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Theorem dchrisum0fno1 25000
 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 (𝜑𝑁 ∈ ℕ)
rpvmasum2.g 𝐺 = (DChr‘𝑁)
rpvmasum2.d 𝐷 = (Base‘𝐺)
rpvmasum2.1 1 = (0g𝐺)
dchrisum0f.f 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))
dchrisum0f.x (𝜑𝑋𝐷)
dchrisum0flb.r (𝜑𝑋:(Base‘𝑍)⟶ℝ)
dchrisum0fno1.a (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹𝑘) / (√‘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.
StepHypRef Expression
1 logno1 24182 . 2 ¬ (𝑥 ∈ ℝ+ ↦ (log‘𝑥)) ∈ 𝑂(1)
2 relogcl 24126 . . . . . . 7 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
32adantl 481 . . . . . 6 ((𝜑𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ)
43recnd 9947 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ)
5 2cnd 10970 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → 2 ∈ ℂ)
6 2ne0 10990 . . . . . 6 2 ≠ 0
76a1i 11 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → 2 ≠ 0)
84, 5, 7divcan2d 10682 . . . 4 ((𝜑𝑥 ∈ ℝ+) → (2 · ((log‘𝑥) / 2)) = (log‘𝑥))
98mpteq2dva 4672 . . 3 (𝜑 → (𝑥 ∈ ℝ+ ↦ (2 · ((log‘𝑥) / 2))) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥)))
103rehalfcld 11156 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → ((log‘𝑥) / 2) ∈ ℝ)
1110recnd 9947 . . . 4 ((𝜑𝑥 ∈ ℝ+) → ((log‘𝑥) / 2) ∈ ℂ)
12 rpssre 11719 . . . . . 6 + ⊆ ℝ
13 2cn 10968 . . . . . 6 2 ∈ ℂ
14 o1const 14198 . . . . . 6 ((ℝ+ ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ ℝ+ ↦ 2) ∈ 𝑂(1))
1512, 13, 14mp2an 704 . . . . 5 (𝑥 ∈ ℝ+ ↦ 2) ∈ 𝑂(1)
1615a1i 11 . . . 4 (𝜑 → (𝑥 ∈ ℝ+ ↦ 2) ∈ 𝑂(1))
17 1red 9934 . . . . 5 (𝜑 → 1 ∈ ℝ)
18 dchrisum0fno1.a . . . . 5 (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹𝑘) / (√‘𝑘))) ∈ 𝑂(1))
19 sumex 14266 . . . . . 6 Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹𝑘) / (√‘𝑘)) ∈ V
2019a1i 11 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹𝑘) / (√‘𝑘)) ∈ V)
2110adantrr 749 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) / 2) ∈ ℝ)
222ad2antrl 760 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
23 log1 24136 . . . . . . . . 9 (log‘1) = 0
24 simprr 792 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
25 1rp 11712 . . . . . . . . . . 11 1 ∈ ℝ+
26 simprl 790 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ+)
27 logleb 24153 . . . . . . . . . . 11 ((1 ∈ ℝ+𝑥 ∈ ℝ+) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
2825, 26, 27sylancr 694 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
2924, 28mpbid 221 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘1) ≤ (log‘𝑥))
3023, 29syl5eqbrr 4619 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ (log‘𝑥))
31 2re 10967 . . . . . . . . 9 2 ∈ ℝ
3231a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 2 ∈ ℝ)
33 2pos 10989 . . . . . . . . 9 0 < 2
3433a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 < 2)
35 divge0 10771 . . . . . . . 8 ((((log‘𝑥) ∈ ℝ ∧ 0 ≤ (log‘𝑥)) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 ≤ ((log‘𝑥) / 2))
3622, 30, 32, 34, 35syl22anc 1319 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘𝑥) / 2))
3721, 36absidd 14009 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘((log‘𝑥) / 2)) = ((log‘𝑥) / 2))
38 fzfid 12634 . . . . . . . 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 (𝜑𝑁 ∈ ℕ)
42 rpvmasum2.g . . . . . . . . . . . 12 𝐺 = (DChr‘𝑁)
43 rpvmasum2.d . . . . . . . . . . . 12 𝐷 = (Base‘𝐺)
44 rpvmasum2.1 . . . . . . . . . . . 12 1 = (0g𝐺)
