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Theorem dchrisum0fno1 27555

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 = (0_g‘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 26678	. 2 ⊢ ¬ (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1)
2		relogcl 26617	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
3	2	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
4	3	recnd 11289	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
5		2cnd 12344	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
6		2ne0 12370	. . . . . 6 ⊢ 2 ≠ 0
7	6	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ≠ 0)
8	4, 5, 7	divcan2d 12045	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · ((log‘𝑥) / 2)) = (log‘𝑥))
9	8	mpteq2dva 5242	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (2 · ((log‘𝑥) / 2))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
10	3	rehalfcld 12513	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / 2) ∈ ℝ)
11	10	recnd 11289	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / 2) ∈ ℂ)
12		rpssre 13042	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
13		2cn 12341	. . . . . 6 ⊢ 2 ∈ ℂ
14		o1const 15656	. . . . . 6 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1))
15	12, 13, 14	mp2an 692	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1)
16	15	a1i 11	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1))
17		1red 11262	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
18		dchrisum0fno1.a	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘))) ∈ 𝑂(1))
19		sumex 15724	. . . . . 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 717	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) / 2) ∈ ℝ)
22	2	ad2antrl 728	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
23		log1 26627	. . . . . . . . 9 ⊢ (log‘1) = 0
24		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
25		1rp 13038	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ⁺
26		simprl 771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ⁺)
27		logleb 26645	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
28	25, 26, 27	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
29	24, 28	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘1) ≤ (log‘𝑥))
30	23, 29	eqbrtrrid 5179	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ (log‘𝑥))
31		2re 12340	. . . . . . . . 9 ⊢ 2 ∈ ℝ
32	31	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 2 ∈ ℝ)
33		2pos 12369	. . . . . . . . 9 ⊢ 0 < 2
34	33	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 < 2)
35		divge0 12137	. . . . . . . 8 ⊢ ((((log‘𝑥) ∈ ℝ ∧ 0 ≤ (log‘𝑥)) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 ≤ ((log‘𝑥) / 2))
36	22, 30, 32, 34, 35	syl22anc 839	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘𝑥) / 2))
37	21, 36	absidd 15461	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘((log‘𝑥) / 2)) = ((log‘𝑥) / 2))
38		fzfid 14014	. . . . . . . 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 = (0_g‘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 27551	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
49	48	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝐹:ℕ⟶ℝ)
50		elfznn 13593	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(⌊‘𝑥)) → 𝑘 ∈ ℕ)
51		ffvelcdm 7101	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶ℝ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
52	49, 50, 51	syl2an 596	. . . . . . . . 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 13075	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℝ⁺)
55	54	rpsqrtcld 15450	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (√‘𝑘) ∈ ℝ⁺)
56	52, 55	rerpdivcld 13108	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((𝐹‘𝑘) / (√‘𝑘)) ∈ ℝ)
57	38, 56	fsumrecl 15770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘)) ∈ ℝ)
58	57	recnd 11289	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘)) ∈ ℂ)
59	58	abscld 15475	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘))) ∈ ℝ)
60		fzfid 14014	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1...(⌊‘(√‘𝑥))) ∈ Fin)
61		elfznn 13593	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(⌊‘(√‘𝑥))) → 𝑖 ∈ ℕ)
62	61	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑖 ∈ ℕ)
63	62	nnrecred 12317	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → (1 / 𝑖) ∈ ℝ)
64	60, 63	fsumrecl 15770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖) ∈ ℝ)
65		logsqrt 26746	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (log‘(√‘𝑥)) = ((log‘𝑥) / 2))
66	65	ad2antrl 728	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘(√‘𝑥)) = ((log‘𝑥) / 2))
67		rpsqrtcl 15303	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ∈ ℝ⁺)
68	67	ad2antrl 728	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (√‘𝑥) ∈ ℝ⁺)
69		harmoniclbnd 27052	. . . . . . . . . 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 5167	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) / 2) ≤ Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖))
72		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) = (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))
73		ovex 7464	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚↑2) ∈ V
74	72, 73	elrnmpti 5973	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) ↔ ∃𝑚 ∈ (1...(⌊‘(√‘𝑥)))𝑘 = (𝑚↑2))
75		elfznn 13593	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ (1...(⌊‘(√‘𝑥))) → 𝑚 ∈ ℕ)
76	75	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑚 ∈ ℕ)
