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Theorem log2sumbnd 25454
Description: Bound on the difference between Σ𝑛𝐴, 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 12980 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (1...(⌊‘𝐴)) ∈ Fin)
2 elfznn 12577 . . . . . . . . . . . 12 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
32adantl 467 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
43nnrpd 12073 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ+)
54relogcld 24590 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℝ)
65resqcld 13242 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((log‘𝑛)↑2) ∈ ℝ)
71, 6fsumrecl 14673 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) ∈ ℝ)
8 rpre 12042 . . . . . . . . 9 (𝐴 ∈ ℝ+𝐴 ∈ ℝ)
98adantr 466 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ)
10 relogcl 24543 . . . . . . . . . . 11 (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ)
1110adantr 466 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℝ)
1211resqcld 13242 . . . . . . . . 9 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((log‘𝐴)↑2) ∈ ℝ)
13 2re 11292 . . . . . . . . . 10 2 ∈ ℝ
14 remulcl 10223 . . . . . . . . . . 11 ((2 ∈ ℝ ∧ (log‘𝐴) ∈ ℝ) → (2 · (log‘𝐴)) ∈ ℝ)
1513, 11, 14sylancr 575 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (2 · (log‘𝐴)) ∈ ℝ)
16 resubcl 10547 . . . . . . . . . 10 ((2 ∈ ℝ ∧ (2 · (log‘𝐴)) ∈ ℝ) → (2 − (2 · (log‘𝐴))) ∈ ℝ)
1713, 15, 16sylancr 575 . . . . . . . . 9 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (2 − (2 · (log‘𝐴))) ∈ ℝ)
1812, 17readdcld 10271 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))) ∈ ℝ)
199, 18remulcld 10272 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))) ∈ ℝ)
207, 19resubcld 10660 . . . . . 6 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) ∈ ℝ)
2120recnd 10270 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) ∈ ℂ)
2221abscld 14383 . . . 4 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) ∈ ℝ)
23 resubcl 10547 . . . 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 574 . . 3 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − 2) ∈ ℝ)
25 2cn 11293 . . . . . 6 2 ∈ ℂ
2625negcli 10551 . . . . 5 -2 ∈ ℂ
27 subcl 10482 . . . . 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 574 . . . 4 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2) ∈ ℂ)
2928abscld 14383 . . 3 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2)) ∈ ℝ)
3025absnegi 14347 . . . . . 6 (abs‘-2) = (abs‘2)
31 0le2 11313 . . . . . . 7 0 ≤ 2
32 absid 14244 . . . . . . 7 ((2 ∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2)
3313, 31, 32mp2an 672 . . . . . 6 (abs‘2) = 2
3430, 33eqtri 2793 . . . . 5 (abs‘-2) = 2
3534oveq2i 6804 . . . 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 14280 . . . . 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 574 . . . 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, 37syl5eqbrr 4822 . . 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 6332 . . . . . . . . . . 11 (𝑥 = 𝐴 → (⌊‘𝑥) = (⌊‘𝐴))
4039oveq2d 6809 . . . . . . . . . 10 (𝑥 = 𝐴 → (1...(⌊‘𝑥)) = (1...(⌊‘𝐴)))
4140sumeq1d 14639 . . . . . . . . 9 (𝑥 = 𝐴 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2))
42 id 22 . . . . . . . . . 10 (𝑥 = 𝐴𝑥 = 𝐴)
43 fveq2 6332 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
4443oveq1d 6808 . . . . . . . . . . 11 (𝑥 = 𝐴 → ((log‘𝑥)↑2) = ((log‘𝐴)↑2))
4543oveq2d 6809 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (2 · (log‘𝑥)) = (2 · (log‘𝐴)))
4645oveq2d 6809 . . . . . . . . . . 11 (𝑥 = 𝐴 → (2 − (2 · (log‘𝑥))) = (2 − (2 · (log‘𝐴))))
4744, 46oveq12d 6811 . . . . . . . . . 10 (𝑥 = 𝐴 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))