45 dchrisum0f.f . . . . . . . . . . . 12 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞𝑏} (𝑋‘(𝐿𝑣)))
46 dchrisum0f.x . . . . . . . . . . . 12 (𝜑𝑋𝐷)
47 dchrisum0flb.r . . . . . . . . . . . 12 (𝜑𝑋:(Base‘𝑍)⟶ℝ)
4839, 40, 41, 42, 43, 44, 45, 46, 47dchrisum0ff 24996 . . . . . . . . . . 11 (𝜑𝐹:ℕ⟶ℝ)
4948adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝐹:ℕ⟶ℝ)
50 elfznn 12241 . . . . . . . . . 10 (𝑘 ∈ (1...(⌊‘𝑥)) → 𝑘 ∈ ℕ)
51 ffvelrn 6265 . . . . . . . . . 10 ((𝐹:ℕ⟶ℝ ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℝ)
5249, 50, 51syl2an 493 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (𝐹𝑘) ∈ ℝ)
5350adantl 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℕ)
5453nnrpd 11746 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℝ+)
5554rpsqrtcld 13998 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (√‘𝑘) ∈ ℝ+)
5652, 55rerpdivcld 11779 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((𝐹𝑘) / (√‘𝑘)) ∈ ℝ)
5738, 56fsumrecl 14312 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹𝑘) / (√‘𝑘)) ∈ ℝ)
5857recnd 9947 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹𝑘) / (√‘𝑘)) ∈ ℂ)
5958abscld 14023 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹𝑘) / (√‘𝑘))) ∈ ℝ)
60 fzfid 12634 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (1...(⌊‘(√‘𝑥))) ∈ Fin)
61 elfznn 12241 . . . . . . . . . . 11 (𝑖 ∈ (1...(⌊‘(√‘𝑥))) → 𝑖 ∈ ℕ)
6261adantl 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑖 ∈ ℕ)
6362nnrecred 10943 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → (1 / 𝑖) ∈ ℝ)
6460, 63fsumrecl 14312 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖) ∈ ℝ)
65 logsqrt 24250 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (log‘(√‘𝑥)) = ((log‘𝑥) / 2))
6665ad2antrl 760 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘(√‘𝑥)) = ((log‘𝑥) / 2))
67 rpsqrtcl 13853 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (√‘𝑥) ∈ ℝ+)
6867ad2antrl 760 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (√‘𝑥) ∈ ℝ+)
69 harmoniclbnd 24535 . . . . . . . . . 10 ((√‘𝑥) ∈ ℝ+ → (log‘(√‘𝑥)) ≤ Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖))
7068, 69syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘(√‘𝑥)) ≤ Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖))
7166, 70eqbrtrrd 4607 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) / 2) ≤ Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖))
72 eqid 2610 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) = (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))
73 ovex 6577 . . . . . . . . . . . . . . . . 17 (𝑚↑2) ∈ V
7472, 73elrnmpti 5297 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) ↔ ∃𝑚 ∈ (1...(⌊‘(√‘𝑥)))𝑘 = (𝑚↑2))
75 elfznn 12241 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ (1...(⌊‘(√‘𝑥))) → 𝑚 ∈ ℕ)
7675adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑚 ∈ ℕ)
7776nnrpd 11746 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑚 ∈ ℝ+)
7877rprege0d 11755 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚 ∈ ℝ ∧ 0 ≤ 𝑚))
79 sqrtsq 13858 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) → (√‘(𝑚↑2)) = 𝑚)
8078, 79syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (√‘(𝑚↑2)) = 𝑚)
8180, 76eqeltrd 2688 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (√‘(𝑚↑2)) ∈ ℕ)
82 fveq2 6103 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑚↑2) → (√‘𝑘) = (√‘(𝑚↑2)))
8382eleq1d 2672 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑚↑2) → ((√‘𝑘) ∈ ℕ ↔ (√‘(𝑚↑2)) ∈ ℕ))
