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Theorem dchrisum0fno1 27839

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 26965	. 2 ⊢ ¬ (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1)
2		relogcl 26904	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
3	2	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
4	3	recnd 11338	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
5		2cnd 12422	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
6		2ne0 12450	. . . . . 6 ⊢ 2 ≠ 0
7	6	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ≠ 0)
8	4, 5, 7	divcan2d 12096	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · ((log‘𝑥) / 2)) = (log‘𝑥))
9	8	mpteq2dva 5197	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (2 · ((log‘𝑥) / 2))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
10	3	rehalfcld 12594	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / 2) ∈ ℝ)
11	10	recnd 11338	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) / 2) ∈ ℂ)
12		rpssre 13129	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
13		2cn 12419	. . . . . 6 ⊢ 2 ∈ ℂ
14		o1const 15788	. . . . . 6 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1))
15	12, 13, 14	mp2an 705	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1)
16	15	a1i 11	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1))
17		1red 11310	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
18		dchrisum0fno1.a	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘))) ∈ 𝑂(1))
19		sumex 15856	. . . . . 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 730	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) / 2) ∈ ℝ)
22	2	ad2antrl 741	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
23		log1 26914	. . . . . . . . 9 ⊢ (log‘1) = 0
24		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
25		1rp 13125	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ⁺
26		simprl 783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ⁺)
27		logleb 26932	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
28	25, 26, 27	sylancr 599	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
29	24, 28	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘1) ≤ (log‘𝑥))
30	23, 29	eqbrtrrid 5140	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ (log‘𝑥))
31		2re 12418	. . . . . . . . 9 ⊢ 2 ∈ ℝ
32	31	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 2 ∈ ℝ)
33		2pos 12448	. . . . . . . . 9 ⊢ 0 < 2
34	33	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 < 2)
35		divge0 12187	. . . . . . . 8 ⊢ ((((log‘𝑥) ∈ ℝ ∧ 0 ≤ (log‘𝑥)) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 ≤ ((log‘𝑥) / 2))
36	22, 30, 32, 34, 35	syl22anc 852	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘𝑥) / 2))
37	21, 36	absidd 15591	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘((log‘𝑥) / 2)) = ((log‘𝑥) / 2))
38		fzfid 14117	. . . . . . . 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 27835	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
49	48	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝐹:ℕ⟶ℝ)
50		elfznn 13688	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(⌊‘𝑥)) → 𝑘 ∈ ℕ)
51		ffvelcdm 7081	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶ℝ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
52	49, 50, 51	syl2an 608	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (𝐹‘𝑘) ∈ ℝ)
53	50	adantl 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℕ)
54	53	nnrpd 13163	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℝ⁺)
55	54	rpsqrtcld 15580	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (√‘𝑘) ∈ ℝ⁺)
56	52, 55	rerpdivcld 13196	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((𝐹‘𝑘) / (√‘𝑘)) ∈ ℝ)
57	38, 56	fsumrecl 15901	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘)) ∈ ℝ)
58	57	recnd 11338	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘)) ∈ ℂ)
59	58	abscld 15607	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘))) ∈ ℝ)
60		fzfid 14117	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1...(⌊‘(√‘𝑥))) ∈ Fin)
61		elfznn 13688	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...(⌊‘(√‘𝑥))) → 𝑖 ∈ ℕ)
62	61	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑖 ∈ ℕ)
63	62	nnrecred 12390	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → (1 / 𝑖) ∈ ℝ)
64	60, 63	fsumrecl 15901	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖) ∈ ℝ)
