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Theorem log2sumbnd 26232
 Description: Bound on the difference between Σ𝑛 ≤ 𝐴, log↑2(𝑛) and the equivalent integral. (Contributed by Mario Carneiro, 20-May-2016.)
Assertion
Ref Expression
log2sumbnd ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) ≤ (((log‘𝐴)↑2) + 2))
Distinct variable group:   𝐴,𝑛

Proof of Theorem log2sumbnd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzfid 13395 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (1...(⌊‘𝐴)) ∈ Fin)
2 elfznn 12990 . . . . . . . . . . . 12 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
32adantl 485 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
43nnrpd 12475 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ+)
54relogcld 25318 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℝ)
65resqcld 13666 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((log‘𝑛)↑2) ∈ ℝ)
71, 6fsumrecl 15144 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) ∈ ℝ)
8 rpre 12443 . . . . . . . . 9 (𝐴 ∈ ℝ+𝐴 ∈ ℝ)
98adantr 484 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ)
10 relogcl 25271 . . . . . . . . . . 11 (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ)
1110adantr 484 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℝ)
1211resqcld 13666 . . . . . . . . 9 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((log‘𝐴)↑2) ∈ ℝ)
13 2re 11753 . . . . . . . . . 10 2 ∈ ℝ
14 remulcl 10665 . . . . . . . . . . 11 ((2 ∈ ℝ ∧ (log‘𝐴) ∈ ℝ) → (2 · (log‘𝐴)) ∈ ℝ)
1513, 11, 14sylancr 590 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (2 · (log‘𝐴)) ∈ ℝ)
16 resubcl 10993 . . . . . . . . . 10 ((2 ∈ ℝ ∧ (2 · (log‘𝐴)) ∈ ℝ) → (2 − (2 · (log‘𝐴))) ∈ ℝ)
1713, 15, 16sylancr 590 . . . . . . . . 9 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (2 − (2 · (log‘𝐴))) ∈ ℝ)
1812, 17readdcld 10713 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))) ∈ ℝ)
199, 18remulcld 10714 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))) ∈ ℝ)
207, 19resubcld 11111 . . . . . 6 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) ∈ ℝ)
2120recnd 10712 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) ∈ ℂ)
2221abscld 14849 . . . 4 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) ∈ ℝ)
23 resubcl 10993 . . . 4 (((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) ∈ ℝ ∧ 2 ∈ ℝ) → ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − 2) ∈ ℝ)
2422, 13, 23sylancl 589 . . 3 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − 2) ∈ ℝ)
25 2cn 11754 . . . . . 6 2 ∈ ℂ
2625negcli 10997 . . . . 5 -2 ∈ ℂ
27 subcl 10928 . . . . 5 (((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) ∈ ℂ ∧ -2 ∈ ℂ) → ((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2) ∈ ℂ)
2821, 26, 27sylancl 589 . . . 4 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2) ∈ ℂ)
2928abscld 14849 . . 3 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2)) ∈ ℝ)
3025absnegi 14813 . . . . . 6 (abs‘-2) = (abs‘2)
31 0le2 11781 . . . . . . 7 0 ≤ 2
32 absid 14709 . . . . . . 7 ((2 ∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2)
3313, 31, 32mp2an 691 . . . . . 6 (abs‘2) = 2
3430, 33eqtri 2781 . . . . 5 (abs‘-2) = 2
3534oveq2i 7166 . . . 4 ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − (abs‘-2)) = ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − 2)
36 abs2dif 14745 . . . . 5 (((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) ∈ ℂ ∧ -2 ∈ ℂ) → ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − (abs‘-2)) ≤ (abs‘((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2)))
3721, 26, 36sylancl 589 . . . 4 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − (abs‘-2)) ≤ (abs‘((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2)))
