)) ∈ ℝ)
161	160	recnd 11254	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → (1 / ((⌊‘𝑥) + 1)) ∈ ℂ)
162		peano2cn 11399	. . . . . . . . . . . . . . 15 ⊢ ((ψ‘𝑥) ∈ ℂ → ((ψ‘𝑥) + 1) ∈ ℂ)
163	154, 162	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘𝑥) + 1) ∈ ℂ)
164	161, 163, 155	subdid 11687	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · (((ψ‘𝑥) + 1) − ((⌊‘𝑥) + 1))) = (((1 / ((⌊‘𝑥) + 1)) · ((ψ‘𝑥) + 1)) − ((1 / ((⌊‘𝑥) + 1)) · ((⌊‘𝑥) + 1))))
165	25	nnne0d 12303	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ≠ 0)
166	163, 155, 165	divrec2d 12012	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) = ((1 / ((⌊‘𝑥) + 1)) · ((ψ‘𝑥) + 1)))
167	166	eqcomd 2771	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · ((ψ‘𝑥) + 1)) = (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)))
168	155, 165	recid2d 12004	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · ((⌊‘𝑥) + 1)) = 1)
169	167, 168	oveq12d 7437	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1[,)+∞) → (((1 / ((⌊‘𝑥) + 1)) · ((ψ‘𝑥) + 1)) − ((1 / ((⌊‘𝑥) + 1)) · ((⌊‘𝑥) + 1))) = ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))
170	159, 164, 169	3eqtrd 2804	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) = ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))
171	28	mul01i 11417	. . . . . . . . . . . . 13 ⊢ (1 · 0) = 0
172	171	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → (1 · 0) = 0)
173	170, 172	oveq12d 7437	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → (((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) − (1 · 0)) = (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) − 0))
174	30	subid1d 11575	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) − 0) = ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))
175	173, 174	eqtrd 2800	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → (((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) − (1 · 0)) = ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))
176	97, 41	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (𝑛 + 1)) ∈ ℕ)
177	176	nnrecred 12304	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 · (𝑛 + 1))) ∈ ℝ)
178	177	recnd 11254	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 · (𝑛 + 1))) ∈ ℂ)
179	97, 38	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘𝑛) ∈ ℝ)
180	179	recnd 11254	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘𝑛) ∈ ℂ)
181	178, 180	mulneg1d 11684	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (-(1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)) = -((1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)))
182	98, 112	mulcld 11246	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · 1) ∈ ℂ)
183	98, 131	mulcld 11246	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (𝑛 + 1)) ∈ ℂ)
184	176	nnne0d 12303	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (𝑛 + 1)) ≠ 0)
185	131, 182, 183, 184	divsubdird 12047	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − (𝑛 · 1)) / (𝑛 · (𝑛 + 1))) = (((𝑛 + 1) / (𝑛 · (𝑛 + 1))) − ((𝑛 · 1) / (𝑛 · (𝑛 + 1)))))
186	98	mulridd 11243	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · 1) = 𝑛)
187	186	oveq2d 7435	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − (𝑛 · 1)) = ((𝑛 + 1) − 𝑛))
188	187, 134	eqtrd 2800	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − (𝑛 · 1)) = 1)
189	188	oveq1d 7434	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − (𝑛 · 1)) / (𝑛 · (𝑛 + 1))) = (1 / (𝑛 · (𝑛 + 1))))
190	131	mulridd 11243	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) · 1) = (𝑛 + 1))
191	131, 98	mulcomd 11247	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) · 𝑛) = (𝑛 · (𝑛 + 1)))
192	190, 191	oveq12d 7437	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) · 1) / ((𝑛 + 1) · 𝑛)) = ((𝑛 + 1) / (𝑛 · (𝑛 + 1))))
193	97, 39	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ∈ ℕ)
194	193	nnne0d 12303	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ≠ 0)
195	112, 98, 131, 142, 194	divcan5d 12034	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) · 1) / ((𝑛 + 1) · 𝑛)) = (1 / 𝑛))
196	192, 195	eqtr3d 2802	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) / (𝑛 · (𝑛 + 1))) = (1 / 𝑛))
197	112, 131, 98, 194, 142	divcan5d 12034	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 · 1) / (𝑛 · (𝑛 + 1))) = (1 / (𝑛 + 1)))
198	196, 197	oveq12d 7437	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) / (𝑛 · (𝑛 + 1))) − ((𝑛 · 1) / (𝑛 · (𝑛 + 1)))) = ((1 / 𝑛) − (1 / (𝑛 + 1))))
199	185, 189, 198	3eqtr3d 2808	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 · (𝑛 + 1))) = ((1 / 𝑛) − (1 / (𝑛 + 1))))
200	199	negeqd 11468	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → -(1 / (𝑛 · (𝑛 + 1))) = -((1 / 𝑛) − (1 / (𝑛 + 1))))
201	97	nnrecred 12304	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
202	201	recnd 11254	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
