Theorem dchrisum0fno1 27247

Description: The sum Σ𝑘 ≤ 𝑥, 𝐹(𝑥) / √𝑘 is divergent (i.e. not eventually bounded). Equation 9.4.30 of [Shapiro], p. 383. (Contributed by Mario Carneiro, 5-May-2016.)

Hypotheses

Ref	Expression
rpvmasum.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
rpvmasum.l	⊢ 𝐿 = (ℤRHom‘𝑍)
rpvmasum.a	⊢ (𝜑 → 𝑁 ∈ ℕ)
rpvmasum2.g	⊢ 𝐺 = (DChr‘𝑁)
rpvmasum2.d	⊢ 𝐷 = (Base‘𝐺)
rpvmasum2.1	⊢ 1 = (0g‘𝐺)
dchrisum0f.f	⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣)))
dchrisum0f.x	⊢ (𝜑 → 𝑋 ∈ 𝐷)
dchrisum0flb.r	⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)
dchrisum0fno1.a	⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘))) ∈ 𝑂(1))

Assertion

Ref	Expression
dchrisum0fno1	⊢ ¬ 𝜑

Distinct variable groups:   𝑥,𝑘, 1   𝑘,𝐹,𝑥   𝑘,𝑏,𝑞,𝑣,𝑥   𝑘,𝑁,𝑞,𝑥   𝜑,𝑘,𝑥   𝑘,𝑍,𝑥   𝐷,𝑘,𝑥   𝐿,𝑏,𝑘,𝑣,𝑥   𝑋,𝑏,𝑘,𝑣,𝑥

Allowed substitution hints:   𝜑(𝑣,𝑞,𝑏)   𝐷(𝑣,𝑞,𝑏)   1 (𝑣,𝑞,𝑏)   𝐹(𝑣,𝑞,𝑏)   𝐺(𝑥,𝑣,𝑘,𝑞,𝑏)   𝐿(𝑞)   𝑁(𝑣,𝑏)   𝑋(𝑞)   𝑍(𝑣,𝑞,𝑏)