4842, 47oveq12d 6811 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))
4941, 48oveq12d 6811 . . . . . . . 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 2771 . . . . . . . 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 6823 . . . . . . . 8 𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) ∈ V
5249, 50, 51fvmpt3i 6429 . . . . . . 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 466 . . . . . 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 12039 . . . . . . 7 1 ∈ ℝ+
55 fveq2 6332 . . . . . . . . . . . . . 14 (𝑥 = 1 → (⌊‘𝑥) = (⌊‘1))
56 1z 11609 . . . . . . . . . . . . . . 15 1 ∈ ℤ
57 flid 12817 . . . . . . . . . . . . . . 15 (1 ∈ ℤ → (⌊‘1) = 1)
5856, 57ax-mp 5 . . . . . . . . . . . . . 14 (⌊‘1) = 1
5955, 58syl6eq 2821 . . . . . . . . . . . . 13 (𝑥 = 1 → (⌊‘𝑥) = 1)
6059oveq2d 6809 . . . . . . . . . . . 12 (𝑥 = 1 → (1...(⌊‘𝑥)) = (1...1))
6160sumeq1d 14639 . . . . . . . . . . 11 (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = Σ𝑛 ∈ (1...1)((log‘𝑛)↑2))
62 0cn 10234 . . . . . . . . . . . 12 0 ∈ ℂ
63 fveq2 6332 . . . . . . . . . . . . . . 15 (𝑛 = 1 → (log‘𝑛) = (log‘1))
64 log1 24553 . . . . . . . . . . . . . . 15 (log‘1) = 0
6563, 64syl6eq 2821 . . . . . . . . . . . . . 14 (𝑛 = 1 → (log‘𝑛) = 0)
6665sq0id 13164 . . . . . . . . . . . . 13 (𝑛 = 1 → ((log‘𝑛)↑2) = 0)
6766fsum1 14684 . . . . . . . . . . . 12 ((1 ∈ ℤ ∧ 0 ∈ ℂ) → Σ𝑛 ∈ (1...1)((log‘𝑛)↑2) = 0)
6856, 62, 67mp2an 672 . . . . . . . . . . 11 Σ𝑛 ∈ (1...1)((log‘𝑛)↑2) = 0
6961, 68syl6eq 2821 . . . . . . . . . 10 (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = 0)
70 id 22 . . . . . . . . . . . 12 (𝑥 = 1 → 𝑥 = 1)
71 fveq2 6332 . . . . . . . . . . . . . . . 16 (𝑥 = 1 → (log‘𝑥) = (log‘1))
7271, 64syl6eq 2821 . . . . . . . . . . . . . . 15 (𝑥 = 1 → (log‘𝑥) = 0)
7372sq0id 13164 . . . . . . . . . . . . . 14 (𝑥 = 1 → ((log‘𝑥)↑2) = 0)
7472oveq2d 6809 . . . . . . . . . . . . . . . . 17 (𝑥 = 1 → (2 · (log‘𝑥)) = (2 · 0))
75 2t0e0 11385 . . . . . . . . . . . . . . . . 17 (2 · 0) = 0
7674, 75syl6eq 2821 . . . . . . . . . . . . . . . 16 (𝑥 = 1 → (2 · (log‘𝑥)) = 0)
7776oveq2d 6809 . . . . . . . . . . . . . . 15 (𝑥 = 1 → (2 − (2 · (log‘𝑥))) = (2 − 0))
7825subid1i 10555 . . . . . . . . . . . . . . 15 (2 − 0) = 2
7977, 78syl6eq 2821 . . . . . . . . . . . . . 14 (𝑥 = 1 → (2 − (2 · (log‘𝑥))) = 2)
8073, 79oveq12d 6811 . . . . . . . . . . . . 13 (𝑥 = 1 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = (0 + 2))
8125addid2i 10426 . . . . . . . . . . . . 13 (0 + 2) = 2
8280, 81syl6eq 2821 . . . . . . . . . . . 12 (𝑥 = 1 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = 2)
8370, 82oveq12d 6811 . . . . . . . . . . 11 (𝑥 = 1 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (1 · 2))
8425mulid2i 10245 . . . . . . . . . . 11 (1 · 2) = 2
8583, 84syl6eq 2821 . . . . . . . . . 10 (𝑥 = 1 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = 2)
8669, 85oveq12d 6811 . . . . . . . . 9 (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = (0 − 2))
87 df-neg 10471 . . . . . . . . 9 -2 = (0 − 2)
8886, 87syl6eqr 2823 . . . . . . . 8 (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = -2)
8988, 50, 51fvmpt3i 6429 . . . . . . 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 6811 . . . . 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 6336 . . . 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 12456 . . . . . 6 (0(,)+∞) = ℝ+
9493eqcomi 2780 . . . . 5 + = (0(,)+∞)
95 nnuz 11925 . . . . 5 ℕ = (ℤ‘1)
9656a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ∈ ℤ)
97 1red 10257 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ)
98 pnfxr 10294 . . . . . 6 +∞ ∈ ℝ*
9998a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → +∞ ∈ ℝ*)