203	193	nnrecred 12304	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 + 1)) ∈ ℝ)
204	203	recnd 11254	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 + 1)) ∈ ℂ)
205	202, 204	negsubdi2d 11602	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → -((1 / 𝑛) − (1 / (𝑛 + 1))) = ((1 / (𝑛 + 1)) − (1 / 𝑛)))
206	200, 205	eqtr2d 2801	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 / (𝑛 + 1)) − (1 / 𝑛)) = -(1 / (𝑛 · (𝑛 + 1))))
207	97	nnrpd 13076	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
208	100, 207	eqeltrd 2865	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) ∈ ℝ⁺)
209	35	pntrval 27779	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 + 1) − 1) ∈ ℝ⁺ → (𝑅‘((𝑛 + 1) − 1)) = ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)))
210	208, 209	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘((𝑛 + 1) − 1)) = ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)))
211	100	fveq2d 6889	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘((𝑛 + 1) − 1)) = (𝑅‘𝑛))
212	210, 211	eqtr3d 2802	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) = (𝑅‘𝑛))
213	206, 212	oveq12d 7437	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((1 / (𝑛 + 1)) − (1 / 𝑛)) · ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1))) = (-(1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)))
214	180, 183, 184	divrec2d 12012	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) = ((1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)))
215	214	negeqd 11468	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → -((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) = -((1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)))
216	181, 213, 215	3eqtr4d 2810	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((1 / (𝑛 + 1)) − (1 / 𝑛)) · ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1))) = -((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
217	96, 216	sumeq12rdv 15783	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((1 / (𝑛 + 1)) − (1 / 𝑛)) · ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1))) = Σ𝑛 ∈ (1...(⌊‘𝑥))-((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
218		fzfid 14029	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → (1...(⌊‘𝑥)) ∈ Fin)
219	97, 42	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℝ)
220	219	recnd 11254	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
221	218, 220	fsumneg 15863	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1...(⌊‘𝑥))-((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) = -Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
222	217, 221	eqtrd 2800	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((1 / (𝑛 + 1)) − (1 / 𝑛)) · ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1))) = -Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
223	175, 222	oveq12d 7437	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → ((((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) − (1 · 0)) − Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((1 / (𝑛 + 1)) − (1 / 𝑛)) · ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)))) = (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) − -Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
224	92, 145, 223	3eqtr3d 2808	. . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) = (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) − -Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
225	30, 46	subnegd 11593	. . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) − -Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) = (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) + Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
226	224, 225	eqtrd 2800	. . . . . . 7 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) = (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) + Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
227	30, 46, 226	mvrladdd 11644	. . . . . 6 ⊢ (𝑥 ∈ (1[,)+∞) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) − ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1)) = Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
228	227	mpteq2ia 5208	. . . . 5 ⊢ (𝑥 ∈ (1[,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) − ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))) = (𝑥 ∈ (1[,)+∞) ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
229		fzfid 14029	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
230	32	adantl 487	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
231	230, 115	syl 18	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
232	231, 139	syl 18	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) − 1) ∈ ℝ)
233	232, 230	nndivred 12307	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) − 1) / 𝑛) ∈ ℝ)
234	229, 233	fsumrecl 15810	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) ∈ ℝ)