65		logsqrt 27033	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (log‘(√‘𝑥)) = ((log‘𝑥) / 2))
66	65	ad2antrl 741	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘(√‘𝑥)) = ((log‘𝑥) / 2))
67		rpsqrtcl 15432	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ∈ ℝ⁺)
68	67	ad2antrl 741	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (√‘𝑥) ∈ ℝ⁺)
69		harmoniclbnd 27337	. . . . . . . . . 10 ⊢ ((√‘𝑥) ∈ ℝ⁺ → (log‘(√‘𝑥)) ≤ Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖))
70	68, 69	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (log‘(√‘𝑥)) ≤ Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖))
71	66, 70	eqbrtrrd 5128	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) / 2) ≤ Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖))
72		eqid 2760	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) = (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))
73		ovex 7453	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚↑2) ∈ V
74	72, 73	elrnmpti 5943	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) ↔ ∃𝑚 ∈ (1...(⌊‘(√‘𝑥)))𝑘 = (𝑚↑2))
75		elfznn 13688	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ (1...(⌊‘(√‘𝑥))) → 𝑚 ∈ ℕ)
76	75	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑚 ∈ ℕ)
77	76	nnrpd 13163	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑚 ∈ ℝ⁺)
78	77	rprege0d 13172	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚 ∈ ℝ ∧ 0 ≤ 𝑚))
79		sqrtsq 15437	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) → (√‘(𝑚↑2)) = 𝑚)
80	78, 79	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (√‘(𝑚↑2)) = 𝑚)
81	80, 76	eqeltrd 2860	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (√‘(𝑚↑2)) ∈ ℕ)
82		fveq2 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑚↑2) → (√‘𝑘) = (√‘(𝑚↑2)))
83	82	eleq1d 2845	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑚↑2) → ((√‘𝑘) ∈ ℕ ↔ (√‘(𝑚↑2)) ∈ ℕ))
84	81, 83	syl5ibrcom 250	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑘 = (𝑚↑2) → (√‘𝑘) ∈ ℕ))
85	84	rexlimdva 3163	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (∃𝑚 ∈ (1...(⌊‘(√‘𝑥)))𝑘 = (𝑚↑2) → (√‘𝑘) ∈ ℕ))
86	74, 85	biimtrid 245	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) → (√‘𝑘) ∈ ℕ))
87	86	imp 412	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → (√‘𝑘) ∈ ℕ)
88	87	iftrued 4489	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → if((√‘𝑘) ∈ ℕ, 1, 0) = 1)
89	88	oveq1d 7435	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) = (1 / (√‘𝑘)))
90	89	sumeq2dv 15870	. . . . . . . . . . 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 6885	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑖↑2) → (√‘𝑘) = (√‘(𝑖↑2)))
92	91	oveq2d 7436	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑖↑2) → (1 / (√‘𝑘)) = (1 / (√‘(𝑖↑2))))
93	76	nnsqcld 14389	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚↑2) ∈ ℕ)
94	68	rpred 13165	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (√‘𝑥) ∈ ℝ)
95		fznnfl 14003	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((√‘𝑥) ∈ ℝ → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↔ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (√‘𝑥))))
96	94, 95	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↔ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (√‘𝑥))))
97	96	simplbda 505	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑚 ≤ (√‘𝑥))
98	68	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (√‘𝑥) ∈ ℝ⁺)
99	98	rprege0d 13172	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → ((√‘𝑥) ∈ ℝ ∧ 0 ≤ (√‘𝑥)))
100		le2sq 14278	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) ∧ ((√‘𝑥) ∈ ℝ ∧ 0 ≤ (√‘𝑥))) → (𝑚 ≤ (√‘𝑥) ↔ (𝑚↑2) ≤ ((√‘𝑥)↑2)))
101	78, 99, 100	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚 ≤ (√‘𝑥) ↔ (𝑚↑2) ≤ ((√‘𝑥)↑2)))
102	97, 101	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚↑2) ≤ ((√‘𝑥)↑2))
103	26	rpred 13165	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
104	103	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑥 ∈ ℝ)
105	104	recnd 11338	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑥 ∈ ℂ)
106	105	sqsqrtd 15610	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → ((√‘𝑥)↑2) = 𝑥)
107	102, 106	breqtrd 5130	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚↑2) ≤ 𝑥)
108		fznnfl 14003	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ → ((𝑚↑2) ∈ (1...(⌊‘𝑥)) ↔ ((𝑚↑2) ∈ ℕ ∧ (𝑚↑2) ≤ 𝑥)))