8481, 83syl5ibrcom 236 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑘 = (𝑚↑2) → (√‘𝑘) ∈ ℕ))
8584rexlimdva 3013 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (∃𝑚 ∈ (1...(⌊‘(√‘𝑥)))𝑘 = (𝑚↑2) → (√‘𝑘) ∈ ℕ))
8674, 85syl5bi 231 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) → (√‘𝑘) ∈ ℕ))
8786imp 444 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → (√‘𝑘) ∈ ℕ)
8887iftrued 4044 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → if((√‘𝑘) ∈ ℕ, 1, 0) = 1)
8988oveq1d 6564 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) = (1 / (√‘𝑘)))
9089sumeq2dv 14281 . . . . . . . . . . 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 6103 . . . . . . . . . . . . 13 (𝑘 = (𝑖↑2) → (√‘𝑘) = (√‘(𝑖↑2)))
9291oveq2d 6565 . . . . . . . . . . . 12 (𝑘 = (𝑖↑2) → (1 / (√‘𝑘)) = (1 / (√‘(𝑖↑2))))
9376nnsqcld 12891 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚↑2) ∈ ℕ)
9468rpred 11748 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (√‘𝑥) ∈ ℝ)
95 fznnfl 12523 . . . . . . . . . . . . . . . . . . . 20 ((√‘𝑥) ∈ ℝ → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↔ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (√‘𝑥))))
9694, 95syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↔ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (√‘𝑥))))
9796simplbda 652 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑚 ≤ (√‘𝑥))
9868adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (√‘𝑥) ∈ ℝ+)
9998rprege0d 11755 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → ((√‘𝑥) ∈ ℝ ∧ 0 ≤ (√‘𝑥)))
100 le2sq 12800 . . . . . . . . . . . . . . . . . . 19 (((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) ∧ ((√‘𝑥) ∈ ℝ ∧ 0 ≤ (√‘𝑥))) → (𝑚 ≤ (√‘𝑥) ↔ (𝑚↑2) ≤ ((√‘𝑥)↑2)))
10178, 99, 100syl2anc 691 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚 ≤ (√‘𝑥) ↔ (𝑚↑2) ≤ ((√‘𝑥)↑2)))
10297, 101mpbid 221 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚↑2) ≤ ((√‘𝑥)↑2))
10326rpred 11748 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
104103adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑥 ∈ ℝ)
105104recnd 9947 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑥 ∈ ℂ)
106105sqsqrtd 14026 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → ((√‘𝑥)↑2) = 𝑥)
107102, 106breqtrd 4609 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚↑2) ≤ 𝑥)
108 fznnfl 12523 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ → ((𝑚↑2) ∈ (1...(⌊‘𝑥)) ↔ ((𝑚↑2) ∈ ℕ ∧ (𝑚↑2) ≤ 𝑥)))
109104, 108syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → ((𝑚↑2) ∈ (1...(⌊‘𝑥)) ↔ ((𝑚↑2) ∈ ℕ ∧ (𝑚↑2) ≤ 𝑥)))
11093, 107, 109mpbir2and 959 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑚↑2) ∈ (1...(⌊‘𝑥)))
111110ex 449 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) → (𝑚↑2) ∈ (1...(⌊‘𝑥))))
11275nnrpd 11746 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (1...(⌊‘(√‘𝑥))) → 𝑚 ∈ ℝ+)
113112rprege0d 11755 . . . . . . . . . . . . . . . 16 (𝑚 ∈ (1...(⌊‘(√‘𝑥))) → (𝑚 ∈ ℝ ∧ 0 ≤ 𝑚))
11461nnrpd 11746 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...(⌊‘(√‘𝑥))) → 𝑖 ∈ ℝ+)
115114rprege0d 11755 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (1...(⌊‘(√‘𝑥))) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖))
116 sq11 12798 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) ∧ (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖)) → ((𝑚↑2) = (𝑖↑2) ↔ 𝑚 = 𝑖))
117113, 115, 116syl2an 493 . . . . . . . . . . . . . . 15 ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → ((𝑚↑2) = (𝑖↑2) ↔ 𝑚 = 𝑖))