235		rpre 13043	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
236	235	adantl 487	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
237	236, 17	syl 18	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (ψ‘𝑥) ∈ ℝ)
238	237, 19	syl 18	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) + 1) ∈ ℝ)
239		rprege0 13050	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
240	239, 22	syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℕ₀)
241	240	adantl 487	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ∈ ℕ₀)
242	241, 24	syl 18	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ ℕ)
243	238, 242	nndivred 12307	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℝ)
244		peano2rem 11542	. . . . . . . 8 ⊢ ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℝ → ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) ∈ ℝ)
245	243, 244	syl 18	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) ∈ ℝ)
246		reex 11208	. . . . . . . . . . . 12 ⊢ ℝ ∈ V
247	246, 14	ssexi 5295	. . . . . . . . . . 11 ⊢ ℝ⁺ ∈ V
248	247	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ℝ⁺ ∈ V)
249	231, 230	nndivred 12307	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
250	249	recnd 11254	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
251	229, 250	fsumcl 15809	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
252		relogcl 26793	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
253	252	adantl 487	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
254	253	recnd 11254	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
255	251, 254	subcld 11586	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℂ)
256	230	nnrecred 12304	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
257	229, 256	fsumrecl 15810	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
258	257, 253	resubcld 11659	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ∈ ℝ)
259		eqidd 2766	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))))
260		eqidd 2766	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))))
261	248, 255, 258, 259, 260	offval2 7704	. . . . . . . . 9 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))))
262	256	recnd 11254	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
263	229, 250, 262	fsumsub 15864	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) − (1 / 𝑛)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)))
264	231	recnd 11254	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
265		1cnd 11219	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
266	230	nncnd 12266	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
267	230	nnne0d 12303	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
268	264, 265, 266, 267	divsubdird 12047	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) − 1) / 𝑛) = (((Λ‘𝑛) / 𝑛) − (1 / 𝑛)))
269	268	sumeq2dv 15779	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) − (1 / 𝑛)))
270	257	recnd 11254	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℂ)
271	251, 270, 254	nnncan2d 11621	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)))
272	263, 269, 271	3eqtr4rd 2811	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛))
273	272	mpteq2dva 5206	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛)))
274	261, 273	eqtrd 2800	. . . . . . . 8 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛)))
275		vmadivsum 27699	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)
276	14	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
277	258	recnd 11254	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ∈ ℂ)
278		1red 11226	. . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℝ)
279		harmoniclbnd 27226	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛))
280	279	adantl 487	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛))
281	253, 257, 280	abssubge0d 15511	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))
282	281	adantrr 730	. . . . . . . . . . 11 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))
283	235	ad2antrl 741	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
284		simprr 785	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
285		harmonicubnd 27227	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
286	283, 284, 285	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
287		1red 11226	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℝ)
288	257, 253, 287	lesubadd2d 11830	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ≤ 1 ↔ Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1)))
289	288	adantrr 730	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ≤ 1 ↔ Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1)))
290	286, 289	mpbird 260	. . . . . . . . . . 11 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ≤ 1)
291	282, 290	eqbrtrd 5135	. . . . . . . . . 10 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) ≤ 1)
