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Theorem dchrisum0fno1 27493

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 26616	. 2 ⊢ ¬ (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1)
2		relogcl 26555	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
3	2	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
4	3	recnd 11162	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
5		2cnd 12248	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
6		2ne0 12274	. . . . . 6 ⊢ 2 ≠ 0
7	6	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ≠ 0)
8	4, 5, 7	divcan2d 11922	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · ((log‘𝑥) / 2)) = (log‘𝑥))
9	8	mpteq2dva 5179	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (2 · ((log‘𝑥) / 2))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
10	3	rehalfcld 12413	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / 2) ∈ ℝ)
11	10	recnd 11162	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / 2) ∈ ℂ)
12		rpssre 12939	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
13		2cn 12245	. . . . . 6 ⊢ 2 ∈ ℂ
14		o1const 15571	. . . . . 6 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1))
15	12, 13, 14	mp2an 693	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1)
16	15	a1i 11	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1))
17		1red 11134	. . . . 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 718	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) / 2) ∈ ℝ)
22	2	ad2antrl 729	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
23		log1 26565	. . . . . . . . 9 ⊢ (log‘1) = 0
24		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
25		1rp 12935	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ⁺
26		simprl 771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ⁺)
27		logleb 26583	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
28	25, 26, 27	sylancr 588	. . . . . . . . . 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 5122	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ (log‘𝑥))
31		2re 12244	. . . . . . . . 9 ⊢ 2 ∈ ℝ
32	31	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 2 ∈ ℝ)
33		2pos 12273	. . . . . . . . 9 ⊢ 0 < 2
34	33	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 < 2)
35		divge0 12014	. . . . . . . 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 15374	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘((log‘𝑥) / 2)) = ((log‘𝑥) / 2))
38		fzfid 13924	. . . . . . . 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 27489	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
49	48	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝐹:ℕ⟶ℝ)
50		elfznn 13496	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(⌊‘𝑥)) → 𝑘 ∈ ℕ)
51		ffvelcdm 7025	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶ℝ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
52	49, 50, 51	syl2an 597	. . . . . . . . 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 12973	. . . . . . . . . 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 13006	. . . . . . . 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 11162	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘)) ∈ ℂ)
59	58	abscld 15390	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘))) ∈ ℝ)
60		fzfid 13924	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1...(⌊‘(√‘𝑥))) ∈ Fin)
61		elfznn 13496	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(⌊‘(√‘𝑥))) → 𝑖 ∈ ℕ)
62	61	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑖 ∈ ℕ)
63	62	nnrecred 12217	. . . . . . . . 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 26684	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (log‘(√‘𝑥)) = ((log‘𝑥) / 2))
66	65	ad2antrl 729	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘(√‘𝑥)) = ((log‘𝑥) / 2))
67		rpsqrtcl 15215	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ∈ ℝ⁺)
68	67	ad2antrl 729	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (√‘𝑥) ∈ ℝ⁺)
69		harmoniclbnd 26990	. . . . . . . . . 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 5110	. . . . . . . 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 7391	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚↑2) ∈ V
74	72, 73	elrnmpti 5909	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) ↔ ∃𝑚 ∈ (1...(⌊‘(√‘𝑥)))𝑘 = (𝑚↑2))
75		elfznn 13496	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ (1...(⌊‘(√‘𝑥))) → 𝑚 ∈ ℕ)
76	75	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑚 ∈ ℕ)
77	76	nnrpd 12973	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑚 ∈ ℝ⁺)
78	77	rprege0d 12982	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚 ∈ ℝ ∧ 0 ≤ 𝑚))
79		sqrtsq 15220	. . . . . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (√‘(𝑚↑2)) ∈ ℕ)
82		fveq2 6832	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑚↑2) → (√‘𝑘) = (√‘(𝑚↑2)))
83	82	eleq1d 2822	. . . . . . . . . . . . . . . . . 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 3139	. . . . . . . . . . . . . . . 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 4475	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → if((√‘𝑘) ∈ ℕ, 1, 0) = 1)
89	88	oveq1d 7373	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) = (1 / (√‘𝑘)))
90	89	sumeq2dv 15653	. . . . . . . . . . 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 6832	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑖↑2) → (√‘𝑘) = (√‘(𝑖↑2)))