3835, 37eqbrtrrid 5071 . . 3 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − 2) ≤ (abs‘((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2)))
39 fveq2 6662 . . . . . . . . . . 11 (𝑥 = 𝐴 → (⌊‘𝑥) = (⌊‘𝐴))
4039oveq2d 7171 . . . . . . . . . 10 (𝑥 = 𝐴 → (1...(⌊‘𝑥)) = (1...(⌊‘𝐴)))
4140sumeq1d 15111 . . . . . . . . 9 (𝑥 = 𝐴 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2))
42 id 22 . . . . . . . . . 10 (𝑥 = 𝐴𝑥 = 𝐴)
43 fveq2 6662 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
4443oveq1d 7170 . . . . . . . . . . 11 (𝑥 = 𝐴 → ((log‘𝑥)↑2) = ((log‘𝐴)↑2))
4543oveq2d 7171 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (2 · (log‘𝑥)) = (2 · (log‘𝐴)))
4645oveq2d 7171 . . . . . . . . . . 11 (𝑥 = 𝐴 → (2 − (2 · (log‘𝑥))) = (2 − (2 · (log‘𝐴))))
4744, 46oveq12d 7173 . . . . . . . . . 10 (𝑥 = 𝐴 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))
4842, 47oveq12d 7173 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))
4941, 48oveq12d 7173 . . . . . . . 8 (𝑥 = 𝐴 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))))
50 eqid 2758 . . . . . . . 8 (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))))) = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))
51 ovex 7188 . . . . . . . 8 𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) ∈ V
5249, 50, 51fvmpt3i 6768 . . . . . . 7 (𝐴 ∈ ℝ+ → ((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘𝐴) = (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))))
5352adantr 484 . . . . . 6 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘𝐴) = (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))))
54 1rp 12439 . . . . . . 7 1 ∈ ℝ+
55 fveq2 6662 . . . . . . . . . . . . . 14 (𝑥 = 1 → (⌊‘𝑥) = (⌊‘1))
56 1z 12056 . . . . . . . . . . . . . . 15 1 ∈ ℤ
57 flid 13232 . . . . . . . . . . . . . . 15 (1 ∈ ℤ → (⌊‘1) = 1)
5856, 57ax-mp 5 . . . . . . . . . . . . . 14 (⌊‘1) = 1
5955, 58eqtrdi 2809 . . . . . . . . . . . . 13 (𝑥 = 1 → (⌊‘𝑥) = 1)
6059oveq2d 7171 . . . . . . . . . . . 12 (𝑥 = 1 → (1...(⌊‘𝑥)) = (1...1))
6160sumeq1d 15111 . . . . . . . . . . 11 (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = Σ𝑛 ∈ (1...1)((log‘𝑛)↑2))
62 0cn 10676 . . . . . . . . . . . 12 0 ∈ ℂ
63 fveq2 6662 . . . . . . . . . . . . . . 15 (𝑛 = 1 → (log‘𝑛) = (log‘1))
64 log1 25281 . . . . . . . . . . . . . . 15 (log‘1) = 0
6563, 64eqtrdi 2809 . . . . . . . . . . . . . 14 (𝑛 = 1 → (log‘𝑛) = 0)
6665sq0id 13612 . . . . . . . . . . . . 13 (𝑛 = 1 → ((log‘𝑛)↑2) = 0)
6766fsum1 15155 . . . . . . . . . . . 12 ((1 ∈ ℤ ∧ 0 ∈ ℂ) → Σ𝑛 ∈ (1...1)((log‘𝑛)↑2) = 0)
6856, 62, 67mp2an 691 . . . . . . . . . . 11 Σ𝑛 ∈ (1...1)((log‘𝑛)↑2) = 0
6961, 68eqtrdi 2809 . . . . . . . . . 10 (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = 0)
70 id 22 . . . . . . . . . . . 12 (𝑥 = 1 → 𝑥 = 1)
71 fveq2 6662 . . . . . . . . . . . . . . . 16 (𝑥 = 1 → (log‘𝑥) = (log‘1))
7271, 64eqtrdi 2809 . . . . . . . . . . . . . . 15 (𝑥 = 1 → (log‘𝑥) = 0)
7372sq0id 13612 . . . . . . . . . . . . . 14 (𝑥 = 1 → ((log‘𝑥)↑2) = 0)
7472oveq2d 7171 . . . . . . . . . . . . . . . . 17 (𝑥 = 1 → (2 · (log‘𝑥)) = (2 · 0))
75 2t0e0 11848 . . . . . . . . . . . . . . . . 17 (2 · 0) = 0
7674, 75eqtrdi 2809 . . . . . . . . . . . . . . . 16 (𝑥 = 1 → (2 · (log‘𝑥)) = 0)
7776oveq2d 7171 . . . . . . . . . . . . . . 15 (𝑥 = 1 → (2 − (2 · (log‘𝑥))) = (2 − 0))
7825subid1i 11001 . . . . . . . . . . . . . . 15 (2 − 0) = 2
7977, 78eqtrdi 2809 . . . . . . . . . . . . . 14 (𝑥 = 1 → (2 − (2 · (log‘𝑥))) = 2)
8073, 79oveq12d 7173 . . . . . . . . . . . . 13 (𝑥 = 1 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = (0 + 2))
8125addid2i 10871 . . . . . . . . . . . . 13 (0 + 2) = 2
8280, 81eqtrdi 2809 . . . . . . . . . . . 12 (𝑥 = 1 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = 2)
8370, 82oveq12d 7173 . . . . . . . . . . 11 (𝑥 = 1 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (1 · 2))