Proof of Theorem dchrisum0fno1

Dummy variables 𝑚 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		logno1 26377	. 2 ⊢ ¬ (𝑥 ∈ ℝ+ ↦ (log‘𝑥)) ∈ 𝑂(1)
2		relogcl 26317	. . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
3	2	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ)
4	3	recnd 11247	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ)
5		2cnd 12295	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 2 ∈ ℂ)
6		2ne0 12321	. . . . . 6 ⊢ 2 ≠ 0
7	6	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 2 ≠ 0)
8	4, 5, 7	divcan2d 11997	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (2 · ((log‘𝑥) / 2)) = (log‘𝑥))
9	8	mpteq2dva 5249	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (2 · ((log‘𝑥) / 2))) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥)))
10	3	rehalfcld 12464	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥) / 2) ∈ ℝ)
11	10	recnd 11247	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥) / 2) ∈ ℂ)
12		rpssre 12986	. . . . . 6 ⊢ ℝ+ ⊆ ℝ
13		2cn 12292	. . . . . 6 ⊢ 2 ∈ ℂ
14		o1const 15569	. . . . . 6 ⊢ ((ℝ+ ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ ℝ+ ↦ 2) ∈ 𝑂(1))
15	12, 13, 14	mp2an 689	. . . . 5 ⊢ (𝑥 ∈ ℝ+ ↦ 2) ∈ 𝑂(1)
16	15	a1i 11	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ 2) ∈ 𝑂(1))
17		1red 11220	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
18		dchrisum0fno1.a	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘))) ∈ 𝑂(1))
19		sumex 15639	. . . . . 6 ⊢ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘)) ∈ V
20	19	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘)) ∈ V)
21	10	adantrr 714	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) / 2) ∈ ℝ)
22	2	ad2antrl 725	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
23		log1 26327	. . . . . . . . 9 ⊢ (log‘1) = 0
24		simprr 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
25		1rp 12983	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ+
26		simprl 768	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ+)
27		logleb 26344	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
28	25, 26, 27	sylancr 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
29	24, 28	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘1) ≤ (log‘𝑥))
30	23, 29	eqbrtrrid 5185	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ (log‘𝑥))
31		2re 12291	. . . . . . . . 9 ⊢ 2 ∈ ℝ
32	31	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 2 ∈ ℝ)
33		2pos 12320	. . . . . . . . 9 ⊢ 0 < 2
34	33	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 < 2)
35		divge0 12088	. . . . . . . 8 ⊢ ((((log‘𝑥) ∈ ℝ ∧ 0 ≤ (log‘𝑥)) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 ≤ ((log‘𝑥) / 2))
36	22, 30, 32, 34, 35	syl22anc 836	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘𝑥) / 2))
37	21, 36	absidd 15374	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘((log‘𝑥) / 2)) = ((log‘𝑥) / 2))
38		fzfid 13943	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
39		rpvmasum.z	. . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
40		rpvmasum.l	. . . . . . . . . . . 12 ⊢ 𝐿 = (ℤRHom‘𝑍)
41		rpvmasum.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
42		rpvmasum2.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (DChr‘𝑁)
43		rpvmasum2.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (Base‘𝐺)
44		rpvmasum2.1	. . . . . . . . . . . 12 ⊢ 1 = (0g‘𝐺)
45		dchrisum0f.f	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣)))
46		dchrisum0f.x	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
47		dchrisum0flb.r	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)
48	39, 40, 41, 42, 43, 44, 45, 46, 47	dchrisum0ff 27243	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
49	48	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝐹:ℕ⟶ℝ)
50		elfznn 13535	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(⌊‘𝑥)) → 𝑘 ∈ ℕ)
51		ffvelcdm 7084	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶ℝ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
52	49, 50, 51	syl2an 595	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (𝐹‘𝑘) ∈ ℝ)
53	50	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℕ)
54	53	nnrpd 13019	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℝ+)
55	54	rpsqrtcld 15363	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (√‘𝑘) ∈ ℝ+)
56	52, 55	rerpdivcld 13052	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((𝐹‘𝑘) / (√‘𝑘)) ∈ ℝ)
57	38, 56	fsumrecl 15685	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘)) ∈ ℝ)
58	57	recnd 11247	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘)) ∈ ℂ)
59	58	abscld 15388	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘))) ∈ ℝ)
60		fzfid 13943	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (1...(⌊‘(√‘𝑥))) ∈ Fin)
61		elfznn 13535	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(⌊‘(√‘𝑥))) → 𝑖 ∈ ℕ)
62	61	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑖 ∈ ℕ)
63	62	nnrecred 12268	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → (1 / 𝑖) ∈ ℝ)
64	60, 63	fsumrecl 15685	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖) ∈ ℝ)
65		logsqrt 26445	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → (log‘(√‘𝑥)) = ((log‘𝑥) / 2))