100 1re 10241 . . . . . . 7 1 ∈ ℝ
101 1nn0 11510 . . . . . . 7 1 ∈ ℕ0
102100, 101nn0addge1i 11543 . . . . . 6 1 ≤ (1 + 1)
103102a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ≤ (1 + 1))
104 0red 10243 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 0 ∈ ℝ)
105 rpre 12042 . . . . . . 7 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
106105adantl 467 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ)
107 simpr 471 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
108107relogcld 24590 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ)
109108resqcld 13242 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥)↑2) ∈ ℝ)
110 remulcl 10223 . . . . . . . . 9 ((2 ∈ ℝ ∧ (log‘𝑥) ∈ ℝ) → (2 · (log‘𝑥)) ∈ ℝ)
11113, 108, 110sylancr 575 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 · (log‘𝑥)) ∈ ℝ)
112 resubcl 10547 . . . . . . . 8 ((2 ∈ ℝ ∧ (2 · (log‘𝑥)) ∈ ℝ) → (2 − (2 · (log‘𝑥))) ∈ ℝ)
11313, 111, 112sylancr 575 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 − (2 · (log‘𝑥))) ∈ ℝ)
114109, 113readdcld 10271 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) ∈ ℝ)
115106, 114remulcld 10272 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) ∈ ℝ)
116 nnrp 12045 . . . . . 6 (𝑥 ∈ ℕ → 𝑥 ∈ ℝ+)
117116, 109sylan2 580 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℕ) → ((log‘𝑥)↑2) ∈ ℝ)
118 reelprrecn 10230 . . . . . . . 8 ℝ ∈ {ℝ, ℂ}
119118a1i 11 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℝ ∈ {ℝ, ℂ})
120106recnd 10270 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ)
121 1red 10257 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 1 ∈ ℝ)
122 recn 10228 . . . . . . . . 9 (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
123122adantl 467 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
124 1red 10257 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
125119dvmptid 23940 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
126 rpssre 12046 . . . . . . . . 9 + ⊆ ℝ
127126a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℝ+ ⊆ ℝ)
128 eqid 2771 . . . . . . . . 9 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
129128tgioo2 22826 . . . . . . . 8 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
130 iooretop 22789 . . . . . . . . . 10 (0(,)+∞) ∈ (topGen‘ran (,))
13193, 130eqeltrri 2847 . . . . . . . . 9 + ∈ (topGen‘ran (,))
132131a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℝ+ ∈ (topGen‘ran (,)))
133119, 123, 124, 125, 127, 129, 128, 132dvmptres 23946 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+𝑥)) = (𝑥 ∈ ℝ+ ↦ 1))
134114recnd 10270 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) ∈ ℂ)
135 resubcl 10547 . . . . . . . . 9 (((2 · (log‘𝑥)) ∈ ℝ ∧ 2 ∈ ℝ) → ((2 · (log‘𝑥)) − 2) ∈ ℝ)
136111, 13, 135sylancl 574 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((2 · (log‘𝑥)) − 2) ∈ ℝ)
137136, 107rerpdivcld 12106 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) − 2) / 𝑥) ∈ ℝ)
138109recnd 10270 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥)↑2) ∈ ℂ)
139111recnd 10270 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 · (log‘𝑥)) ∈ ℂ)
140107rpreccld 12085 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ+)
141140rpcnd 12077 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℂ)
142139, 141mulcld 10262 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((2 · (log‘𝑥)) · (1 / 𝑥)) ∈ ℂ)
143 cnelprrecn 10231 . . . . . . . . . . 11 ℂ ∈ {ℝ, ℂ}
144143a1i 11 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℂ ∈ {ℝ, ℂ})
145108recnd 10270 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ)
146 sqcl 13132 . . . . . . . . . . 11 (𝑦 ∈ ℂ → (𝑦↑2) ∈ ℂ)