8425mulid2i 10689 . . . . . . . . . . 11 (1 · 2) = 2
8583, 84eqtrdi 2809 . . . . . . . . . 10 (𝑥 = 1 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = 2)
8669, 85oveq12d 7173 . . . . . . . . 9 (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = (0 − 2))
87 df-neg 10916 . . . . . . . . 9 -2 = (0 − 2)
8886, 87eqtr4di 2811 . . . . . . . 8 (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = -2)
8988, 50, 51fvmpt3i 6768 . . . . . . 7 (1 ∈ ℝ+ → ((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘1) = -2)
9054, 89mp1i 13 . . . . . 6 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘1) = -2)
9153, 90oveq12d 7173 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘𝐴) − ((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘1)) = ((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2))
9291fveq2d 6666 . . . 4 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘(((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘𝐴) − ((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘1))) = (abs‘((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2)))
93 ioorp 12862 . . . . . 6 (0(,)+∞) = ℝ+
9493eqcomi 2767 . . . . 5 + = (0(,)+∞)
95 nnuz 12326 . . . . 5 ℕ = (ℤ‘1)
9656a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ∈ ℤ)
97 1red 10685 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ)
98 pnfxr 10738 . . . . . 6 +∞ ∈ ℝ*
9998a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → +∞ ∈ ℝ*)
100 1re 10684 . . . . . . 7 1 ∈ ℝ
101 1nn0 11955 . . . . . . 7 1 ∈ ℕ0
102100, 101nn0addge1i 11987 . . . . . 6 1 ≤ (1 + 1)
103102a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ≤ (1 + 1))
104 0red 10687 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 0 ∈ ℝ)
105 rpre 12443 . . . . . . 7 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
106105adantl 485 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ)
107 simpr 488 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
108107relogcld 25318 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ)
109108resqcld 13666 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥)↑2) ∈ ℝ)
110 remulcl 10665 . . . . . . . . 9 ((2 ∈ ℝ ∧ (log‘𝑥) ∈ ℝ) → (2 · (log‘𝑥)) ∈ ℝ)
11113, 108, 110sylancr 590 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 · (log‘𝑥)) ∈ ℝ)
112 resubcl 10993 . . . . . . . 8 ((2 ∈ ℝ ∧ (2 · (log‘𝑥)) ∈ ℝ) → (2 − (2 · (log‘𝑥))) ∈ ℝ)
11313, 111, 112sylancr 590 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 − (2 · (log‘𝑥))) ∈ ℝ)
114109, 113readdcld 10713 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) ∈ ℝ)
115106, 114remulcld 10714 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) ∈ ℝ)
116 nnrp 12446 . . . . . 6 (𝑥 ∈ ℕ → 𝑥 ∈ ℝ+)
117116, 109sylan2 595 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℕ) → ((log‘𝑥)↑2) ∈ ℝ)
118 reelprrecn 10672 . . . . . . . 8 ℝ ∈ {ℝ, ℂ}
119118a1i 11 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℝ ∈ {ℝ, ℂ})
120106recnd 10712 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ)
121 1red 10685 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 1 ∈ ℝ)
122 recn 10670 . . . . . . . . 9 (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
123122adantl 485 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
124 1red 10685 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
125119dvmptid 24661 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
126 rpssre 12442 . . . . . . . . 9 + ⊆ ℝ
127126a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℝ+ ⊆ ℝ)
128 eqid 2758 . . . . . . . . 9 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
129128tgioo2 23509 . . . . . . . 8 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
130 iooretop 23472 . . . . . . . . . 10 (0(,)+∞) ∈ (topGen‘ran (,))
13193, 130eqeltrri 2849 . . . . . . . . 9 + ∈ (topGen‘ran (,))
