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Theorem log2sumbnd 25033
 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 12634 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (1...(⌊‘𝐴)) ∈ Fin)
2 elfznn 12241 . . . . . . . . . . . 12 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
32adantl 481 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
43nnrpd 11746 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ+)
54relogcld 24173 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℝ)
65resqcld 12897 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((log‘𝑛)↑2) ∈ ℝ)
71, 6fsumrecl 14312 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) ∈ ℝ)
8 rpre 11715 . . . . . . . . 9 (𝐴 ∈ ℝ+𝐴 ∈ ℝ)
98adantr 480 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ)
10 relogcl 24126 . . . . . . . . . . 11 (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ)
1110adantr 480 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℝ)
1211resqcld 12897 . . . . . . . . 9 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((log‘𝐴)↑2) ∈ ℝ)
13 2re 10967 . . . . . . . . . 10 2 ∈ ℝ
14 remulcl 9900 . . . . . . . . . . 11 ((2 ∈ ℝ ∧ (log‘𝐴) ∈ ℝ) → (2 · (log‘𝐴)) ∈ ℝ)
1513, 11, 14sylancr 694 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (2 · (log‘𝐴)) ∈ ℝ)
16 resubcl 10224 . . . . . . . . . 10 ((2 ∈ ℝ ∧ (2 · (log‘𝐴)) ∈ ℝ) → (2 − (2 · (log‘𝐴))) ∈ ℝ)
1713, 15, 16sylancr 694 . . . . . . . . 9 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (2 − (2 · (log‘𝐴))) ∈ ℝ)
1812, 17readdcld 9948 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))) ∈ ℝ)
199, 18remulcld 9949 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))) ∈ ℝ)
207, 19resubcld 10337 . . . . . 6 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) ∈ ℝ)
2120recnd 9947 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) ∈ ℂ)
2221abscld 14023 . . . 4 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) ∈ ℝ)
23 resubcl 10224 . . . 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 693 . . 3 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − 2) ∈ ℝ)
25 2cn 10968 . . . . . 6 2 ∈ ℂ
2625negcli 10228 . . . . 5 -2 ∈ ℂ
27 subcl 10159 . . . . 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 693 . . . 4 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2) ∈ ℂ)
2928abscld 14023 . . 3 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2)) ∈ ℝ)
3025absnegi 13987 . . . . . 6 (abs‘-2) = (abs‘2)
31 0le2 10988 . . . . . . 7 0 ≤ 2
32 absid 13884 . . . . . . 7 ((2 ∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2)
3313, 31, 32mp2an 704 . . . . . 6 (abs‘2) = 2
3430, 33eqtri 2632 . . . . 5 (abs‘-2) = 2
3534oveq2i 6560 . . . 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 13920 . . . . 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 693 . . . 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 4619 . . 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 6103 . . . . . . . . . . 11 (𝑥 = 𝐴 → (⌊‘𝑥) = (⌊‘𝐴))
4039oveq2d 6565 . . . . . . . . . 10 (𝑥 = 𝐴 → (1...(⌊‘𝑥)) = (1...(⌊‘𝐴)))
4140sumeq1d 14279 . . . . . . . . 9 (𝑥 = 𝐴 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2))
42 id 22 . . . . . . . . . 10 (𝑥 = 𝐴𝑥 = 𝐴)
43 fveq2 6103 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
4443oveq1d 6564 . . . . . . . . . . 11 (𝑥 = 𝐴 → ((log‘𝑥)↑2) = ((log‘𝐴)↑2))
4543oveq2d 6565 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (2 · (log‘𝑥)) = (2 · (log‘𝐴)))
4645oveq2d 6565 . . . . . . . . . . 11 (𝑥 = 𝐴 → (2 − (2 · (log‘𝑥))) = (2 − (2 · (log‘𝐴))))
4744, 46oveq12d 6567 . . . . . . . . . 10 (𝑥 = 𝐴 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))