147146adantl 467 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → (𝑦↑2) ∈ ℂ)
148 simpr 471 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
149 mulcl 10222 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (2 · 𝑦) ∈ ℂ)
15025, 148, 149sylancr 575 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → (2 · 𝑦) ∈ ℂ)
151 dvrelog 24604 . . . . . . . . . . 11 (ℝ D (log ↾ ℝ+)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))
152 relogf1o 24534 . . . . . . . . . . . . . . 15 (log ↾ ℝ+):ℝ+1-1-onto→ℝ
153 f1of 6278 . . . . . . . . . . . . . . 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 6391 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ ((log ↾ ℝ+)‘𝑥)))
156 fvres 6348 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+ → ((log ↾ ℝ+)‘𝑥) = (log‘𝑥))
157156mpteq2ia 4874 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ ↦ ((log ↾ ℝ+)‘𝑥)) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))
158155, 157syl6eq 2821 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥)))
159158oveq2d 6809 . . . . . . . . . . 11 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (log ↾ ℝ+)) = (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))))
160151, 159syl5reqr 2820 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)))
161 2nn 11387 . . . . . . . . . . . 12 2 ∈ ℕ
162 dvexp 23936 . . . . . . . . . . . 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 11337 . . . . . . . . . . . . . . 15 (2 − 1) = 1
165164oveq2i 6804 . . . . . . . . . . . . . 14 (𝑦↑(2 − 1)) = (𝑦↑1)
166 exp1 13073 . . . . . . . . . . . . . 14 (𝑦 ∈ ℂ → (𝑦↑1) = 𝑦)
167165, 166syl5eq 2817 . . . . . . . . . . . . 13 (𝑦 ∈ ℂ → (𝑦↑(2 − 1)) = 𝑦)
168167oveq2d 6809 . . . . . . . . . . . 12 (𝑦 ∈ ℂ → (2 · (𝑦↑(2 − 1))) = (2 · 𝑦))
169168mpteq2ia 4874 . . . . . . . . . . 11 (𝑦 ∈ ℂ ↦ (2 · (𝑦↑(2 − 1)))) = (𝑦 ∈ ℂ ↦ (2 · 𝑦))
170163, 169syl6eq 2821 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑2))) = (𝑦 ∈ ℂ ↦ (2 · 𝑦)))
171 oveq1 6800 . . . . . . . . . 10 (𝑦 = (log‘𝑥) → (𝑦↑2) = ((log‘𝑥)↑2))
172 oveq2 6801 . . . . . . . . . 10 (𝑦 = (log‘𝑥) → (2 · 𝑦) = (2 · (log‘𝑥)))
173119, 144, 145, 140, 147, 150, 160, 170, 171, 172dvmptco 23955 . . . . . . . . 9 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ ((log‘𝑥)↑2))) = (𝑥 ∈ ℝ+ ↦ ((2 · (log‘𝑥)) · (1 / 𝑥))))
174113recnd 10270 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 − (2 · (log‘𝑥))) ∈ ℂ)
175 ovexd 6825 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (0 − (2 · (1 / 𝑥))) ∈ V)
176 2cnd 11295 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 2 ∈ ℂ)
177 0red 10243 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 0 ∈ ℝ)
178 2cnd 11295 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 2 ∈ ℂ)
179 0red 10243 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
180 2cnd 11295 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 2 ∈ ℂ)
181119, 180dvmptc 23941 . . . . . . . . . . 11 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 2)) = (𝑥 ∈ ℝ ↦ 0))
182119, 178, 179, 181, 127, 129, 128, 132dvmptres 23946 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ 2)) = (𝑥 ∈ ℝ+ ↦ 0))
183 mulcl 10222 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ (1 / 𝑥) ∈ ℂ) → (2 · (1 / 𝑥)) ∈ ℂ)
18425, 141, 183sylancr 575 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 · (1 / 𝑥)) ∈ ℂ)
185119, 145, 140, 160, 180dvmptcmul 23947 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (2 · (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ (2 · (1 / 𝑥))))
186119, 176, 177, 182, 139, 184, 185dvmptsub 23950 . . . . . . . . 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 23943 . . . . . . . 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 10689 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) − 2) · (1 / 𝑥)) = (((2 · (log‘𝑥)) · (1 / 𝑥)) − (2 · (1 / 𝑥))))