77	76	nnrpd 13075	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑚 ∈ ℝ⁺)
78	77	rprege0d 13084	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚 ∈ ℝ ∧ 0 ≤ 𝑚))
79		sqrtsq 15308	. . . . . . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (√‘(𝑚↑2)) ∈ ℕ)
82		fveq2 6906	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑚↑2) → (√‘𝑘) = (√‘(𝑚↑2)))
83	82	eleq1d 2826	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑚↑2) → ((√‘𝑘) ∈ ℕ ↔ (√‘(𝑚↑2)) ∈ ℕ))
84	81, 83	syl5ibrcom 247	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑘 = (𝑚↑2) → (√‘𝑘) ∈ ℕ))
85	84	rexlimdva 3155	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (∃𝑚 ∈ (1...(⌊‘(√‘𝑥)))𝑘 = (𝑚↑2) → (√‘𝑘) ∈ ℕ))
86	74, 85	biimtrid 242	. . . . . . . . . . . . . . 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 4533	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → if((√‘𝑘) ∈ ℕ, 1, 0) = 1)
89	88	oveq1d 7446	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) = (1 / (√‘𝑘)))
90	89	sumeq2dv 15738	. . . . . . . . . . 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 6906	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑖↑2) → (√‘𝑘) = (√‘(𝑖↑2)))
92	91	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑖↑2) → (1 / (√‘𝑘)) = (1 / (√‘(𝑖↑2))))
93	76	nnsqcld 14283	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚↑2) ∈ ℕ)
94	68	rpred 13077	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (√‘𝑥) ∈ ℝ)
95		fznnfl 13902	. . . . . . . . . . . . . . . . . . . 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 13084	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → ((√‘𝑥) ∈ ℝ ∧ 0 ≤ (√‘𝑥)))
100		le2sq 14174	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) ∧ ((√‘𝑥) ∈ ℝ ∧ 0 ≤ (√‘𝑥))) → (𝑚 ≤ (√‘𝑥) ↔ (𝑚↑2) ≤ ((√‘𝑥)↑2)))
101	78, 99, 100	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚 ≤ (√‘𝑥) ↔ (𝑚↑2) ≤ ((√‘𝑥)↑2)))
102	97, 101	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚↑2) ≤ ((√‘𝑥)↑2))
103	26	rpred 13077	. . . . . . . . . . . . . . . . . . . 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 11289	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑥 ∈ ℂ)
106	105	sqsqrtd 15478	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → ((√‘𝑥)↑2) = 𝑥)
107	102, 106	breqtrd 5169	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚↑2) ≤ 𝑥)
108		fznnfl 13902	. . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . 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 13075	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (1...(⌊‘(√‘𝑥))) → 𝑚 ∈ ℝ⁺)
113	112	rprege0d 13084	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1...(⌊‘(√‘𝑥))) → (𝑚 ∈ ℝ ∧ 0 ≤ 𝑚))
114	61	nnrpd 13075	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...(⌊‘(√‘𝑥))) → 𝑖 ∈ ℝ⁺)
115	114	rprege0d 13084	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (1...(⌊‘(√‘𝑥))) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖))
116		sq11 14171	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) ∧ (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖)) → ((𝑚↑2) = (𝑖↑2) ↔ 𝑚 = 𝑖))
117	113, 115, 116	syl2an 596	. . . . . . . . . . . . . . 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 9032	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))–1-1→(1...(⌊‘𝑥)))
120		f1f1orn 6859	. . . . . . . . . . . . 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 7438	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑖 → (𝑚↑2) = (𝑖↑2))
123	122, 72, 73	fvmpt3i 7021	. . . . . . . . . . . . 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 6804	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))–1-1→(1...(⌊‘𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))⟶(1...(⌊‘𝑥)))
126		frn 6743	. . . . . . . . . . . . . . . 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 11261	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
130		0re 11263	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
131	129, 130	ifcli 4573	. . . . . . . . . . . . . . . 16 ⊢ if((√‘𝑘) ∈ ℕ, 1, 0) ∈ ℝ
132		rerpdivcl 13065	. . . . . . . . . . . . . . . 16 ⊢ ((if((√‘𝑘) ∈ ℕ, 1, 0) ∈ ℝ ∧ (√‘𝑘) ∈ ℝ⁺) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℝ)
133	131, 55, 132	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℝ)
134	133	recnd 11289	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℂ)
135	128, 134	syldan 591	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℂ)
136	89, 135	eqeltrrd 2842	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → (1 / (√‘𝑘)) ∈ ℂ)
137	92, 60, 121, 124, 136	fsumf1o 15759	. . . . . . . . . . 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 2777	. . . . . . . . . 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 3961	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) ↔ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ ¬ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))))
140	50	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ∈ ℕ)
141	140	nncnd 12282	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ∈ ℂ)
142	141	sqsqrtd 15478	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → ((√‘𝑘)↑2) = 𝑘)
143		simprr 773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑘) ∈ ℕ)
144		fznnfl 13902	. . . . . . . . . . . . . . . . . . . . . . . . 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 717	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ≤ 𝑥)
148	140	nnrpd 13075	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ∈ ℝ⁺)