4842, 47oveq12d 6567 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))
4941, 48oveq12d 6567 . . . . . . . 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 2610 . . . . . . . 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 6577 . . . . . . . 8 𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) ∈ V
5249, 50, 51fvmpt3i 6196 . . . . . . 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 480 . . . . . 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 11712 . . . . . . 7 1 ∈ ℝ+
55 fveq2 6103 . . . . . . . . . . . . . 14 (𝑥 = 1 → (⌊‘𝑥) = (⌊‘1))
56 1z 11284 . . . . . . . . . . . . . . 15 1 ∈ ℤ
57 flid 12471 . . . . . . . . . . . . . . 15 (1 ∈ ℤ → (⌊‘1) = 1)
5856, 57ax-mp 5 . . . . . . . . . . . . . 14 (⌊‘1) = 1
5955, 58syl6eq 2660 . . . . . . . . . . . . 13 (𝑥 = 1 → (⌊‘𝑥) = 1)
6059oveq2d 6565 . . . . . . . . . . . 12 (𝑥 = 1 → (1...(⌊‘𝑥)) = (1...1))
6160sumeq1d 14279 . . . . . . . . . . 11 (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = Σ𝑛 ∈ (1...1)((log‘𝑛)↑2))
62 0cn 9911 . . . . . . . . . . . 12 0 ∈ ℂ
63 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑛 = 1 → (log‘𝑛) = (log‘1))
64 log1 24136 . . . . . . . . . . . . . . 15 (log‘1) = 0
6563, 64syl6eq 2660 . . . . . . . . . . . . . 14 (𝑛 = 1 → (log‘𝑛) = 0)
6665sq0id 12819 . . . . . . . . . . . . 13 (𝑛 = 1 → ((log‘𝑛)↑2) = 0)
6766fsum1 14320 . . . . . . . . . . . 12 ((1 ∈ ℤ ∧ 0 ∈ ℂ) → Σ𝑛 ∈ (1...1)((log‘𝑛)↑2) = 0)
6856, 62, 67mp2an 704 . . . . . . . . . . 11 Σ𝑛 ∈ (1...1)((log‘𝑛)↑2) = 0
6961, 68syl6eq 2660 . . . . . . . . . 10 (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = 0)
70 id 22 . . . . . . . . . . . 12 (𝑥 = 1 → 𝑥 = 1)
71 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑥 = 1 → (log‘𝑥) = (log‘1))
7271, 64syl6eq 2660 . . . . . . . . . . . . . . 15 (𝑥 = 1 → (log‘𝑥) = 0)
7372sq0id 12819 . . . . . . . . . . . . . 14 (𝑥 = 1 → ((log‘𝑥)↑2) = 0)
7472oveq2d 6565 . . . . . . . . . . . . . . . . 17 (𝑥 = 1 → (2 · (log‘𝑥)) = (2 · 0))
75 2t0e0 11060 . . . . . . . . . . . . . . . . 17 (2 · 0) = 0
7674, 75syl6eq 2660 . . . . . . . . . . . . . . . 16 (𝑥 = 1 → (2 · (log‘𝑥)) = 0)
7776oveq2d 6565 . . . . . . . . . . . . . . 15 (𝑥 = 1 → (2 − (2 · (log‘𝑥))) = (2 − 0))
7825subid1i 10232 . . . . . . . . . . . . . . 15 (2 − 0) = 2
7977, 78syl6eq 2660 . . . . . . . . . . . . . 14 (𝑥 = 1 → (2 − (2 · (log‘𝑥))) = 2)
8073, 79oveq12d 6567 . . . . . . . . . . . . 13 (𝑥 = 1 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = (0 + 2))
8125addid2i 10103 . . . . . . . . . . . . 13 (0 + 2) = 2
8280, 81syl6eq 2660 . . . . . . . . . . . 12 (𝑥 = 1 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = 2)
8370, 82oveq12d 6567 . . . . . . . . . . 11 (𝑥 = 1 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (1 · 2))
8425mulid2i 9922 . . . . . . . . . . 11 (1 · 2) = 2
8583, 84syl6eq 2660 . . . . . . . . . 10 (𝑥 = 1 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = 2)
8669, 85oveq12d 6567 . . . . . . . . 9 (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = (0 − 2))
87 df-neg 10148 . . . . . . . . 9 -2 = (0 − 2)
8886, 87syl6eqr 2662 . . . . . . . 8 (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = -2)
8988, 50, 51fvmpt3i 6196 . . . . . . 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 6567 . . . . 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 6107 . . . 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 12122 . . . . . 6 (0(,)+∞) = ℝ+
9493eqcomi 2619 . . . . 5 + = (0(,)+∞)
95 nnuz 11599 . . . . 5 ℕ = (ℤ‘1)
9656a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ∈ ℤ)
97 1red 9934 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ)
98 pnfxr 9971 . . . . . 6 +∞ ∈ ℝ*
9998a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → +∞ ∈ ℝ*)