189136recnd 10270 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((2 · (log‘𝑥)) − 2) ∈ ℂ)
190 rpne0 12051 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+𝑥 ≠ 0)
191190adantl 467 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0)
192189, 120, 191divrecd 11006 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) − 2) / 𝑥) = (((2 · (log‘𝑥)) − 2) · (1 / 𝑥)))
193 df-neg 10471 . . . . . . . . . . . 12 -(2 · (1 / 𝑥)) = (0 − (2 · (1 / 𝑥)))
194193oveq2i 6804 . . . . . . . . . . 11 (((2 · (log‘𝑥)) · (1 / 𝑥)) + -(2 · (1 / 𝑥))) = (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥))))
195142, 184negsubd 10600 . . . . . . . . . . 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 2819 . . . . . . . . . 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 2816 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥)))) = (((2 · (log‘𝑥)) − 2) / 𝑥))
198197mpteq2dva 4878 . . . . . . . 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 2805 . . . . . . 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 23944 . . . . . 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 10260 . . . . . . . . . 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 10623 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑2) − ((2 · (log‘𝑥)) − 2)) = (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))
203201, 202eqtr4d 2808 . . . . . . . . 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 11004 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((((2 · (log‘𝑥)) − 2) / 𝑥) · 𝑥) = ((2 · (log‘𝑥)) − 2))
205203, 204oveq12d 6811 . . . . . . . 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 10598 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((((log‘𝑥)↑2) − ((2 · (log‘𝑥)) − 2)) + ((2 · (log‘𝑥)) − 2)) = ((log‘𝑥)↑2))
207205, 206eqtrd 2805 . . . . . . 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 4878 . . . . . 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 2805 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))))) = (𝑥 ∈ ℝ+ ↦ ((log‘𝑥)↑2)))
210 fveq2 6332 . . . . . 6 (𝑥 = 𝑛 → (log‘𝑥) = (log‘𝑛))
211210oveq1d 6808 . . . . 5 (𝑥 = 𝑛 → ((log‘𝑥)↑2) = ((log‘𝑛)↑2))
212 simp32 1252 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑥𝑛)
213 simp2l 1241 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑥 ∈ ℝ+)
214 simp2r 1242 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑛 ∈ ℝ+)
215213, 214logled 24594 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (𝑥𝑛 ↔ (log‘𝑥) ≤ (log‘𝑛)))
216212, 215mpbid 222 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (log‘𝑥) ≤ (log‘𝑛))
217213relogcld 24590 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (log‘𝑥) ∈ ℝ)
218214relogcld 24590 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (log‘𝑛) ∈ ℝ)
219 simp31 1251 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 1 ≤ 𝑥)
220 logleb 24570 . . . . . . . . . 10 ((1 ∈ ℝ+𝑥 ∈ ℝ+) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
22154, 213, 220sylancr 575 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
222219, 221mpbid 222 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (log‘1) ≤ (log‘𝑥))
22364, 222syl5eqbrr 4822 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 0 ≤ (log‘𝑥))
224214rpred 12075 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑛 ∈ ℝ)
225 1red 10257 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 1 ∈ ℝ)
226213rpred 12075 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑥 ∈ ℝ)
227225, 226, 224, 219, 212letrd 10396 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 1 ≤ 𝑛)
228224, 227logge0d 24597 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 0 ≤ (log‘𝑛))