118117a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → ((𝑚↑2) = (𝑖↑2) ↔ 𝑚 = 𝑖)))
119111, 118dom2lem 7881 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))–1-1→(1...(⌊‘𝑥)))
120 f1f1orn 6061 . . . . . . . . . . . . 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)))
121119, 120syl 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 6556 . . . . . . . . . . . . . 14 (𝑚 = 𝑖 → (𝑚↑2) = (𝑖↑2))
123122, 72, 73fvmpt3i 6196 . . . . . . . . . . . . 13 (𝑖 ∈ (1...(⌊‘(√‘𝑥))) → ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))‘𝑖) = (𝑖↑2))
124123adantl 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))‘𝑖) = (𝑖↑2))
125 f1f 6014 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))–1-1→(1...(⌊‘𝑥)) → (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))⟶(1...(⌊‘𝑥)))
126 frn 5966 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)):(1...(⌊‘(√‘𝑥)))⟶(1...(⌊‘𝑥)) → ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) ⊆ (1...(⌊‘𝑥)))
127119, 125, 1263syl 18 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) ⊆ (1...(⌊‘𝑥)))
128127sselda 3568 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → 𝑘 ∈ (1...(⌊‘𝑥)))
129 1re 9918 . . . . . . . . . . . . . . . . 17 1 ∈ ℝ
130 0re 9919 . . . . . . . . . . . . . . . . 17 0 ∈ ℝ
131129, 130keepel 4105 . . . . . . . . . . . . . . . 16 if((√‘𝑘) ∈ ℕ, 1, 0) ∈ ℝ
132 rerpdivcl 11737 . . . . . . . . . . . . . . . 16 ((if((√‘𝑘) ∈ ℕ, 1, 0) ∈ ℝ ∧ (√‘𝑘) ∈ ℝ+) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℝ)
133131, 55, 132sylancr 694 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℝ)
134133recnd 9947 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℂ)
135128, 134syldan 486 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ∈ ℂ)
13689, 135eqeltrrd 2689 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → (1 / (√‘𝑘)) ∈ ℂ)
13792, 60, 121, 124, 136fsumf1o 14301 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))(1 / (√‘𝑘)) = Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / (√‘(𝑖↑2))))
13890, 137eqtrd 2644 . . . . . . . . . 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 3550 . . . . . . . . . . . . . . 15 (𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) ↔ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ ¬ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))))
14050ad2antrl 760 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ∈ ℕ)
141140nncnd 10913 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ∈ ℂ)
142141sqsqrtd 14026 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → ((√‘𝑘)↑2) = 𝑘)
143 simprr 792 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑘) ∈ ℕ)
144 fznnfl 12523 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ ℝ → (𝑘 ∈ (1...(⌊‘𝑥)) ↔ (𝑘 ∈ ℕ ∧ 𝑘𝑥)))
145103, 144syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑘 ∈ (1...(⌊‘𝑥)) ↔ (𝑘 ∈ ℕ ∧ 𝑘𝑥)))
146145simplbda 652 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘𝑥)
147146adantrr 749 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘𝑥)
148140nnrpd 11746 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ∈ ℝ+)
149148rprege0d 11755 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (𝑘 ∈ ℝ ∧ 0 ≤ 𝑘))
15026adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑥 ∈ ℝ+)
151150rprege0d 11755 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
152 sqrtle 13849 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (𝑘𝑥 ↔ (√‘𝑘) ≤ (√‘𝑥)))