100 1re 9918 . . . . . . 7 1 ∈ ℝ
101 1nn0 11185 . . . . . . 7 1 ∈ ℕ0
102100, 101nn0addge1i 11218 . . . . . 6 1 ≤ (1 + 1)
103102a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ≤ (1 + 1))
104 0red 9920 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 0 ∈ ℝ)
105 rpre 11715 . . . . . . 7 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
106105adantl 481 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ)
107 simpr 476 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
108107relogcld 24173 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ)
109108resqcld 12897 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥)↑2) ∈ ℝ)
110 remulcl 9900 . . . . . . . . 9 ((2 ∈ ℝ ∧ (log‘𝑥) ∈ ℝ) → (2 · (log‘𝑥)) ∈ ℝ)
11113, 108, 110sylancr 694 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 · (log‘𝑥)) ∈ ℝ)
112 resubcl 10224 . . . . . . . 8 ((2 ∈ ℝ ∧ (2 · (log‘𝑥)) ∈ ℝ) → (2 − (2 · (log‘𝑥))) ∈ ℝ)
11313, 111, 112sylancr 694 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 − (2 · (log‘𝑥))) ∈ ℝ)
114109, 113readdcld 9948 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) ∈ ℝ)
115106, 114remulcld 9949 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) ∈ ℝ)
116 nnrp 11718 . . . . . 6 (𝑥 ∈ ℕ → 𝑥 ∈ ℝ+)
117116, 109sylan2 490 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℕ) → ((log‘𝑥)↑2) ∈ ℝ)
118 reelprrecn 9907 . . . . . . . 8 ℝ ∈ {ℝ, ℂ}
119118a1i 11 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℝ ∈ {ℝ, ℂ})
120106recnd 9947 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ)
121 1red 9934 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 1 ∈ ℝ)
122 recn 9905 . . . . . . . . 9 (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
123122adantl 481 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
124 1red 9934 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
125119dvmptid 23526 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
126 rpssre 11719 . . . . . . . . 9 + ⊆ ℝ
127126a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℝ+ ⊆ ℝ)
128 eqid 2610 . . . . . . . . 9 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
129128tgioo2 22414 . . . . . . . 8 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
130 iooretop 22379 . . . . . . . . . 10 (0(,)+∞) ∈ (topGen‘ran (,))
13193, 130eqeltrri 2685 . . . . . . . . 9 + ∈ (topGen‘ran (,))
132131a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℝ+ ∈ (topGen‘ran (,)))
133119, 123, 124, 125, 127, 129, 128, 132dvmptres 23532 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+𝑥)) = (𝑥 ∈ ℝ+ ↦ 1))
134114recnd 9947 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) ∈ ℂ)
135 resubcl 10224 . . . . . . . . 9 (((2 · (log‘𝑥)) ∈ ℝ ∧ 2 ∈ ℝ) → ((2 · (log‘𝑥)) − 2) ∈ ℝ)
136111, 13, 135sylancl 693 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((2 · (log‘𝑥)) − 2) ∈ ℝ)
137136, 107rerpdivcld 11779 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) − 2) / 𝑥) ∈ ℝ)
138109recnd 9947 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥)↑2) ∈ ℂ)
139111recnd 9947 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 · (log‘𝑥)) ∈ ℂ)
140107rpreccld 11758 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ+)
141140rpcnd 11750 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℂ)
142139, 141mulcld 9939 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((2 · (log‘𝑥)) · (1 / 𝑥)) ∈ ℂ)
143 cnelprrecn 9908 . . . . . . . . . . 11 ℂ ∈ {ℝ, ℂ}
144143a1i 11 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℂ ∈ {ℝ, ℂ})
145108recnd 9947 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ)
146 sqcl 12787 . . . . . . . . . . 11 (𝑦 ∈ ℂ → (𝑦↑2) ∈ ℂ)
147146adantl 481 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → (𝑦↑2) ∈ ℂ)