229217, 218, 223, 228le2sqd 13251 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → ((log‘𝑥) ≤ (log‘𝑛) ↔ ((log‘𝑥)↑2) ≤ ((log‘𝑛)↑2)))
230216, 229mpbid 222 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → ((log‘𝑥)↑2) ≤ ((log‘𝑛)↑2))
231 relogcl 24543 . . . . . . 7 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
232231ad2antrl 707 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
233232sqge0d 13243 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘𝑥)↑2))
23454a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ+)
235 simpl 468 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ+)
236 1le1 10857 . . . . . 6 1 ≤ 1
237236a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ≤ 1)
238 simpr 471 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ≤ 𝐴)
2399rexrd 10291 . . . . . 6 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ*)
240 pnfge 12169 . . . . . 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 24017 . . . 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 4810 . . 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 10396 . 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 10826 . 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 222 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 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  Vcvv 3351  wss 3723  {cpr 4318   class class class wbr 4786  cmpt 4863  ran crn 5250  cres 5251  wf 6027  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793  cc 10136  cr 10137  0cc0 10138  1c1 10139   + caddc 10141   · cmul 10143  +∞cpnf 10273  *cxr 10275  cle 10277  cmin 10468  -cneg 10469   / cdiv 10886  cn 11222  2c2 11272  cz 11579  +crp 12035  (,)cioo 12380  ...cfz 12533  cfl 12799  cexp 13067  abscabs 14182  Σcsu 14624  TopOpenctopn 16290  topGenctg 16306  fldccnfld 19961   D cdv 23847  logclog 24522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216  ax-addf 10217  ax-mulf 10218
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fsupp 8432  df-fi 8473  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-z 11580  df-dec 11696  df-uz 11889  df-q 11992  df-rp 12036  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-ioo 12384  df-ioc 12385  df-ico 12386  df-icc 12387  df-fz 12534  df-fzo 12674  df-fl 12801  df-mod 12877  df-seq 13009  df-exp 13068  df-fac 13265  df-bc 13294  df-hash 13322  df-shft 14015  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-limsup 14410  df-clim 14427  df-rlim 14428  df-sum 14625  df-ef 15004  df-sin 15006  df-cos 15007  df-pi 15009  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-mulr 16163  df-starv 16164  df-sca 16165  df-vsca 16166  df-ip 16167  df-tset 16168  df-ple 16169  df-ds 16172  df-unif 16173  df-hom 16174  df-cco 16175  df-rest 16291  df-topn 16292  df-0g 16310  df-gsum 16311  df-topgen 16312  df-pt 16313  df-prds 16316  df-xrs 16370  df-qtop 16375  df-imas 16376  df-xps 16378  df-mre 16454  df-mrc 16455  df-acs 16457  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-submnd 17544  df-mulg 17749  df-cntz 17957  df-cmn 18402  df-psmet 19953  df-xmet 19954  df-met 19955  df-bl 19956  df-mopn 19957  df-fbas 19958  df-fg 19959  df-cnfld 19962  df-top 20919  df-topon 20936  df-topsp 20958  df-bases 20971  df-cld 21044  df-ntr 21045  df-cls 21046  df-nei 21123  df-lp 21161  df-perf 21162  df-cn 21252  df-cnp 21253  df-haus 21340  df-cmp 21411  df-tx 21586  df-hmeo 21779  df-fil 21870  df-fm 21962  df-flim 21963  df-flf 21964  df-xms 22345  df-ms 22346  df-tms 22347  df-cncf 22901  df-limc 23850  df-dv 23851  df-log 24524
This theorem is referenced by:  selberglem2  25456
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