292	276, 277, 278, 278, 291	elo1d 15613	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
293		o1sub 15693	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1) ∧ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)) → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))) ∈ 𝑂(1))
294	275, 292, 293	sylancr 599	. . . . . . . 8 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))) ∈ 𝑂(1))
295	274, 294	eqeltrrd 2866	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛)) ∈ 𝑂(1))
296	243	recnd 11254	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℂ)
297		1cnd 11219	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
298	237	recnd 11254	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (ψ‘𝑥) ∈ ℂ)
299		rpcnne0 13053	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
300	299	adantl 487	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
301		divdir 11914	. . . . . . . . . . . 12 ⊢ (((ψ‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((ψ‘𝑥) + 1) / 𝑥) = (((ψ‘𝑥) / 𝑥) + (1 / 𝑥)))
302	298, 297, 300, 301	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) = (((ψ‘𝑥) / 𝑥) + (1 / 𝑥)))
303	302	mpteq2dva 5206	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) + 1) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) + (1 / 𝑥))))
304		simpr 490	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
305	237, 304	rerpdivcld 13109	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
306		rpreccl 13062	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
307	306	adantl 487	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
308		eqidd 2766	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)))
309		eqidd 2766	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
310	248, 305, 307, 308, 309	offval2 7704	. . . . . . . . . . 11 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∘_f + (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) + (1 / 𝑥))))
311		chpo1ub 27697	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
312		divrcnv 15931	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0)
313	28, 312	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0
314		rlimo1 15694	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0 → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ∈ 𝑂(1))
315	313, 314	mp1i 14	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ∈ 𝑂(1))
316		o1add 15691	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1) ∧ (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ∈ 𝑂(1)) → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∘_f + (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))) ∈ 𝑂(1))
317	311, 315, 316	sylancr 599	. . . . . . . . . . 11 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∘_f + (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))) ∈ 𝑂(1))
318	310, 317	eqeltrrd 2866	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) + (1 / 𝑥))) ∈ 𝑂(1))
319	303, 318	eqeltrd 2865	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) + 1) / 𝑥)) ∈ 𝑂(1))
320	238, 304	rerpdivcld 13109	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) ∈ ℝ)
321		chpge0 27343	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → 0 ≤ (ψ‘𝑥))
322	236, 321	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ (ψ‘𝑥))
323	237, 322	ge0p1rpd 13108	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) + 1) ∈ ℝ⁺)
324	323	rprege0d 13085	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) ∈ ℝ ∧ 0 ≤ ((ψ‘𝑥) + 1)))
325	242	nnrpd 13076	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ ℝ⁺)
326	325	rpregt0d 13084	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((⌊‘𝑥) + 1) ∈ ℝ ∧ 0 < ((⌊‘𝑥) + 1)))
327		divge0 12101	. . . . . . . . . . . . 13 ⊢ (((((ψ‘𝑥) + 1) ∈ ℝ ∧ 0 ≤ ((ψ‘𝑥) + 1)) ∧ (((⌊‘𝑥) + 1) ∈ ℝ ∧ 0 < ((⌊‘𝑥) + 1))) → 0 ≤ (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)))
328	324, 326, 327	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)))
329	243, 328	absidd 15500	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘(((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1))) = (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)))
330	320	recnd 11254	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) ∈ ℂ)
331	330	abscld 15516	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘(((ψ‘𝑥) + 1) / 𝑥)) ∈ ℝ)
332		fllep1 13854	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
333	236, 332	syl 18	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≤ ((⌊‘𝑥) + 1))
334		rpregt0 13049	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
335	334	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
336	323	rpregt0d 13084	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) ∈ ℝ ∧ 0 < ((ψ‘𝑥) + 1)))
337		lediv2 12122	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ (((⌊‘𝑥) + 1) ∈ ℝ ∧ 0 < ((⌊‘𝑥) + 1)) ∧ (((ψ‘𝑥) + 1) ∈ ℝ ∧ 0 < ((ψ‘𝑥) + 1))) → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ≤ (((ψ‘𝑥) + 1) / 𝑥)))