109	104, 108	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → ((𝑚↑2) ∈ (1...(⌊‘𝑥)) ↔ ((𝑚↑2) ∈ ℕ ∧ (𝑚↑2) ≤ 𝑥)))
110	93, 107, 109	mpbir2and 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚↑2) ∈ (1...(⌊‘𝑥)))
111	110	ex 418	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) → (𝑚↑2) ∈ (1...(⌊‘𝑥))))
112	75	nnrpd 13163	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (1...(⌊‘(√‘𝑥))) → 𝑚 ∈ ℝ⁺)
113	112	rprege0d 13172	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1...(⌊‘(√‘𝑥))) → (𝑚 ∈ ℝ ∧ 0 ≤ 𝑚))
114	61	nnrpd 13163	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...(⌊‘(√‘𝑥))) → 𝑖 ∈ ℝ⁺)
115	114	rprege0d 13172	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (1...(⌊‘(√‘𝑥))) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖))
116		sq11 14275	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) ∧ (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖)) → ((𝑚↑2) = (𝑖↑2) ↔ 𝑚 = 𝑖))
117	113, 115, 116	syl2an 608	. . . . . . . . . . . . . . 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 9020	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))–1-1→(1...(⌊‘𝑥)))
120		f1f1orn 6836	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))–1-1-onto→ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))
122		oveq1 7427	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑖 → (𝑚↑2) = (𝑖↑2))
123	122, 72, 73	fvmpt3i 6999	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...(⌊‘(√‘𝑥))) → ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))‘𝑖) = (𝑖↑2))
124	123	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))‘𝑖) = (𝑖↑2))
125		f1f 6778	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))–1-1→(1...(⌊‘𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))⟶(1...(⌊‘𝑥)))
126		frn 6717	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))⟶(1...(⌊‘𝑥)) → ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) ⊆ (1...(⌊‘𝑥)))
127	119, 125, 126	3syl 19	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) ⊆ (1...(⌊‘𝑥)))
128	127	sselda 3908	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → 𝑘 ∈ (1...(⌊‘𝑥)))
129		1re 11309	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
130		0re 11311	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
131	129, 130	ifcli 4529	. . . . . . . . . . . . . . . 16 ⊢ if((√‘𝑘) ∈ ℕ, 1, 0) ∈ ℝ
132		rerpdivcl 13153	. . . . . . . . . . . . . . . 16 ⊢ ((if((√‘𝑘) ∈ ℕ, 1, 0) ∈ ℝ ∧ (√‘𝑘) ∈ ℝ⁺) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℝ)
133	131, 55, 132	sylancr 599	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℝ)
134	133	recnd 11338	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℂ)
135	128, 134	syldan 603	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℂ)
136	89, 135	eqeltrrd 2861	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → (1 / (√‘𝑘)) ∈ ℂ)
137	92, 60, 121, 124, 136	fsumf1o 15890	. . . . . . . . . . 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 2795	. . . . . . . . . 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 4049	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) ↔ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ ¬ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))))
140	50	ad2antrl 741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ∈ ℕ)
141	140	nncnd 12352	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ∈ ℂ)
142	141	sqsqrtd 15610	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → ((√‘𝑘)↑2) = 𝑘)
143		simprr 785	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑘) ∈ ℕ)
144		fznnfl 14003	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ ℝ → (𝑘 ∈ (1...(⌊‘𝑥)) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≤ 𝑥)))
145	103, 144	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑘 ∈ (1...(⌊‘𝑥)) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≤ 𝑥)))
146	145	simplbda 505	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ≤ 𝑥)
147	146	adantrr 730	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ≤ 𝑥)
148	140	nnrpd 13163	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ∈ ℝ⁺)
149	148	rprege0d 13172	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (𝑘 ∈ ℝ ∧ 0 ≤ 𝑘))
150	26	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑥 ∈ ℝ⁺)
151	150	rprege0d 13172	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
152		sqrtle 15428	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (𝑘 ≤ 𝑥 ↔ (√‘𝑘) ≤ (√‘𝑥)))