153149, 151, 152syl2anc 691 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (𝑘𝑥 ↔ (√‘𝑘) ≤ (√‘𝑥)))
154147, 153mpbid 221 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑘) ≤ (√‘𝑥))
15568adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑥) ∈ ℝ+)
156155rpred 11748 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑥) ∈ ℝ)
157 fznnfl 12523 . . . . . . . . . . . . . . . . . . . . . 22 ((√‘𝑥) ∈ ℝ → ((√‘𝑘) ∈ (1...(⌊‘(√‘𝑥))) ↔ ((√‘𝑘) ∈ ℕ ∧ (√‘𝑘) ≤ (√‘𝑥))))
158156, 157syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → ((√‘𝑘) ∈ (1...(⌊‘(√‘𝑥))) ↔ ((√‘𝑘) ∈ ℕ ∧ (√‘𝑘) ≤ (√‘𝑥))))
159143, 154, 158mpbir2and 959 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → (√‘𝑘) ∈ (1...(⌊‘(√‘𝑥))))
160142, 140eqeltrd 2688 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → ((√‘𝑘)↑2) ∈ ℕ)
161 oveq1 6556 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = (√‘𝑘) → (𝑚↑2) = ((√‘𝑘)↑2))
16272, 161elrnmpt1s 5294 . . . . . . . . . . . . . . . . . . . 20 (((√‘𝑘) ∈ (1...(⌊‘(√‘𝑥))) ∧ ((√‘𝑘)↑2) ∈ ℕ) → ((√‘𝑘)↑2) ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))
163159, 160, 162syl2anc 691 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → ((√‘𝑘)↑2) ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))
164142, 163eqeltrrd 2689 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ (√‘𝑘) ∈ ℕ)) → 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))
165164expr 641 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((√‘𝑘) ∈ ℕ → 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))))
166165con3d 147 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (¬ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)) → ¬ (√‘𝑘) ∈ ℕ))
167166impr 647 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ ¬ 𝑘 ∈ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → ¬ (√‘𝑘) ∈ ℕ)
168139, 167sylan2b 491 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → ¬ (√‘𝑘) ∈ ℕ)
169168iffalsed 4047 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → if((√‘𝑘) ∈ ℕ, 1, 0) = 0)
170169oveq1d 6564 . . . . . . . . . . . 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 3694 . . . . . . . . . . . . . . 15 (𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2))) → 𝑘 ∈ (1...(⌊‘𝑥)))
172171, 55sylan2 490 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → (√‘𝑘) ∈ ℝ+)
173172rpcnne0d 11757 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → ((√‘𝑘) ∈ ℂ ∧ (√‘𝑘) ≠ 0))
174 div0 10594 . . . . . . . . . . . . 13 (((√‘𝑘) ∈ ℂ ∧ (√‘𝑘) ≠ 0) → (0 / (√‘𝑘)) = 0)
175173, 174syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → (0 / (√‘𝑘)) = 0)
176170, 175eqtrd 2644 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ ((1...(⌊‘𝑥)) ∖ ran (𝑚 ∈ (1...(⌊‘(√‘𝑥))) ↦ (𝑚↑2)))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) = 0)
177127, 135, 176, 38fsumss 14303 . . . . . . . . . 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)))
17862nnrpd 11746 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → 𝑖 ∈ ℝ+)
179178rprege0d 11755 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖))
180 sqrtsq 13858 . . . . . . . . . . . . 13 ((𝑖 ∈ ℝ ∧ 0 ≤ 𝑖) → (√‘(𝑖↑2)) = 𝑖)
181179, 180syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → (√‘(𝑖↑2)) = 𝑖)
182181oveq2d 6565 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑖 ∈ (1...(⌊‘(√‘𝑥)))) → (1 / (√‘(𝑖↑2))) = (1 / 𝑖))
183182sumeq2dv 14281 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / (√‘(𝑖↑2))) = Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖))
