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Theorem log2sumbnd 25214
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 12755 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (1...(⌊‘𝐴)) ∈ Fin)
2 elfznn 12355 . . . . . . . . . . . 12 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
32adantl 482 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
43nnrpd 11855 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ+)
54relogcld 24350 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℝ)
65resqcld 13018 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((log‘𝑛)↑2) ∈ ℝ)
71, 6fsumrecl 14446 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) ∈ ℝ)
8 rpre 11824 . . . . . . . . 9 (𝐴 ∈ ℝ+𝐴 ∈ ℝ)
98adantr 481 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ)
10 relogcl 24303 . . . . . . . . . . 11 (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ)
1110adantr 481 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℝ)
1211resqcld 13018 . . . . . . . . 9 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((log‘𝐴)↑2) ∈ ℝ)
13 2re 11075 . . . . . . . . . 10 2 ∈ ℝ
14 remulcl 10006 . . . . . . . . . . 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 10330 . . . . . . . . . 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 10054 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))) ∈ ℝ)
199, 18remulcld 10055 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))) ∈ ℝ)
207, 19resubcld 10443 . . . . . 6 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) ∈ ℝ)
2120recnd 10053 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) ∈ ℂ)
2221abscld 14156 . . . 4 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) ∈ ℝ)
23 resubcl 10330 . . . 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 11076 . . . . . 6 2 ∈ ℂ
2625negcli 10334 . . . . 5 -2 ∈ ℂ
27 subcl 10265 . . . . 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 14156 . . 3 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2)) ∈ ℝ)
3025absnegi 14120 . . . . . 6 (abs‘-2) = (abs‘2)
31 0le2 11096 . . . . . . 7 0 ≤ 2
32 absid 14017 . . . . . . 7 ((2 ∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2)
3313, 31, 32mp2an 707 . . . . . 6 (abs‘2) = 2
3430, 33eqtri 2642 . . . . 5 (abs‘-2) = 2
3534oveq2i 6646 . . . 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 14053 . . . . 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 4680 . . 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 6178 . . . . . . . . . . 11 (𝑥 = 𝐴 → (⌊‘𝑥) = (⌊‘𝐴))
4039oveq2d 6651 . . . . . . . . . 10 (𝑥 = 𝐴 → (1...(⌊‘𝑥)) = (1...(⌊‘𝐴)))
4140sumeq1d 14412 . . . . . . . . 9 (𝑥 = 𝐴 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2))
42 id 22 . . . . . . . . . 10 (𝑥 = 𝐴𝑥 = 𝐴)
43 fveq2 6178 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
4443oveq1d 6650 . . . . . . . . . . 11 (𝑥 = 𝐴 → ((log‘𝑥)↑2) = ((log‘𝐴)↑2))
4543oveq2d 6651 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (2 · (log‘𝑥)) = (2 · (log‘𝐴)))
4645oveq2d 6651 . . . . . . . . . . 11 (𝑥 = 𝐴 → (2 − (2 · (log‘𝑥))) = (2 − (2 · (log‘𝐴))))
4744, 46oveq12d 6653 . . . . . . . . . 10 (𝑥 = 𝐴 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))
4842, 47oveq12d 6653 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))
4941, 48oveq12d 6653 . . . . . . . 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 2620 . . . . . . . 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 6663 . . . . . . . 8 𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) ∈ V
5249, 50, 51fvmpt3i 6274 . . . . . . 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 481 . . . . . 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 11821 . . . . . . 7 1 ∈ ℝ+
55 fveq2 6178 . . . . . . . . . . . . . 14 (𝑥 = 1 → (⌊‘𝑥) = (⌊‘1))
56 1z 11392 . . . . . . . . . . . . . . 15 1 ∈ ℤ
57 flid 12592 . . . . . . . . . . . . . . 15 (1 ∈ ℤ → (⌊‘1) = 1)
5856, 57ax-mp 5 . . . . . . . . . . . . . 14 (⌊‘1) = 1
5955, 58syl6eq 2670 . . . . . . . . . . . . 13 (𝑥 = 1 → (⌊‘𝑥) = 1)
6059oveq2d 6651 . . . . . . . . . . . 12 (𝑥 = 1 → (1...(⌊‘𝑥)) = (1...1))
6160sumeq1d 14412 . . . . . . . . . . 11 (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = Σ𝑛 ∈ (1...1)((log‘𝑛)↑2))
62 0cn 10017 . . . . . . . . . . . 12 0 ∈ ℂ
63 fveq2 6178 . . . . . . . . . . . . . . 15 (𝑛 = 1 → (log‘𝑛) = (log‘1))
64 log1 24313 . . . . . . . . . . . . . . 15 (log‘1) = 0
6563, 64syl6eq 2670 . . . . . . . . . . . . . 14 (𝑛 = 1 → (log‘𝑛) = 0)
6665sq0id 12940 . . . . . . . . . . . . 13 (𝑛 = 1 → ((log‘𝑛)↑2) = 0)
6766fsum1 14457 . . . . . . . . . . . 12 ((1 ∈ ℤ ∧ 0 ∈ ℂ) → Σ𝑛 ∈ (1...1)((log‘𝑛)↑2) = 0)
6856, 62, 67mp2an 707 . . . . . . . . . . 11 Σ𝑛 ∈ (1...1)((log‘𝑛)↑2) = 0
6961, 68syl6eq 2670 . . . . . . . . . 10 (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = 0)
70 id 22 . . . . . . . . . . . 12 (𝑥 = 1 → 𝑥 = 1)
71 fveq2 6178 . . . . . . . . . . . . . . . 16 (𝑥 = 1 → (log‘𝑥) = (log‘1))