148 simpr 476 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
149 mulcl 9899 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (2 · 𝑦) ∈ ℂ)
15025, 148, 149sylancr 694 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → (2 · 𝑦) ∈ ℂ)
151 dvrelog 24183 . . . . . . . . . . 11 (ℝ D (log ↾ ℝ+)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))
152 relogf1o 24117 . . . . . . . . . . . . . . 15 (log ↾ ℝ+):ℝ+1-1-onto→ℝ
153 f1of 6050 . . . . . . . . . . . . . . 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 6159 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ ((log ↾ ℝ+)‘𝑥)))
156 fvres 6117 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+ → ((log ↾ ℝ+)‘𝑥) = (log‘𝑥))
157156mpteq2ia 4668 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ ↦ ((log ↾ ℝ+)‘𝑥)) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))
158155, 157syl6eq 2660 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥)))
159158oveq2d 6565 . . . . . . . . . . 11 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (log ↾ ℝ+)) = (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))))
160151, 159syl5reqr 2659 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)))
161 2nn 11062 . . . . . . . . . . . 12 2 ∈ ℕ
162 dvexp 23522 . . . . . . . . . . . 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 11012 . . . . . . . . . . . . . . 15 (2 − 1) = 1
165164oveq2i 6560 . . . . . . . . . . . . . 14 (𝑦↑(2 − 1)) = (𝑦↑1)
166 exp1 12728 . . . . . . . . . . . . . 14 (𝑦 ∈ ℂ → (𝑦↑1) = 𝑦)
167165, 166syl5eq 2656 . . . . . . . . . . . . 13 (𝑦 ∈ ℂ → (𝑦↑(2 − 1)) = 𝑦)
168167oveq2d 6565 . . . . . . . . . . . 12 (𝑦 ∈ ℂ → (2 · (𝑦↑(2 − 1))) = (2 · 𝑦))
169168mpteq2ia 4668 . . . . . . . . . . 11 (𝑦 ∈ ℂ ↦ (2 · (𝑦↑(2 − 1)))) = (𝑦 ∈ ℂ ↦ (2 · 𝑦))
170163, 169syl6eq 2660 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑2))) = (𝑦 ∈ ℂ ↦ (2 · 𝑦)))
171 oveq1 6556 . . . . . . . . . 10 (𝑦 = (log‘𝑥) → (𝑦↑2) = ((log‘𝑥)↑2))
172 oveq2 6557 . . . . . . . . . 10 (𝑦 = (log‘𝑥) → (2 · 𝑦) = (2 · (log‘𝑥)))
173119, 144, 145, 140, 147, 150, 160, 170, 171, 172dvmptco 23541 . . . . . . . . 9 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ ((log‘𝑥)↑2))) = (𝑥 ∈ ℝ+ ↦ ((2 · (log‘𝑥)) · (1 / 𝑥))))
174113recnd 9947 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 − (2 · (log‘𝑥))) ∈ ℂ)
175 ovex 6577 . . . . . . . . . 10 (0 − (2 · (1 / 𝑥))) ∈ V
176175a1i 11 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (0 − (2 · (1 / 𝑥))) ∈ V)
177 2cnd 10970 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 2 ∈ ℂ)
178 0red 9920 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 0 ∈ ℝ)
179 2cnd 10970 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 2 ∈ ℂ)
180 0red 9920 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
181 2cnd 10970 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 2 ∈ ℂ)
182119, 181dvmptc 23527 . . . . . . . . . . 11 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 2)) = (𝑥 ∈ ℝ ↦ 0))
183119, 179, 180, 182, 127, 129, 128, 132dvmptres 23532 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ 2)) = (𝑥 ∈ ℝ+ ↦ 0))
184 mulcl 9899 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ (1 / 𝑥) ∈ ℂ) → (2 · (1 / 𝑥)) ∈ ℂ)
18525, 141, 184sylancr 694 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 · (1 / 𝑥)) ∈ ℂ)
186119, 145, 140, 160, 181dvmptcmul 23533 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (2 · (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ (2 · (1 / 𝑥))))
187119, 177, 178, 183, 139, 185, 186dvmptsub 23536 . . . . . . . . 9 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (2 − (2 · (log‘𝑥))))) = (𝑥 ∈ ℝ+ ↦ (0 − (2 · (1 / 𝑥)))))