184138, 177, 1833eqtr3d 2652 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))(if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) = Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖))
185131a1i 11 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → if((√‘𝑘) ∈ ℕ, 1, 0) ∈ ℝ)
18641ad2antrr 758 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑁 ∈ ℕ)
18746ad2antrr 758 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑋𝐷)
18847ad2antrr 758 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑋:(Base‘𝑍)⟶ℝ)
18939, 40, 186, 42, 43, 44, 45, 187, 188, 53dchrisum0flb 24999 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → if((√‘𝑘) ∈ ℕ, 1, 0) ≤ (𝐹𝑘))
190185, 52, 55, 189lediv1dd 11806 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ≤ ((𝐹𝑘) / (√‘𝑘)))
19138, 133, 56, 190fsumle 14372 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))(if((√‘𝑘) ∈ ℕ, 1, 0) / (√‘𝑘)) ≤ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹𝑘) / (√‘𝑘)))
192184, 191eqbrtrrd 4607 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑖 ∈ (1...(⌊‘(√‘𝑥)))(1 / 𝑖) ≤ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹𝑘) / (√‘𝑘)))
19321, 64, 57, 71, 192letrd 10073 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) / 2) ≤ Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹𝑘) / (√‘𝑘)))
19457leabsd 14001 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹𝑘) / (√‘𝑘)) ≤ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹𝑘) / (√‘𝑘))))
19521, 57, 59, 193, 194letrd 10073 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘𝑥) / 2) ≤ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹𝑘) / (√‘𝑘))))
19637, 195eqbrtrd 4605 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘((log‘𝑥) / 2)) ≤ (abs‘Σ𝑘 ∈ (1...(⌊‘𝑥))((𝐹𝑘) / (√‘𝑘))))
19717, 18, 20, 11, 196o1le 14231 . . . 4 (𝜑 → (𝑥 ∈ ℝ+ ↦ ((log‘𝑥) / 2)) ∈ 𝑂(1))
1985, 11, 16, 197o1mul2 14203 . . 3 (𝜑 → (𝑥 ∈ ℝ+ ↦ (2 · ((log‘𝑥) / 2))) ∈ 𝑂(1))
1999, 198eqeltrrd 2689 . 2 (𝜑 → (𝑥 ∈ ℝ+ ↦ (log‘𝑥)) ∈ 𝑂(1))
2001, 199mto 187 1 ¬ 𝜑
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   · cmul 9820   < clt 9953   ≤ cle 9954   / cdiv 10563  ℕcn 10897  2c2 10947  ℝ+crp 11708  ...cfz 12197  ⌊cfl 12453  ↑cexp 12722  √csqrt 13821  abscabs 13822  𝑂(1)co1 14065  Σcsu 14264   ∥ cdvds 14821  Basecbs 15695  0gc0g 15923  ℤRHomczrh 19667  ℤ/nℤczn 19670  logclog 24105  DChrcdchr 24757 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-o1 14069  df-lo1 14070  df-sum 14265  df-ef 14637  df-e 14638  df-sin 14639  df-cos 14640  df-pi 14642  df-dvds 14822  df-gcd 15055  df-prm 15224  df-numer 15281  df-denom 15282  df-pc 15380  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-qus 15992  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-nsg 17415  df-eqg 17416  df-ghm 17481  df-cntz 17573  df-od 17771  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-rnghom 18538  df-drng 18572  df-subrg 18601  df-lmod 18688  df-lss 18754  df-lsp 18793  df-sra 18993  df-rgmod 18994  df-lidl 18995  df-rsp 18996  df-2idl 19053  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-zring 19638  df-zrh 19671  df-zn 19674  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-haus 20929  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-limc 23436  df-dv 23437  df-log 24107  df-cxp 24108  df-em 24519  df-dchr 24758 This theorem is referenced by:  dchrisum0  25009
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