132131a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℝ+ ∈ (topGen‘ran (,)))
133119, 123, 124, 125, 127, 129, 128, 132dvmptres 24667 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+𝑥)) = (𝑥 ∈ ℝ+ ↦ 1))
134114recnd 10712 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) ∈ ℂ)
135 resubcl 10993 . . . . . . . . 9 (((2 · (log‘𝑥)) ∈ ℝ ∧ 2 ∈ ℝ) → ((2 · (log‘𝑥)) − 2) ∈ ℝ)
136111, 13, 135sylancl 589 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((2 · (log‘𝑥)) − 2) ∈ ℝ)
137136, 107rerpdivcld 12508 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) − 2) / 𝑥) ∈ ℝ)
138109recnd 10712 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥)↑2) ∈ ℂ)
139111recnd 10712 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 · (log‘𝑥)) ∈ ℂ)
140107rpreccld 12487 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ+)
141140rpcnd 12479 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℂ)
142139, 141mulcld 10704 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((2 · (log‘𝑥)) · (1 / 𝑥)) ∈ ℂ)
143 cnelprrecn 10673 . . . . . . . . . . 11 ℂ ∈ {ℝ, ℂ}
144143a1i 11 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℂ ∈ {ℝ, ℂ})
145108recnd 10712 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ)
146 sqcl 13539 . . . . . . . . . . 11 (𝑦 ∈ ℂ → (𝑦↑2) ∈ ℂ)
147146adantl 485 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → (𝑦↑2) ∈ ℂ)
148 simpr 488 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
149 mulcl 10664 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (2 · 𝑦) ∈ ℂ)
15025, 148, 149sylancr 590 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → (2 · 𝑦) ∈ ℂ)
151 dvrelog 25332 . . . . . . . . . . 11 (ℝ D (log ↾ ℝ+)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))
152 relogf1o 25262 . . . . . . . . . . . . . . 15 (log ↾ ℝ+):ℝ+1-1-onto→ℝ
153 f1of 6606 . . . . . . . . . . . . . . 15 ((log ↾ ℝ+):ℝ+1-1-onto→ℝ → (log ↾ ℝ+):ℝ+⟶ℝ)
154152, 153mp1i 13 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log ↾ ℝ+):ℝ+⟶ℝ)
155154feqmptd 6725 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ ((log ↾ ℝ+)‘𝑥)))
156 fvres 6681 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+ → ((log ↾ ℝ+)‘𝑥) = (log‘𝑥))
157156mpteq2ia 5126 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ ↦ ((log ↾ ℝ+)‘𝑥)) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))
158155, 157eqtrdi 2809 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥)))
159158oveq2d 7171 . . . . . . . . . . 11 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (log ↾ ℝ+)) = (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))))
160151, 159syl5reqr 2808 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)))
161 2nn 11752 . . . . . . . . . . . 12 2 ∈ ℕ
162 dvexp 24657 . . . . . . . . . . . 12 (2 ∈ ℕ → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑2))) = (𝑦 ∈ ℂ ↦ (2 · (𝑦↑(2 − 1)))))
163161, 162mp1i 13 . . . . . . . . . . 11 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑2))) = (𝑦 ∈ ℂ ↦ (2 · (𝑦↑(2 − 1)))))
164 2m1e1 11805 . . . . . . . . . . . . . . 15 (2 − 1) = 1
165164oveq2i 7166 . . . . . . . . . . . . . 14 (𝑦↑(2 − 1)) = (𝑦↑1)
166 exp1 13490 . . . . . . . . . . . . . 14 (𝑦 ∈ ℂ → (𝑦↑1) = 𝑦)
167165, 166syl5eq 2805 . . . . . . . . . . . . 13 (𝑦 ∈ ℂ → (𝑦↑(2 − 1)) = 𝑦)
168167oveq2d 7171 . . . . . . . . . . . 12 (𝑦 ∈ ℂ → (2 · (𝑦↑(2 − 1))) = (2 · 𝑦))
169168mpteq2ia 5126 . . . . . . . . . . 11 (𝑦 ∈ ℂ ↦ (2 · (𝑦↑(2 − 1)))) = (𝑦 ∈ ℂ ↦ (2 · 𝑦))
170163, 169eqtrdi 2809 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑2))) = (𝑦 ∈ ℂ ↦ (2 · 𝑦)))
171 oveq1 7162 . . . . . . . . . 10 (𝑦 = (log‘𝑥) → (𝑦↑2) = ((log‘𝑥)↑2))
172 oveq2 7163 . . . . . . . . . 10 (𝑦 = (log‘𝑥) → (2 · 𝑦) = (2 · (log‘𝑥)))
173119, 144, 145, 140, 147, 150, 160, 170, 171, 172dvmptco 24676 . . . . . . . . 9 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ ((log‘𝑥)↑2))) = (𝑥 ∈ ℝ+ ↦ ((2 · (log‘𝑥)) · (1 / 𝑥))))