66	65	ad2antrl 725	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘(√‘𝑥)) = ((log‘𝑥) / 2))
67		rpsqrtcl 15216	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → (√‘𝑥) ∈ ℝ+)
68	67	ad2antrl 725	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (√‘𝑥) ∈ ℝ+)
69		harmoniclbnd 26746	. . . . . . . . . 10 ⊢ ((√‘𝑥) ∈ ℝ+ → (log‘(√‘𝑥)) ≤ Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖))
70	68, 69	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘(√‘𝑥)) ≤ Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖))
71	66, 70	eqbrtrrd 5173	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) / 2) ≤ Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖))
72		eqid 2731	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) = (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))
73		ovex 7445	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚↑2) ∈ V
74	72, 73	elrnmpti 5960	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) ↔ ∃𝑚 ∈ (1...(⌊‘(√‘𝑥)))𝑘 = (𝑚↑2))
75		elfznn 13535	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ (1...(⌊‘(√‘𝑥))) → 𝑚 ∈ ℕ)
76	75	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑚 ∈ ℕ)
77	76	nnrpd 13019	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑚 ∈ ℝ+)
78	77	rprege0d 13028	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚 ∈ ℝ ∧ 0 ≤ 𝑚))
79		sqrtsq 15221	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) → (√‘(𝑚↑2)) = 𝑚)
80	78, 79	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (√‘(𝑚↑2)) = 𝑚)
81	80, 76	eqeltrd 2832	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (√‘(𝑚↑2)) ∈ ℕ)
82		fveq2 6892	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑚↑2) → (√‘𝑘) = (√‘(𝑚↑2)))
83	82	eleq1d 2817	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑚↑2) → ((√‘𝑘) ∈ ℕ ↔ (√‘(𝑚↑2)) ∈ ℕ))
84	81, 83	syl5ibrcom 246	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑘 = (𝑚↑2) → (√‘𝑘) ∈ ℕ))
85	84	rexlimdva 3154	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (∃𝑚 ∈ (1...(⌊‘(√‘𝑥)))𝑘 = (𝑚↑2) → (√‘𝑘) ∈ ℕ))
86	74, 85	biimtrid 241	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) → (√‘𝑘) ∈ ℕ))
87	86	imp 406	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → (√‘𝑘) ∈ ℕ)
88	87	iftrued 4537	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → if((√‘𝑘) ∈ ℕ, 1, 0) = 1)
89	88	oveq1d 7427	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) = (1 / (√‘𝑘)))
90	89	sumeq2dv 15654	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))(if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) = Σ𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))(1 / (√‘𝑘)))
91		fveq2 6892	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑖↑2) → (√‘𝑘) = (√‘(𝑖↑2)))
92	91	oveq2d 7428	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑖↑2) → (1 / (√‘𝑘)) = (1 / (√‘(𝑖↑2))))
93	76	nnsqcld 14212	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚↑2) ∈ ℕ)
94	68	rpred 13021	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (√‘𝑥) ∈ ℝ)
95		fznnfl 13832	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((√‘𝑥) ∈ ℝ → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↔ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (√‘𝑥))))
96	94, 95	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↔ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (√‘𝑥))))
97	96	simplbda 499	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑚 ≤ (√‘𝑥))
98	68	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (√‘𝑥) ∈ ℝ+)
99	98	rprege0d 13028	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → ((√‘𝑥) ∈ ℝ ∧ 0 ≤ (√‘𝑥)))
100		le2sq 14104	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) ∧ ((√‘𝑥) ∈ ℝ ∧ 0 ≤ (√‘𝑥))) → (𝑚 ≤ (√‘𝑥) ↔ (𝑚↑2) ≤ ((√‘𝑥)↑2)))
101	78, 99, 100	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚 ≤ (√‘𝑥) ↔ (𝑚↑2) ≤ ((√‘𝑥)↑2)))
102	97, 101	mpbid 231	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚↑2) ≤ ((√‘𝑥)↑2))
103	26	rpred 13021	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
104	103	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑥 ∈ ℝ)
105	104	recnd 11247	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑥 ∈ ℂ)
106	105	sqsqrtd 15391	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → ((√‘𝑥)↑2) = 𝑥)
107	102, 106	breqtrd 5175	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚↑2) ≤ 𝑥)
108		fznnfl 13832	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ → ((𝑚↑2) ∈ (1...(⌊‘𝑥)) ↔ ((𝑚↑2) ∈ ℕ ∧ (𝑚↑2) ≤ 𝑥)))
109	104, 108	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → ((𝑚↑2) ∈ (1...(⌊‘𝑥)) ↔ ((𝑚↑2) ∈ ℕ ∧ (𝑚↑2) ≤ 𝑥)))
110	93, 107, 109	mpbir2and 710	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚↑2) ∈ (1...(⌊‘𝑥)))
111	110	ex 412	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) → (𝑚↑2) ∈ (1...(⌊‘𝑥))))
112	75	nnrpd 13019	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (1...(⌊‘(√‘𝑥))) → 𝑚 ∈ ℝ+)
113	112	rprege0d 13028	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1...(⌊‘(√‘𝑥))) → (𝑚 ∈ ℝ ∧ 0 ≤ 𝑚))
114	61	nnrpd 13019	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...(⌊‘(√‘𝑥))) → 𝑖 ∈ ℝ+)
115	114	rprege0d 13028	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (1...(⌊‘(√‘𝑥))) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖))