188119, 138, 142, 173, 174, 176, 187dvmptadd 23529 . . . . . . . 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))))))
189139, 177, 141subdird 10366 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) − 2) · (1 / 𝑥)) = (((2 · (log‘𝑥)) · (1 / 𝑥)) − (2 · (1 / 𝑥))))
190136recnd 9947 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((2 · (log‘𝑥)) − 2) ∈ ℂ)
191 rpne0 11724 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+𝑥 ≠ 0)
192191adantl 481 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0)
193190, 120, 192divrecd 10683 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) − 2) / 𝑥) = (((2 · (log‘𝑥)) − 2) · (1 / 𝑥)))
194 df-neg 10148 . . . . . . . . . . . 12 -(2 · (1 / 𝑥)) = (0 − (2 · (1 / 𝑥)))
195194oveq2i 6560 . . . . . . . . . . 11 (((2 · (log‘𝑥)) · (1 / 𝑥)) + -(2 · (1 / 𝑥))) = (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥))))
196142, 185negsubd 10277 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) · (1 / 𝑥)) + -(2 · (1 / 𝑥))) = (((2 · (log‘𝑥)) · (1 / 𝑥)) − (2 · (1 / 𝑥))))
197195, 196syl5eqr 2658 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥)))) = (((2 · (log‘𝑥)) · (1 / 𝑥)) − (2 · (1 / 𝑥))))
198189, 193, 1973eqtr4rd 2655 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥)))) = (((2 · (log‘𝑥)) − 2) / 𝑥))
199198mpteq2dva 4672 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (𝑥 ∈ ℝ+ ↦ (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥))))) = (𝑥 ∈ ℝ+ ↦ (((2 · (log‘𝑥)) − 2) / 𝑥)))
200188, 199eqtrd 2644 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = (𝑥 ∈ ℝ+ ↦ (((2 · (log‘𝑥)) − 2) / 𝑥)))
201119, 120, 121, 133, 134, 137, 200dvmptmul 23530 . . . . . 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))))
202134mulid2d 9937 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (1 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))
203138, 139, 177subsub2d 10300 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑2) − ((2 · (log‘𝑥)) − 2)) = (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))
204202, 203eqtr4d 2647 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (1 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (((log‘𝑥)↑2) − ((2 · (log‘𝑥)) − 2)))
205190, 120, 192divcan1d 10681 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((((2 · (log‘𝑥)) − 2) / 𝑥) · 𝑥) = ((2 · (log‘𝑥)) − 2))
206204, 205oveq12d 6567 . . . . . . . 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)))
207138, 190npcand 10275 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((((log‘𝑥)↑2) − ((2 · (log‘𝑥)) − 2)) + ((2 · (log‘𝑥)) − 2)) = ((log‘𝑥)↑2))
208206, 207eqtrd 2644 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((1 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) + ((((2 · (log‘𝑥)) − 2) / 𝑥) · 𝑥)) = ((log‘𝑥)↑2))
209208mpteq2dva 4672 . . . . . 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)))
210201, 209eqtrd 2644 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))))) = (𝑥 ∈ ℝ+ ↦ ((log‘𝑥)↑2)))
211 fveq2 6103 . . . . . 6 (𝑥 = 𝑛 → (log‘𝑥) = (log‘𝑛))
212211oveq1d 6564 . . . . 5 (𝑥 = 𝑛 → ((log‘𝑥)↑2) = ((log‘𝑛)↑2))
213 simp32 1091 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑥𝑛)
214 simp2l 1080 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑥 ∈ ℝ+)
215 simp2r 1081 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑛 ∈ ℝ+)
216214, 215logled 24177 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (𝑥𝑛 ↔ (log‘𝑥) ≤ (log‘𝑛)))
217213, 216mpbid 221 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (log‘𝑥) ≤ (log‘𝑛))
218214relogcld 24173 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (log‘𝑥) ∈ ℝ)
219215relogcld 24173 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (log‘𝑛) ∈ ℝ)