174113recnd 10712 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 − (2 · (log‘𝑥))) ∈ ℂ)
175 ovexd 7190 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (0 − (2 · (1 / 𝑥))) ∈ V)
176 2cnd 11757 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 2 ∈ ℂ)
177 0red 10687 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 0 ∈ ℝ)
178 2cnd 11757 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 2 ∈ ℂ)
179 0red 10687 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
180 2cnd 11757 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 2 ∈ ℂ)
181119, 180dvmptc 24662 . . . . . . . . . . 11 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 2)) = (𝑥 ∈ ℝ ↦ 0))
182119, 178, 179, 181, 127, 129, 128, 132dvmptres 24667 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ 2)) = (𝑥 ∈ ℝ+ ↦ 0))
183 mulcl 10664 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ (1 / 𝑥) ∈ ℂ) → (2 · (1 / 𝑥)) ∈ ℂ)
18425, 141, 183sylancr 590 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 · (1 / 𝑥)) ∈ ℂ)
185119, 145, 140, 160, 180dvmptcmul 24668 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (2 · (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ (2 · (1 / 𝑥))))
186119, 176, 177, 182, 139, 184, 185dvmptsub 24671 . . . . . . . . 9 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (2 − (2 · (log‘𝑥))))) = (𝑥 ∈ ℝ+ ↦ (0 − (2 · (1 / 𝑥)))))
187119, 138, 142, 173, 174, 175, 186dvmptadd 24664 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = (𝑥 ∈ ℝ+ ↦ (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥))))))
188139, 176, 141subdird 11140 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) − 2) · (1 / 𝑥)) = (((2 · (log‘𝑥)) · (1 / 𝑥)) − (2 · (1 / 𝑥))))
189136recnd 10712 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((2 · (log‘𝑥)) − 2) ∈ ℂ)
190 rpne0 12451 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+𝑥 ≠ 0)
191190adantl 485 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0)
192189, 120, 191divrecd 11462 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) − 2) / 𝑥) = (((2 · (log‘𝑥)) − 2) · (1 / 𝑥)))
193 df-neg 10916 . . . . . . . . . . . 12 -(2 · (1 / 𝑥)) = (0 − (2 · (1 / 𝑥)))
194193oveq2i 7166 . . . . . . . . . . 11 (((2 · (log‘𝑥)) · (1 / 𝑥)) + -(2 · (1 / 𝑥))) = (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥))))
195142, 184negsubd 11046 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) · (1 / 𝑥)) + -(2 · (1 / 𝑥))) = (((2 · (log‘𝑥)) · (1 / 𝑥)) − (2 · (1 / 𝑥))))
196194, 195syl5eqr 2807 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥)))) = (((2 · (log‘𝑥)) · (1 / 𝑥)) − (2 · (1 / 𝑥))))
197188, 192, 1963eqtr4rd 2804 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥)))) = (((2 · (log‘𝑥)) − 2) / 𝑥))
198197mpteq2dva 5130 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (𝑥 ∈ ℝ+ ↦ (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥))))) = (𝑥 ∈ ℝ+ ↦ (((2 · (log‘𝑥)) − 2) / 𝑥)))
199187, 198eqtrd 2793 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = (𝑥 ∈ ℝ+ ↦ (((2 · (log‘𝑥)) − 2) / 𝑥)))
200119, 120, 121, 133, 134, 137, 199dvmptmul 24665 . . . . . 6 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))))) = (𝑥 ∈ ℝ+ ↦ ((1 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) + ((((2 · (log‘𝑥)) − 2) / 𝑥) · 𝑥))))
201134mulid2d 10702 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (1 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))
202138, 139, 176subsub2d 11069 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑2) − ((2 · (log‘𝑥)) − 2)) = (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))
203201, 202eqtr4d 2796 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (1 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (((log‘𝑥)↑2) − ((2 · (log‘𝑥)) − 2)))
204189, 120, 191divcan1d 11460 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((((2 · (log‘𝑥)) − 2) / 𝑥) · 𝑥) = ((2 · (log‘𝑥)) − 2))