153	149, 151, 152	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (𝑘 ≤ 𝑥 ↔ (√‘𝑘) ≤ (√‘𝑥)))
154	147, 153	mpbid 235	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑘) ≤ (√‘𝑥))
155	68	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑥) ∈ ℝ⁺)
156	155	rpred 13165	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑥) ∈ ℝ)
157		fznnfl 14003	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((√‘𝑥) ∈ ℝ → ((√‘𝑘) ∈ (1...(⌊‘(√‘𝑥))) ↔ ((√‘𝑘) ∈ ℕ ∧ (√‘𝑘) ≤ (√‘𝑥))))
158	156, 157	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → ((√‘𝑘) ∈ (1...(⌊‘(√‘𝑥))) ↔ ((√‘𝑘) ∈ ℕ ∧ (√‘𝑘) ≤ (√‘𝑥))))
159	143, 154, 158	mpbir2and 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑘) ∈ (1...(⌊‘(√‘𝑥))))
160	142, 140	eqeltrd 2860	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → ((√‘𝑘)↑2) ∈ ℕ)
161		oveq1 7427	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = (√‘𝑘) → (𝑚↑2) = ((√‘𝑘)↑2))
162	72, 161	elrnmpt1s 5940	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((√‘𝑘) ∈ (1...(⌊‘(√‘𝑥))) ∧ ((√‘𝑘)↑2) ∈ ℕ) → ((√‘𝑘)↑2) ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))
163	159, 160, 162	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → ((√‘𝑘)↑2) ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))
164	142, 163	eqeltrrd 2861	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))
165	164	expr 462	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((√‘𝑘) ∈ ℕ → 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))))
166	165	con3d 153	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (¬ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) → ¬ (√‘𝑘) ∈ ℕ))
167	166	impr 460	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ ¬ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → ¬ (√‘𝑘) ∈ ℕ)
168	139, 167	sylan2b 606	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → ¬ (√‘𝑘) ∈ ℕ)
169	168	iffalsed 4492	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → if((√‘𝑘) ∈ ℕ, 1, 0) = 0)
170	169	oveq1d 7435	. . . . . . . . . . . 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 4067	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → 𝑘 ∈ (1...(⌊‘𝑥)))
172	171, 55	sylan2 605	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → (√‘𝑘) ∈ ℝ⁺)
173	172	rpcnne0d 13174	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → ((√‘𝑘) ∈ ℂ ∧ (√‘𝑘) ≠ 0))
174		div0 12006	. . . . . . . . . . . . 13 ⊢ (((√‘𝑘) ∈ ℂ ∧ (√‘𝑘) ≠ 0) → (0 / (√‘𝑘)) = 0)
175	173, 174	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → (0 / (√‘𝑘)) = 0)
176	170, 175	eqtrd 2795	. . . . . . . . . . 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 15892	. . . . . . . . . 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 13163	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑖 ∈ ℝ⁺)
179	178	rprege0d 13172	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖))
180		sqrtsq 15437	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ ℝ ∧ 0 ≤ 𝑖) → (√‘(𝑖↑2)) = 𝑖)
181	179, 180	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → (√‘(𝑖↑2)) = 𝑖)
182	181	oveq2d 7436	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → (1 / (√‘(𝑖↑2))) = (1 / 𝑖))
183	182	sumeq2dv 15870	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / (√‘(𝑖↑2))) = Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖))
184	138, 177, 183	3eqtr3d 2803	. . . . . . . . 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 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑁 ∈ ℕ)
187	46	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
188	47	ad2antrr 739	. . . . . . . . . . . 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 27838	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → if((√‘𝑘) ∈ ℕ, 1, 0) ≤ (𝐹‘𝑘))
190	185, 52, 55, 189	lediv1dd 13223	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ≤ ((𝐹‘𝑘) / (√‘𝑘)))
191	38, 133, 56, 190	fsumle 15967	. . . . . . . . 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 5128	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖) ≤ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘)))
193	21, 64, 57, 71, 192	letrd 11468	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) / 2) ≤ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘)))
194	57	leabsd 15583	. . . . . . 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 11468	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) / 2) ≤ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘))))