92	91	oveq2d 7374	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑖↑2) → (1 / (√‘𝑘)) = (1 / (√‘(𝑖↑2))))
93	76	nnsqcld 14195	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚↑2) ∈ ℕ)
94	68	rpred 12975	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (√‘𝑥) ∈ ℝ)
95		fznnfl 13810	. . . . . . . . . . . . . . . . . . . 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 12982	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → ((√‘𝑥) ∈ ℝ ∧ 0 ≤ (√‘𝑥)))
100		le2sq 14085	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) ∧ ((√‘𝑥) ∈ ℝ ∧ 0 ≤ (√‘𝑥))) → (𝑚 ≤ (√‘𝑥) ↔ (𝑚↑2) ≤ ((√‘𝑥)↑2)))
101	78, 99, 100	syl2anc 585	. . . . . . . . . . . . . . . . . 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 12975	. . . . . . . . . . . . . . . . . . . 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 11162	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑥 ∈ ℂ)
106	105	sqsqrtd 15393	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → ((√‘𝑥)↑2) = 𝑥)
107	102, 106	breqtrd 5112	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚↑2) ≤ 𝑥)
108		fznnfl 13810	. . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . 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 12973	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (1...(⌊‘(√‘𝑥))) → 𝑚 ∈ ℝ⁺)
113	112	rprege0d 12982	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1...(⌊‘(√‘𝑥))) → (𝑚 ∈ ℝ ∧ 0 ≤ 𝑚))
114	61	nnrpd 12973	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...(⌊‘(√‘𝑥))) → 𝑖 ∈ ℝ⁺)
115	114	rprege0d 12982	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (1...(⌊‘(√‘𝑥))) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖))
116		sq11 14082	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) ∧ (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖)) → ((𝑚↑2) = (𝑖↑2) ↔ 𝑚 = 𝑖))
117	113, 115, 116	syl2an 597	. . . . . . . . . . . . . . 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 8930	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))–1-1→(1...(⌊‘𝑥)))
120		f1f1orn 6783	. . . . . . . . . . . . 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 7365	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑖 → (𝑚↑2) = (𝑖↑2))
123	122, 72, 73	fvmpt3i 6945	. . . . . . . . . . . . 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 6728	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))–1-1→(1...(⌊‘𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))⟶(1...(⌊‘𝑥)))
126		frn 6667	. . . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → 𝑘 ∈ (1...(⌊‘𝑥)))
129		1re 11133	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
130		0re 11135	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
131	129, 130	ifcli 4515	. . . . . . . . . . . . . . . 16 ⊢ if((√‘𝑘) ∈ ℕ, 1, 0) ∈ ℝ
132		rerpdivcl 12963	. . . . . . . . . . . . . . . 16 ⊢ ((if((√‘𝑘) ∈ ℕ, 1, 0) ∈ ℝ ∧ (√‘𝑘) ∈ ℝ⁺) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℝ)
133	131, 55, 132	sylancr 588	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℝ)
134	133	recnd 11162	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℂ)
135	128, 134	syldan 592	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℂ)
136	89, 135	eqeltrrd 2838	. . . . . . . . . . . 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 2772	. . . . . . . . . 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 3900	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) ↔ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ ¬ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))))
140	50	ad2antrl 729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ∈ ℕ)
141	140	nncnd 12179	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ∈ ℂ)
142	141	sqsqrtd 15393	. . . . . . . . . . . . . . . . . . 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 13810	. . . . . . . . . . . . . . . . . . . . . . . . 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 718	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ≤ 𝑥)
148	140	nnrpd 12973	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ∈ ℝ⁺)
149	148	rprege0d 12982	. . . . . . . . . . . . . . . . . . . . . . 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 12982	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
152		sqrtle 15211	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (𝑘 ≤ 𝑥 ↔ (√‘𝑘) ≤ (√‘𝑥)))
153	149, 151, 152	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . 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 12975	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑥) ∈ ℝ)
157		fznnfl 13810	. . . . . . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑘) ∈ (1...(⌊‘(√‘𝑥))))
160	142, 140	eqeltrd 2837	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → ((√‘𝑘)↑2) ∈ ℕ)
161		oveq1 7365	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = (√‘𝑘) → (𝑚↑2) = ((√‘𝑘)↑2))
162	72, 161	elrnmpt1s 5906	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((√‘𝑘) ∈ (1...(⌊‘(√‘𝑥))) ∧ ((√‘𝑘)↑2) ∈ ℕ) → ((√‘𝑘)↑2) ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))
163	159, 160, 162	syl2anc 585	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → ((√‘𝑘)↑2) ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))
164	142, 163	eqeltrrd 2838	. . . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → ¬ (√‘𝑘) ∈ ℕ)