205203, 204oveq12d 7173 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((1 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) + ((((2 · (log‘𝑥)) − 2) / 𝑥) · 𝑥)) = ((((log‘𝑥)↑2) − ((2 · (log‘𝑥)) − 2)) + ((2 · (log‘𝑥)) − 2)))
206138, 189npcand 11044 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((((log‘𝑥)↑2) − ((2 · (log‘𝑥)) − 2)) + ((2 · (log‘𝑥)) − 2)) = ((log‘𝑥)↑2))
207205, 206eqtrd 2793 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((1 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) + ((((2 · (log‘𝑥)) − 2) / 𝑥) · 𝑥)) = ((log‘𝑥)↑2))
208207mpteq2dva 5130 . . . . . 6 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (𝑥 ∈ ℝ+ ↦ ((1 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) + ((((2 · (log‘𝑥)) − 2) / 𝑥) · 𝑥))) = (𝑥 ∈ ℝ+ ↦ ((log‘𝑥)↑2)))
209200, 208eqtrd 2793 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))))) = (𝑥 ∈ ℝ+ ↦ ((log‘𝑥)↑2)))
210 fveq2 6662 . . . . . 6 (𝑥 = 𝑛 → (log‘𝑥) = (log‘𝑛))
211210oveq1d 7170 . . . . 5 (𝑥 = 𝑛 → ((log‘𝑥)↑2) = ((log‘𝑛)↑2))
212 simp32 1207 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑥𝑛)
213 simp2l 1196 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑥 ∈ ℝ+)
214 simp2r 1197 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑛 ∈ ℝ+)
215213, 214logled 25322 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (𝑥𝑛 ↔ (log‘𝑥) ≤ (log‘𝑛)))
216212, 215mpbid 235 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (log‘𝑥) ≤ (log‘𝑛))
217213relogcld 25318 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (log‘𝑥) ∈ ℝ)
218214relogcld 25318 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (log‘𝑛) ∈ ℝ)
219 simp31 1206 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 1 ≤ 𝑥)
220 logleb 25298 . . . . . . . . . 10 ((1 ∈ ℝ+𝑥 ∈ ℝ+) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
22154, 213, 220sylancr 590 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
222219, 221mpbid 235 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (log‘1) ≤ (log‘𝑥))
22364, 222eqbrtrrid 5071 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 0 ≤ (log‘𝑥))
224214rpred 12477 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑛 ∈ ℝ)
225 1red 10685 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 1 ∈ ℝ)
226213rpred 12477 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑥 ∈ ℝ)
227225, 226, 224, 219, 212letrd 10840 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 1 ≤ 𝑛)
228224, 227logge0d 25325 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 0 ≤ (log‘𝑛))
229217, 218, 223, 228le2sqd 13675 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → ((log‘𝑥) ≤ (log‘𝑛) ↔ ((log‘𝑥)↑2) ≤ ((log‘𝑛)↑2)))
230216, 229mpbid 235 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → ((log‘𝑥)↑2) ≤ ((log‘𝑛)↑2))
231 relogcl 25271 . . . . . . 7 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
232231ad2antrl 727 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
233232sqge0d 13667 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘𝑥)↑2))
23454a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ+)
235 simpl 486 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ+)
236 1le1 11311 . . . . . 6 1 ≤ 1
237236a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ≤ 1)
238 simpr 488 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ≤ 𝐴)
2399rexrd 10734 . . . . . 6 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ*)
240 pnfge 12571 . . . . . 6 (𝐴 ∈ ℝ*𝐴 ≤ +∞)
241239, 240syl 17 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ≤ +∞)
24294, 95, 96, 97, 99, 103, 104, 115, 109, 117, 209, 211, 230, 50, 233, 234, 235, 237, 238, 241, 44dvfsum2 24738 . . . 4 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘(((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘𝐴) − ((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))))‘1))) ≤ ((log‘𝐴)↑2))