196	37, 195	eqbrtrd 5126	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘((log‘𝑥) / 2)) ≤ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹‘𝑘) / (√‘𝑘))))
197	17, 18, 20, 11, 196	o1le 15821	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / 2)) ∈ 𝑂(1))
198	5, 11, 16, 197	o1mul2 15793	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (2 · ((log‘𝑥) / 2))) ∈ 𝑂(1))
199	9, 198	eqeltrrd 2861	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) ∈ 𝑂(1))
200	1, 199	mto 200	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∃wrex 3086 {crab 3412 Vcvv 3450 ⊆ wss 3895 ∖ cdif 4045 ifcif 4481 class class class wbr 5102 ↦ cmpt 5185 ran crn 5651 ⟶wf 6534 –1-1→wf1 6535 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7420 ℂcc 11199 ℝcr 11200 0cc0 11201 1c1 11202 · cmul 11206 < clt 11344 ≤ cle 11345 / cdiv 11974 ℕcn 12336 2c2 12398 ℝ⁺crp 13121 ...cfz 13640 ⌊cfl 13931 ↑cexp 14205 √csqrt 15401 abscabs 15402 𝑂(1)co1 15654 Σcsu 15854 ∥ cdvds 16423 Basecbs 17388 0_gc0g 17611 ℤRHomczrh 21806 ℤ/nℤczn 21809 logclog 26883 DChrcdchr 27560

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9643 ax-cnex 11257 ax-resscn 11258 ax-1cn 11259 ax-icn 11260 ax-addcl 11261 ax-addrcl 11262 ax-mulcl 11263 ax-mulrcl 11264 ax-mulcom 11265 ax-addass 11266 ax-mulass 11267 ax-distr 11268 ax-i2m1 11269 ax-1ne0 11270 ax-1rid 11271 ax-rnegex 11272 ax-rrecex 11273 ax-cnre 11274 ax-pre-lttri 11275 ax-pre-lttrn 11276 ax-pre-ltadd 11277 ax-pre-mulgt0 11278 ax-pre-sup 11279 ax-addf 11280 ax-mulf 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-disj 5070 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-tpos 8244 df-frecs 8300 df-wrecs 8331 df-recs 8380 df-rdg 8419 df-1o 8477 df-2o 8478 df-oadd 8481 df-omul 8482 df-er 8718 df-ec 8720 df-qs 8724 df-map 8850 df-pm 8851 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9355 df-fi 9404 df-sup 9435 df-inf 9436 df-oi 9505 df-card 10021 df-acn 10024 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-sub 11544 df-neg 11545 df-div 11975 df-nn 12337 df-2 12406 df-3 12407 df-4 12408 df-5 12409 df-6 12410 df-7 12411 df-8 12412 df-9 12413 df-n0 12608 df-xnn0 12681 df-z 12695 df-dec 12816 df-uz 12967 df-q 13077 df-rp 13122 df-xneg 13242 df-xadd 13243 df-xmul 13244 df-ioo 13481 df-ioc 13482 df-ico 13483 df-icc 13484 df-fz 13641 df-fzo 13790 df-fl 13933 df-mod 14011 df-seq 14146 df-exp 14206 df-fac 14419 df-bc 14448 df-hash 14476 df-shft 15221 df-cj 15267 df-re 15268 df-im 15269 df-sqrt 15403 df-abs 15404 df-limsup 15639 df-clim 15656 df-rlim 15657 df-o1 15658 df-lo1 15659 df-sum 15855 df-ef 16234 df-e 16235 df-sin 16236 df-cos 16237 df-tan 16238 df-pi 16239 df-dvds 16424 df-gcd 16666 df-prm 16848 df-numer 16912 df-denom 16913 df-pc 17016 df-struct 17326 df-sets 17343 df-slot 17361 df-ndx 17373 df-base 17389 df-ress 17410 df-plusg 17442 df-mulr 17443 df-starv 17444 df-sca 17445 df-vsca 17446 df-ip 17447 df-tset 17448 df-ple 17449 df-ds 17451 df-unif 17452 df-hom 17453 df-cco 17454 df-rest 17594 df-topn 17595 df-0g 17613 df-gsum 17614 df-topgen 17615 df-pt 17616 df-prds 17619 df-xrs 17675 df-qtop 17680 df-imas 17681 df-qus 17682 df-xps 17683 df-mre 17757 df-mrc 17758 df-acs 17760 df-mgm 18817 df-sgrp 18909 df-mnd 18925 df-mhm 18979 df-submnd 18980 df-grp 19148 df-minusg 19149 df-sbg 19150 df-mulg 19279 df-subg 19334 df-nsg 19335 df-eqg 19336 df-ghm 19429 df-cntz 19532 df-od 19743 df-cmn 19997 df-abl 19998 df-mgp 20362 df-rng 20376 df-ur 20409 df-ring 20462 df-cring 20463 df-oppr 20568 df-dvdsr 20588 df-unit 20589 df-invr 20619 df-dvr 20632 df-rhm 20703 df-subrng 20799 df-subrg 20823 df-drng 20983 df-lmod 21138 df-lss 21208 df-lsp 21248 df-sra 21449 df-rgmod 21450 df-lidl 21487 df-rsp 21488 df-2idl 21544 df-psmet 21671 df-xmet 21672 df-met 21673 df-bl 21674 df-mopn 21675 df-fbas 21676 df-fg 21677 df-cnfld 21680 df-zring 21754 df-zrh 21810 df-zn 21813 df-top 23213 df-topon 23230 df-topsp 23252 df-bases 23265 df-cld 23338 df-ntr 23339 df-cls 23340 df-nei 23417 df-lp 23455 df-perf 23456 df-cn 23546 df-cnp 23547 df-haus 23634 df-cmp 23706 df-tx 23882 df-hmeo 24075 df-fil 24166 df-fm 24258 df-flim 24259 df-flf 24260 df-xms 24640 df-ms 24641 df-tms 24642 df-cncf 25200 df-limc 26187 df-dv 26188 df-ulm 26705 df-log 26885 df-cxp 26886 df-atan 27196 df-em 27321 df-dchr 27561

This theorem is used by: dchrisum0 27848

W3C validator