169	168	iffalsed 4478	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → if((√‘𝑘) ∈ ℕ, 1, 0) = 0)
170	169	oveq1d 7373	. . . . . . . . . . . 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 4072	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → 𝑘 ∈ (1...(⌊‘𝑥)))
172	171, 55	sylan2 594	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → (√‘𝑘) ∈ ℝ⁺)
173	172	rpcnne0d 12984	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → ((√‘𝑘) ∈ ℂ ∧ (√‘𝑘) ≠ 0))
174		div0 11831	. . . . . . . . . . . . 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 2772	. . . . . . . . . . 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 12973	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑖 ∈ ℝ⁺)
179	178	rprege0d 12982	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖))
180		sqrtsq 15220	. . . . . . . . . . . . 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 7374	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → (1 / (√‘(𝑖↑2))) = (1 / 𝑖))
183	182	sumeq2dv 15653	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / (√‘(𝑖↑2))) = Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖))
184	138, 177, 183	3eqtr3d 2780	. . . . . . . . 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 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑁 ∈ ℕ)
187	46	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
188	47	ad2antrr 727	. . . . . . . . . . . 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 27492	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → if((√‘𝑘) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑘))
190	185, 52, 55, 189	lediv1dd 13033	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ≤ ((𝐹‘𝑘) / (√‘𝑘)))
191	38, 133, 56, 190	fsumle 15751	. . . . . . . . 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 5110	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖) ≤ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘)))
193	21, 64, 57, 71, 192	letrd 11292	. . . . . . 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 11292	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) / 2) ≤ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘))))
196	37, 195	eqbrtrd 5108	. . . . 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 15576	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (2 · ((log‘𝑥) / 2))) ∈ 𝑂(1))
199	9, 198	eqeltrrd 2838	. 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 1542 ∈ wcel 2114 ≠ wne 2933 ∃wrex 3062 {crab 3390 Vcvv 3430 ∖ cdif 3887 ⊆ wss 3890 ifcif 4467 class class class wbr 5086 ↦ cmpt 5167 ran crn 5623 ⟶wf 6486 –1-1→wf1 6487 –1-1-onto→wf1o 6489 ‘cfv 6490 (class class class)co 7358 ℂcc 11025 ℝcr 11026 0cc0 11027 1c1 11028 · cmul 11032 < clt 11168 ≤ cle 11169 / cdiv 11796 ℕcn 12163 2c2 12225 ℝ⁺crp 12931 ...cfz 13450 ⌊cfl 13738 ↑cexp 14012 √csqrt 15184 abscabs 15185 𝑂(1)co1 15437 Σcsu 15637 ∥ cdvds 16210 Basecbs 17168 0_gc0g 17391 ℤRHomczrh 21487 ℤ/nℤczn 21490 logclog 26534 DChrcdchr 27214

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-inf2 9551 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106 ax-mulf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-tpos 8167 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-oadd 8400 df-omul 8401 df-er 8634 df-ec 8636 df-qs 8640 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-card 9852 df-acn 9855 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-xnn0 12500 df-z 12514 df-dec 12634 df-uz 12778 df-q 12888 df-rp 12932 df-xneg 13052 df-xadd 13053 df-xmul 13054 df-ioo 13291 df-ioc 13292 df-ico 13293 df-icc 13294 df-fz 13451 df-fzo 13598 df-fl 13740 df-mod 13818 df-seq 13953 df-exp 14013 df-fac 14225 df-bc 14254 df-hash 14282 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15422 df-clim 15439 df-rlim 15440 df-o1 15441 df-lo1 15442 df-sum 15638 df-ef 16021 df-e 16022 df-sin 16023 df-cos 16024 df-tan 16025 df-pi 16026 df-dvds 16211 df-gcd 16453 df-prm 16630 df-numer 16694 df-denom 16695 df-pc 16797 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-hom 17233 df-cco 17234 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17455 df-qtop 17460 df-imas 17461 df-qus 17462 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18740 df-submnd 18741 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19033 df-subg 19088 df-nsg 19089 df-eqg 19090 df-ghm 19177 df-cntz 19281 df-od 19492 df-cmn 19746 df-abl 19747 df-mgp 20111 df-rng 20123 df-ur 20152 df-ring 20205 df-cring 20206 df-oppr 20306 df-dvdsr 20326 df-unit 20327 df-invr 20357 df-dvr 20370 df-rhm 20441 df-subrng 20512 df-subrg 20536 df-drng 20697 df-lmod 20846 df-lss 20916 df-lsp 20956 df-sra 21158 df-rgmod 21159 df-lidl 21196 df-rsp 21197 df-2idl 21238 df-psmet 21334 df-xmet 21335 df-met 21336 df-bl 21337 df-mopn 21338 df-fbas 21339 df-fg 21340 df-cnfld 21343 df-zring 21435 df-zrh 21491 df-zn 21494 df-top 22868 df-topon 22885 df-topsp 22907 df-bases 22920 df-cld 22993 df-ntr 22994 df-cls 22995 df-nei 23072 df-lp 23110 df-perf 23111 df-cn 23201 df-cnp 23202 df-haus 23289 df-cmp 23361 df-tx 23536 df-hmeo 23729 df-fil 23820 df-fm 23912 df-flim 23913 df-flf 23914 df-xms 24294 df-ms 24295 df-tms 24296 df-cncf 24854 df-limc 25842 df-dv 25843 df-ulm 26357 df-log 26536 df-cxp 26537 df-atan 26848 df-em 26974 df-dchr 27215

This theorem is referenced by: dchrisum0 27502

W3C validator