24392, 242eqbrtrrd 5059 . . 3 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2)) ≤ ((log‘𝐴)↑2))
24424, 29, 12, 38, 243letrd 10840 . 2 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − 2) ≤ ((log‘𝐴)↑2))
24513a1i 11 . . 3 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 2 ∈ ℝ)
24622, 245, 12lesubaddd 11280 . 2 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − 2) ≤ ((log‘𝐴)↑2) ↔ (abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) ≤ (((log‘𝐴)↑2) + 2)))
247244, 246mpbid 235 1 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) ≤ (((log‘𝐴)↑2) + 2))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  Vcvv 3409   ⊆ wss 3860  {cpr 4527   class class class wbr 5035   ↦ cmpt 5115  ran crn 5528   ↾ cres 5529  ⟶wf 6335  –1-1-onto→wf1o 6338  ‘cfv 6339  (class class class)co 7155  ℂcc 10578  ℝcr 10579  0cc0 10580  1c1 10581   + caddc 10583   · cmul 10585  +∞cpnf 10715  ℝ*cxr 10717   ≤ cle 10719   − cmin 10913  -cneg 10914   / cdiv 11340  ℕcn 11679  2c2 11734  ℤcz 12025  ℝ+crp 12435  (,)cioo 12784  ...cfz 12944  ⌊cfl 13214  ↑cexp 13484  abscabs 14646  Σcsu 15095  TopOpenctopn 16758  topGenctg 16774  ℂfldccnfld 20171   D cdv 24567  logclog 25250 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-inf2 9142  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657  ax-pre-sup 10658  ax-addf 10659  ax-mulf 10660 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-iin 4889  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-se 5487  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-of 7410  df-om 7585  df-1st 7698  df-2nd 7699  df-supp 7841  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-1o 8117  df-2o 8118  df-er 8304  df-map 8423  df-pm 8424  df-ixp 8485  df-en 8533  df-dom 8534  df-sdom 8535  df-fin 8536  df-fsupp 8872  df-fi 8913  df-sup 8944  df-inf 8945  df-oi 9012  df-card 9406  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-div 11341  df-nn 11680  df-2 11742  df-3 11743  df-4 11744  df-5 11745  df-6 11746  df-7 11747  df-8 11748  df-9 11749  df-n0 11940  df-z 12026  df-dec 12143  df-uz 12288  df-q 12394  df-rp 12436  df-xneg 12553  df-xadd 12554  df-xmul 12555  df-ioo 12788  df-ioc 12789  df-ico 12790  df-icc 12791  df-fz 12945  df-fzo 13088  df-fl 13216  df-mod 13292  df-seq 13424  df-exp 13485  df-fac 13689  df-bc 13718  df-hash 13746  df-shft 14479  df-cj 14511  df-re 14512  df-im 14513  df-sqrt 14647  df-abs 14648  df-limsup 14881  df-clim 14898  df-rlim 14899  df-sum 15096  df-ef 15474  df-sin 15476  df-cos 15477  df-pi 15479  df-struct 16548  df-ndx 16549  df-slot 16550  df-base 16552  df-sets 16553  df-ress 16554  df-plusg 16641  df-mulr 16642  df-starv 16643  df-sca 16644  df-vsca 16645  df-ip 16646  df-tset 16647  df-ple 16648  df-ds 16650  df-unif 16651  df-hom 16652  df-cco 16653  df-rest 16759  df-topn 16760  df-0g 16778  df-gsum 16779  df-topgen 16780  df-pt 16781  df-prds 16784  df-xrs 16838  df-qtop 16843  df-imas 16844  df-xps 16846  df-mre 16920  df-mrc 16921  df-acs 16923  df-mgm 17923  df-sgrp 17972  df-mnd 17983  df-submnd 18028  df-mulg 18297  df-cntz 18519  df-cmn 18980  df-psmet 20163  df-xmet 20164  df-met 20165  df-bl 20166  df-mopn 20167  df-fbas 20168  df-fg 20169  df-cnfld 20172  df-top 21599  df-topon 21616  df-topsp 21638  df-bases 21651  df-cld 21724  df-ntr 21725  df-cls 21726  df-nei 21803  df-lp 21841  df-perf 21842  df-cn 21932  df-cnp 21933  df-haus 22020  df-cmp 22092  df-tx 22267  df-hmeo 22460  df-fil 22551  df-fm 22643  df-flim 22644  df-flf 22645  df-xms 23027  df-ms 23028  df-tms 23029  df-cncf 23584  df-limc 24570  df-dv 24571  df-log 25252 This theorem is referenced by:  selberglem2  26234
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