149	148	rprege0d 13084	. . . . . . . . . . . . . . . . . . . . . . 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 13084	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
152		sqrtle 15299	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (𝑘 ≤ 𝑥 ↔ (√‘𝑘) ≤ (√‘𝑥)))
153	149, 151, 152	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (𝑘 ≤ 𝑥 ↔ (√‘𝑘) ≤ (√‘𝑥)))
154	147, 153	mpbid 232	. . . . . . . . . . . . . . . . . . . . 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 13077	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑥) ∈ ℝ)
157		fznnfl 13902	. . . . . . . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑘) ∈ (1...(⌊‘(√‘𝑥))))
160	142, 140	eqeltrd 2841	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → ((√‘𝑘)↑2) ∈ ℕ)
161		oveq1 7438	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = (√‘𝑘) → (𝑚↑2) = ((√‘𝑘)↑2))
162	72, 161	elrnmpt1s 5970	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((√‘𝑘) ∈ (1...(⌊‘(√‘𝑥))) ∧ ((√‘𝑘)↑2) ∈ ℕ) → ((√‘𝑘)↑2) ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))
163	159, 160, 162	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → ((√‘𝑘)↑2) ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))
164	142, 163	eqeltrrd 2842	. . . . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → ¬ (√‘𝑘) ∈ ℕ)
169	168	iffalsed 4536	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → if((√‘𝑘) ∈ ℕ, 1, 0) = 0)
170	169	oveq1d 7446	. . . . . . . . . . . 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 4131	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → 𝑘 ∈ (1...(⌊‘𝑥)))
172	171, 55	sylan2 593	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → (√‘𝑘) ∈ ℝ⁺)
173	172	rpcnne0d 13086	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → ((√‘𝑘) ∈ ℂ ∧ (√‘𝑘) ≠ 0))
174		div0 11955	. . . . . . . . . . . . 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 2777	. . . . . . . . . . 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 15761	. . . . . . . . . 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 13075	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑖 ∈ ℝ⁺)
179	178	rprege0d 13084	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖))
180		sqrtsq 15308	. . . . . . . . . . . . 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 7447	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → (1 / (√‘(𝑖↑2))) = (1 / 𝑖))
183	182	sumeq2dv 15738	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / (√‘(𝑖↑2))) = Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖))
184	138, 177, 183	3eqtr3d 2785	. . . . . . . . 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 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑁 ∈ ℕ)
187	46	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
188	47	ad2antrr 726	. . . . . . . . . . . 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 27554	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → if((√‘𝑘) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑘))
190	185, 52, 55, 189	lediv1dd 13135	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ≤ ((𝐹‘𝑘) / (√‘𝑘)))
191	38, 133, 56, 190	fsumle 15835	. . . . . . . . 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 5167	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖) ≤ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘)))
193	21, 64, 57, 71, 192	letrd 11418	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) / 2) ≤ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘)))
194	57	leabsd 15453	. . . . . . 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 11418	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) / 2) ≤ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘))))
196	37, 195	eqbrtrd 5165	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘((log‘𝑥) / 2)) ≤ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘))))
197	17, 18, 20, 11, 196	o1le 15689	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / 2)) ∈ 𝑂(1))
198	5, 11, 16, 197	o1mul2 15661	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (2 · ((log‘𝑥) / 2))) ∈ 𝑂(1))
199	9, 198	eqeltrrd 2842	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1))
200	1, 199	mto 197	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 {crab 3436 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 ⟶wf 6557 –1-1→wf1 6558 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 · cmul 11160 < clt 11295 ≤ cle 11296 / cdiv 11920 ℕcn 12266 2c2 12321 ℝ⁺crp 13034 ...cfz 13547 ⌊cfl 13830 ↑cexp 14102 √csqrt 15272 abscabs 15273 𝑂(1)co1 15522 Σcsu 15722 ∥ cdvds 16290 Basecbs 17247 0_gc0g 17484 ℤRHomczrh 21510 ℤ/nℤczn 21513 logclog 26596 DChrcdchr 27276

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 ax-mulf 11235

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-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-ec 8747 df-qs 8751 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-card 9979 df-acn 9982 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-numer 16772 df-denom 16773 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-qus 17554 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-nsg 19142 df-eqg 19143 df-ghm 19231 df-cntz 19335 df-od 19546 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-rsp 21219 df-2idl 21260 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-zring 21458 df-zrh 21514 df-zn 21517 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-cxp 26599 df-atan 26910 df-em 27036 df-dchr 27277

This theorem is referenced by: dchrisum0 27564

W3C validator