338	335, 326, 336, 337	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ≤ (((ψ‘𝑥) + 1) / 𝑥)))
339	333, 338	mpbid 235	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ≤ (((ψ‘𝑥) + 1) / 𝑥))
340	320	leabsd 15492	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) ≤ (abs‘(((ψ‘𝑥) + 1) / 𝑥)))
341	243, 320, 331, 339, 340	letrd 11384	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ≤ (abs‘(((ψ‘𝑥) + 1) / 𝑥)))
342	329, 341	eqbrtrd 5135	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘(((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1))) ≤ (abs‘(((ψ‘𝑥) + 1) / 𝑥)))
343	342	adantrr 730	. . . . . . . . 9 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1))) ≤ (abs‘(((ψ‘𝑥) + 1) / 𝑥)))
344	278, 319, 320, 296, 343	o1le 15730	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1))) ∈ 𝑂(1))
345		o1const 15697	. . . . . . . . . 10 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
346	14, 28, 345	mp2an 705	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1)
347	346	a1i 11	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
348	296, 297, 344, 347	o1sub2 15703	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1)) ∈ 𝑂(1))
349	234, 245, 295, 348	o1sub2 15703	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) − ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))) ∈ 𝑂(1))
350	13, 349	o1res2 15640	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) − ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))) ∈ 𝑂(1))
351	228, 350	eqeltrrid 2870	. . . 4 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1))
352	16, 351	eqeltrd 2865	. . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ↾ (1[,)+∞)) ∈ 𝑂(1))
353		eqid 2765	. . . . . 6 ⊢ (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
354	353, 45	fmpti 7111	. . . . 5 ⊢ (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))):ℝ⟶ℂ
355	354	a1i 11	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))):ℝ⟶ℂ)
356		ssidd 3961	. . . 4 ⊢ (⊤ → ℝ ⊆ ℝ)
357	355, 356, 278	o1resb 15643	. . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1) ↔ ((𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ↾ (1[,)+∞)) ∈ 𝑂(1)))
358	352, 357	mpbird 260	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1))
359	358	mptru 1577	1 ⊢ (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 = wceq 1570 ⊤wtru 1571 ∈ wcel 2146 ≠ wne 2960 Vcvv 3457 ⊆ wss 3906 class class class wbr 5111 ↦ cmpt 5194 ↾ cres 5665 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ∘_f cof 7682 ℂcc 11115 ℝcr 11116 0cc0 11117 1c1 11118 + caddc 11120 · cmul 11122 +∞cpnf 11257 < clt 11260 ≤ cle 11261 − cmin 11458 -cneg 11459 / cdiv 11888 ℕcn 12250 2c2 12312 ℕ₀cn0 12521 ℤcz 12608 ℤ_≥cuz 12880 ℝ⁺crp 13034 [,)cico 13392 ...cfz 13553 ..^cfzo 13701 ⌊cfl 13843 abscabs 15311 ⇝_𝑟 crli 15562 𝑂(1)co1 15563 Σcsu 15763 logclog 26772 Λcvma 27309 ψcchp 27310

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-pre-sup 11195 ax-addf 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-dju 9903 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-div 11889 df-nn 12251 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12522 df-xnn0 12595 df-z 12609 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13394 df-ioc 13395 df-ico 13396 df-icc 13397 df-fz 13554 df-fzo 13702 df-fl 13845 df-mod 13923 df-seq 14058 df-exp 14118 df-fac 14330 df-bc 14359 df-hash 14387 df-shft 15130 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-limsup 15548 df-clim 15565 df-rlim 15566 df-o1 15567 df-lo1 15568 df-sum 15764 df-ef 16145 df-e 16146 df-sin 16147 df-cos 16148 df-tan 16149 df-pi 16150 df-dvds 16335 df-gcd 16577 df-prm 16754 df-pc 16921 df-struct 17231 df-sets 17248 df-slot 17266 df-ndx 17278 df-base 17294 df-ress 17315 df-plusg 17347 df-mulr 17348 df-starv 17349 df-sca 17350 df-vsca 17351 df-ip 17352 df-tset 17353 df-ple 17354 df-ds 17356 df-unif 17357 df-hom 17358 df-cco 17359 df-rest 17499 df-topn 17500 df-0g 17518 df-gsum 17519 df-topgen 17520 df-pt 17521 df-prds 17524 df-xrs 17580 df-qtop 17585 df-imas 17586 df-xps 17588 df-mre 17662 df-mrc 17663 df-acs 17665 df-mgm 18722 df-sgrp 18811 df-mnd 18827 df-submnd 18881 df-mulg 19180 df-cntz 19433 df-cmn 19898 df-psmet 21566 df-xmet 21567 df-met 21568 df-bl 21569 df-mopn 21570 df-fbas 21571 df-fg 21572 df-cnfld 21575 df-top 23103 df-topon 23120 df-topsp 23142 df-bases 23155 df-cld 23228 df-ntr 23229 df-cls 23230 df-nei 23307 df-lp 23345 df-perf 23346 df-cn 23436 df-cnp 23437 df-haus 23524 df-cmp 23596 df-tx 23772 df-hmeo 23965 df-fil 24056 df-fm 24148 df-flim 24149 df-flf 24150 df-xms 24530 df-ms 24531 df-tms 24532 df-cncf 25090 df-limc 26078 df-dv 26079 df-ulm 26593 df-log 26774 df-cxp 26775 df-atan 27085 df-em 27210 df-cht 27314 df-vma 27315 df-chp 27316 df-ppi 27317

This theorem is used by: pntrsumbnd 27783

W3C validator