220 simp31 1090 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 1 ≤ 𝑥)
221 logleb 24153 . . . . . . . . . 10 ((1 ∈ ℝ+𝑥 ∈ ℝ+) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
22254, 214, 221sylancr 694 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
223220, 222mpbid 221 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (log‘1) ≤ (log‘𝑥))
22464, 223syl5eqbrr 4619 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 0 ≤ (log‘𝑥))
225215rpred 11748 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑛 ∈ ℝ)
226 1red 9934 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 1 ∈ ℝ)
227214rpred 11748 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑥 ∈ ℝ)
228226, 227, 225, 220, 213letrd 10073 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 1 ≤ 𝑛)
229225, 228logge0d 24180 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 0 ≤ (log‘𝑛))
230218, 219, 224, 229le2sqd 12906 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → ((log‘𝑥) ≤ (log‘𝑛) ↔ ((log‘𝑥)↑2) ≤ ((log‘𝑛)↑2)))
231217, 230mpbid 221 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → ((log‘𝑥)↑2) ≤ ((log‘𝑛)↑2))
232 relogcl 24126 . . . . . . 7 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
233232ad2antrl 760 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
234233sqge0d 12898 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘𝑥)↑2))
23554a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ+)
236 simpl 472 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ+)
237 1le1 10534 . . . . . 6 1 ≤ 1
238237a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ≤ 1)
239 simpr 476 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ≤ 𝐴)
2409rexrd 9968 . . . . . 6 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ*)
241 pnfge 11840 . . . . . 6 (𝐴 ∈ ℝ*𝐴 ≤ +∞)
242240, 241syl 17 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ≤ +∞)
24394, 95, 96, 97, 99, 103, 104, 115, 109, 117, 210, 212, 231, 50, 234, 235, 236, 238, 239, 242, 44dvfsum2 23601 . . . 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))
24492, 243eqbrtrrd 4607 . . 3 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2)) ≤ ((log‘𝐴)↑2))
24524, 29, 12, 38, 244letrd 10073 . 2 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − 2) ≤ ((log‘𝐴)↑2))
24613a1i 11 . . 3 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 2 ∈ ℝ)
24722, 246, 12lesubaddd 10503 . 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)))
248245, 247mpbid 221 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ⊆ wss 3540  {cpr 4127   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039   ↾ cres 5040  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  +∞cpnf 9950  ℝ*cxr 9952   ≤ cle 9954   − cmin 10145  -cneg 10146   / cdiv 10563  ℕcn 10897  2c2 10947  ℤcz 11254  ℝ+crp 11708  (,)cioo 12046  ...cfz 12197  ⌊cfl 12453  ↑cexp 12722  abscabs 13822  Σcsu 14264  TopOpenctopn 15905  topGenctg 15921  ℂfldccnfld 19567   D cdv 23433  logclog 24105 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-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-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  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-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-sum 14265  df-ef 14637  df-sin 14639  df-cos 14640  df-pi 14642  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-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  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-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-cmp 21000  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 This theorem is referenced by:  selberglem2  25035
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