116		sq11 14101	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) ∧ (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖)) → ((𝑚↑2) = (𝑖↑2) ↔ 𝑚 = 𝑖))
117	113, 115, 116	syl2an 595	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → ((𝑚↑2) = (𝑖↑2) ↔ 𝑚 = 𝑖))
118	117	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → ((𝑚↑2) = (𝑖↑2) ↔ 𝑚 = 𝑖)))
119	111, 118	dom2lem 8991	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))–1-1→(1...(⌊‘𝑥)))
120		f1f1orn 6845	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))–1-1→(1...(⌊‘𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))–1-1-onto→ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))
121	119, 120	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))–1-1-onto→ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))
122		oveq1 7419	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑖 → (𝑚↑2) = (𝑖↑2))
123	122, 72, 73	fvmpt3i 7004	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...(⌊‘(√‘𝑥))) → ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))‘𝑖) = (𝑖↑2))
124	123	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))‘𝑖) = (𝑖↑2))
125		f1f 6788	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))–1-1→(1...(⌊‘𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))⟶(1...(⌊‘𝑥)))
126		frn 6725	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))⟶(1...(⌊‘𝑥)) → ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) ⊆ (1...(⌊‘𝑥)))
127	119, 125, 126	3syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) ⊆ (1...(⌊‘𝑥)))
128	127	sselda 3983	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → 𝑘 ∈ (1...(⌊‘𝑥)))
129		1re 11219	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
130		0re 11221	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
131	129, 130	ifcli 4576	. . . . . . . . . . . . . . . 16 ⊢ if((√‘𝑘) ∈ ℕ, 1, 0) ∈ ℝ
132		rerpdivcl 13009	. . . . . . . . . . . . . . . 16 ⊢ ((if((√‘𝑘) ∈ ℕ, 1, 0) ∈ ℝ ∧ (√‘𝑘) ∈ ℝ+) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℝ)
133	131, 55, 132	sylancr 586	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℝ)
134	133	recnd 11247	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℂ)
135	128, 134	syldan 590	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℂ)
136	89, 135	eqeltrrd 2833	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → (1 / (√‘𝑘)) ∈ ℂ)
137	92, 60, 121, 124, 136	fsumf1o 15674	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))(1 / (√‘𝑘)) = Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / (√‘(𝑖↑2))))
138	90, 137	eqtrd 2771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))(if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) = Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / (√‘(𝑖↑2))))
139		eldif 3959	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) ↔ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ ¬ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))))
140	50	ad2antrl 725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ∈ ℕ)
141	140	nncnd 12233	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ∈ ℂ)
142	141	sqsqrtd 15391	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → ((√‘𝑘)↑2) = 𝑘)
143		simprr 770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑘) ∈ ℕ)
144		fznnfl 13832	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ ℝ → (𝑘 ∈ (1...(⌊‘𝑥)) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≤ 𝑥)))
145	103, 144	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑘 ∈ (1...(⌊‘𝑥)) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≤ 𝑥)))
146	145	simplbda 499	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ≤ 𝑥)
147	146	adantrr 714	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ≤ 𝑥)
148	140	nnrpd 13019	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ∈ ℝ+)
149	148	rprege0d 13028	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (𝑘 ∈ ℝ ∧ 0 ≤ 𝑘))
150	26	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑥 ∈ ℝ+)
151	150	rprege0d 13028	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
152		sqrtle 15212	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (𝑘 ≤ 𝑥 ↔ (√‘𝑘) ≤ (√‘𝑥)))
153	149, 151, 152	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (𝑘 ≤ 𝑥 ↔ (√‘𝑘) ≤ (√‘𝑥)))
154	147, 153	mpbid 231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑘) ≤ (√‘𝑥))
155	68	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑥) ∈ ℝ+)
156	155	rpred 13021	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑥) ∈ ℝ)
157		fznnfl 13832	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((√‘𝑥) ∈ ℝ → ((√‘𝑘) ∈ (1...(⌊‘(√‘𝑥))) ↔ ((√‘𝑘) ∈ ℕ ∧ (√‘𝑘) ≤ (√‘𝑥))))
158	156, 157	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → ((√‘𝑘) ∈ (1...(⌊‘(√‘𝑥))) ↔ ((√‘𝑘) ∈ ℕ ∧ (√‘𝑘) ≤ (√‘𝑥))))
159	143, 154, 158	mpbir2and 710	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑘) ∈ (1...(⌊‘(√‘𝑥))))
160	142, 140	eqeltrd 2832	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → ((√‘𝑘)↑2) ∈ ℕ)
161		oveq1 7419	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = (√‘𝑘) → (𝑚↑2) = ((√‘𝑘)↑2))
162	72, 161	elrnmpt1s 5957	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((√‘𝑘) ∈ (1...(⌊‘(√‘𝑥))) ∧ ((√‘𝑘)↑2) ∈ ℕ) → ((√‘𝑘)↑2) ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))
163	159, 160, 162	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → ((√‘𝑘)↑2) ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))
164	142, 163	eqeltrrd 2833	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))
165	164	expr 456	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((√‘𝑘) ∈ ℕ → 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))))
166	165	con3d 152	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (¬ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) → ¬ (√‘𝑘) ∈ ℕ))
167	166	impr 454	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ ¬ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → ¬ (√‘𝑘) ∈ ℕ)
168	139, 167	sylan2b 593	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → ¬ (√‘𝑘) ∈ ℕ)
169	168	iffalsed 4540	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → if((√‘𝑘) ∈ ℕ, 1, 0) = 0)
170	169	oveq1d 7427	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) = (0 / (√‘𝑘)))
171		eldifi 4127	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → 𝑘 ∈ (1...(⌊‘𝑥)))
172	171, 55	sylan2 592	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → (√‘𝑘) ∈ ℝ+)
173	172	rpcnne0d 13030	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → ((√‘𝑘) ∈ ℂ ∧ (√‘𝑘) ≠ 0))
174		div0 11907	. . . . . . . . . . . . 13 ⊢ (((√‘𝑘) ∈ ℂ ∧ (√‘𝑘) ≠ 0) → (0 / (√‘𝑘)) = 0)
175	173, 174	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → (0 / (√‘𝑘)) = 0)
176	170, 175	eqtrd 2771	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) = 0)
177	127, 135, 176, 38	fsumss 15676	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))(if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) = Σ𝑘 ∈ (1...(⌊‘𝑥))(if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)))
178	62	nnrpd 13019	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑖 ∈ ℝ+)
179	178	rprege0d 13028	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖))
180		sqrtsq 15221	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ ℝ ∧ 0 ≤ 𝑖) → (√‘(𝑖↑2)) = 𝑖)
181	179, 180	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → (√‘(𝑖↑2)) = 𝑖)
182	181	oveq2d 7428	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → (1 / (√‘(𝑖↑2))) = (1 / 𝑖))
183	182	sumeq2dv 15654	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / (√‘(𝑖↑2))) = Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖))
184	138, 177, 183	3eqtr3d 2779	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))(if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) = Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖))
185	131	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → if((√‘𝑘) ∈ ℕ, 1, 0) ∈ ℝ)
186	41	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑁 ∈ ℕ)
187	46	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
188	47	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑋:(Base‘𝑍)⟶ℝ)
189	39, 40, 186, 42, 43, 44, 45, 187, 188, 53	dchrisum0flb 27246	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → if((√‘𝑘) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑘))
190	185, 52, 55, 189	lediv1dd 13079	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ≤ ((𝐹‘𝑘) / (√‘𝑘)))
191	38, 133, 56, 190	fsumle 15750	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))(if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ≤ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘)))
192	184, 191	eqbrtrrd 5173	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖) ≤ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘)))
193	21, 64, 57, 71, 192	letrd 11376	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) / 2) ≤ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘)))
194	57	leabsd 15366	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘)) ≤ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘))))
195	21, 57, 59, 193, 194	letrd 11376	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) / 2) ≤ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘))))
196	37, 195	eqbrtrd 5171	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘((log‘𝑥) / 2)) ≤ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘))))
197	17, 18, 20, 11, 196	o1le 15604	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ ((log‘𝑥) / 2)) ∈ 𝑂(1))
198	5, 11, 16, 197	o1mul2 15574	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (2 · ((log‘𝑥) / 2))) ∈ 𝑂(1))
199	9, 198	eqeltrrd 2833	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (log‘𝑥)) ∈ 𝑂(1))
200	1, 199	mto 196	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ∃wrex 3069  {crab 3431  Vcvv 3473   ∖ cdif 3946   ⊆ wss 3949  ifcif 4529   class class class wbr 5149   ↦ cmpt 5232  ran crn 5678  ⟶wf 6540  –1-1→wf1 6541  –1-1-onto→wf1o 6543  ‘cfv 6544  (class class class)co 7412  ℂcc 11111  ℝcr 11112  0cc0 11113  1c1 11114   · cmul 11118   < clt 11253   ≤ cle 11254   / cdiv 11876  ℕcn 12217  2c2 12272  ℝ⁺crp 12979  ...cfz 13489  ⌊cfl 13760  ↑cexp 14032  √csqrt 15185  abscabs 15186  𝑂(1)co1 15435  Σcsu 15637   ∥ cdvds 16202  Basecbs 17149  0_gc0g 17390  ℤRHomczrh 21269  ℤ/nℤczn 21272  logclog 26296  DChrcdchr 26968

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-inf2 9639  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190  ax-pre-sup 11191  ax-addf 11192  ax-mulf 11193

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-disj 5115  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7673  df-om 7859  df-1st 7978  df-2nd 7979  df-supp 8150  df-tpos 8214  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-2o 8470  df-oadd 8473  df-omul 8474  df-er 8706  df-ec 8708  df-qs 8712  df-map 8825  df-pm 8826  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fsupp 9365  df-fi 9409  df-sup 9440  df-inf 9441  df-oi 9508  df-card 9937  df-acn 9940  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12478  df-xnn0 12550  df-z 12564  df-dec 12683  df-uz 12828  df-q 12938  df-rp 12980  df-xneg 13097  df-xadd 13098  df-xmul 13099  df-ioo 13333  df-ioc 13334  df-ico 13335  df-icc 13336  df-fz 13490  df-fzo 13633  df-fl 13762  df-mod 13840  df-seq 13972  df-exp 14033  df-fac 14239  df-bc 14268  df-hash 14296  df-shft 15019  df-cj 15051  df-re 15052  df-im 15053  df-sqrt 15187  df-abs 15188  df-limsup 15420  df-clim 15437  df-rlim 15438  df-o1 15439  df-lo1 15440  df-sum 15638  df-ef 16016  df-e 16017  df-sin 16018  df-cos 16019  df-tan 16020  df-pi 16021  df-dvds 16203  df-gcd 16441  df-prm 16614  df-numer 16676  df-denom 16677  df-pc 16775  df-struct 17085  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-plusg 17215  df-mulr 17216  df-starv 17217  df-sca 17218  df-vsca 17219  df-ip 17220  df-tset 17221  df-ple 17222  df-ds 17224  df-unif 17225  df-hom 17226  df-cco 17227  df-rest 17373  df-topn 17374  df-0g 17392  df-gsum 17393  df-topgen 17394  df-pt 17395  df-prds 17398  df-xrs 17453  df-qtop 17458  df-imas 17459  df-qus 17460  df-xps 17461  df-mre 17535  df-mrc 17536  df-acs 17538  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-mhm 18706  df-submnd 18707  df-grp 18859  df-minusg 18860  df-sbg 18861  df-mulg 18988  df-subg 19040  df-nsg 19041  df-eqg 19042  df-ghm 19129  df-cntz 19223  df-od 19438  df-cmn 19692  df-abl 19693  df-mgp 20030  df-rng 20048  df-ur 20077  df-ring 20130  df-cring 20131  df-oppr 20226  df-dvdsr 20249  df-unit 20250  df-invr 20280  df-dvr 20293  df-rhm 20364  df-subrng 20435  df-subrg 20460  df-drng 20503  df-lmod 20617  df-lss 20688  df-lsp 20728  df-sra 20931  df-rgmod 20932  df-lidl 20933  df-rsp 20934  df-2idl 21007  df-psmet 21137  df-xmet 21138  df-met 21139  df-bl 21140  df-mopn 21141  df-fbas 21142  df-fg 21143  df-cnfld 21146  df-zring 21219  df-zrh 21273  df-zn 21276  df-top 22617  df-topon 22634  df-topsp 22656  df-bases 22670  df-cld 22744  df-ntr 22745  df-cls 22746  df-nei 22823  df-lp 22861  df-perf 22862  df-cn 22952  df-cnp 22953  df-haus 23040  df-cmp 23112  df-tx 23287  df-hmeo 23480  df-fil 23571  df-fm 23663  df-flim 23664  df-flf 23665  df-xms 24047  df-ms 24048  df-tms 24049  df-cncf 24619  df-limc 25616  df-dv 25617  df-ulm 26122  df-log 26298  df-cxp 26299  df-atan 26605  df-em 26730  df-dchr 26969

This theorem is referenced by:  dchrisum0  27256