7271, 64syl6eq 2670 . . . . . . . . . . . . . . 15 (𝑥 = 1 → (log‘𝑥) = 0)
7372sq0id 12940 . . . . . . . . . . . . . 14 (𝑥 = 1 → ((log‘𝑥)↑2) = 0)
7472oveq2d 6651 . . . . . . . . . . . . . . . . 17 (𝑥 = 1 → (2 · (log‘𝑥)) = (2 · 0))
75 2t0e0 11168 . . . . . . . . . . . . . . . . 17 (2 · 0) = 0
7674, 75syl6eq 2670 . . . . . . . . . . . . . . . 16 (𝑥 = 1 → (2 · (log‘𝑥)) = 0)
7776oveq2d 6651 . . . . . . . . . . . . . . 15 (𝑥 = 1 → (2 − (2 · (log‘𝑥))) = (2 − 0))
7825subid1i 10338 . . . . . . . . . . . . . . 15 (2 − 0) = 2
7977, 78syl6eq 2670 . . . . . . . . . . . . . 14 (𝑥 = 1 → (2 − (2 · (log‘𝑥))) = 2)
8073, 79oveq12d 6653 . . . . . . . . . . . . 13 (𝑥 = 1 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = (0 + 2))
8125addid2i 10209 . . . . . . . . . . . . 13 (0 + 2) = 2
8280, 81syl6eq 2670 . . . . . . . . . . . 12 (𝑥 = 1 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = 2)
8370, 82oveq12d 6653 . . . . . . . . . . 11 (𝑥 = 1 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (1 · 2))
8425mulid2i 10028 . . . . . . . . . . 11 (1 · 2) = 2
8583, 84syl6eq 2670 . . . . . . . . . 10 (𝑥 = 1 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = 2)
8669, 85oveq12d 6653 . . . . . . . . 9 (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = (0 − 2))
87 df-neg 10254 . . . . . . . . 9 -2 = (0 − 2)
8886, 87syl6eqr 2672 . . . . . . . 8 (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = -2)
8988, 50, 51fvmpt3i 6274 . . . . . . 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 6653 . . . . 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 6182 . . . 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 12236 . . . . . 6 (0(,)+∞) = ℝ+
9493eqcomi 2629 . . . . 5 + = (0(,)+∞)
95 nnuz 11708 . . . . 5 ℕ = (ℤ‘1)
9656a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ∈ ℤ)
97 1red 10040 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ)
98 pnfxr 10077 . . . . . 6 +∞ ∈ ℝ*
9998a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → +∞ ∈ ℝ*)
100 1re 10024 . . . . . . 7 1 ∈ ℝ
101 1nn0 11293 . . . . . . 7 1 ∈ ℕ0
102100, 101nn0addge1i 11326 . . . . . 6 1 ≤ (1 + 1)
103102a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ≤ (1 + 1))
104 0red 10026 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 0 ∈ ℝ)
105 rpre 11824 . . . . . . 7 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
106105adantl 482 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ)
107 simpr 477 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
108107relogcld 24350 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ)
109108resqcld 13018 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥)↑2) ∈ ℝ)
110 remulcl 10006 . . . . . . . . 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 10330 . . . . . . . 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 10054 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) ∈ ℝ)
115106, 114remulcld 10055 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) ∈ ℝ)
116 nnrp 11827 . . . . . 6 (𝑥 ∈ ℕ → 𝑥 ∈ ℝ+)
117116, 109sylan2 491 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℕ) → ((log‘𝑥)↑2) ∈ ℝ)
118 reelprrecn 10013 . . . . . . . 8 ℝ ∈ {ℝ, ℂ}
119118a1i 11 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℝ ∈ {ℝ, ℂ})
120106recnd 10053 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ)
121 1red 10040 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 1 ∈ ℝ)
122 recn 10011 . . . . . . . . 9 (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
123122adantl 482 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
124 1red 10040 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
125119dvmptid 23701 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
126 rpssre 11828 . . . . . . . . 9 + ⊆ ℝ
127126a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℝ+ ⊆ ℝ)
128 eqid 2620 . . . . . . . . 9 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
129128tgioo2 22587 . . . . . . . 8 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
130 iooretop 22550 . . . . . . . . . 10 (0(,)+∞) ∈ (topGen‘ran (,))
13193, 130eqeltrri 2696 . . . . . . . . 9 + ∈ (topGen‘ran (,))
132131a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℝ+ ∈ (topGen‘ran (,)))
133119, 123, 124, 125, 127, 129, 128, 132dvmptres 23707 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+𝑥)) = (𝑥 ∈ ℝ+ ↦ 1))
134114recnd 10053 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) ∈ ℂ)
135 resubcl 10330 . . . . . . . . 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 11888 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) − 2) / 𝑥) ∈ ℝ)
138109recnd 10053 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥)↑2) ∈ ℂ)
139111recnd 10053 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 · (log‘𝑥)) ∈ ℂ)
140107rpreccld 11867 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ+)
141140rpcnd 11859 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℂ)
142139, 141mulcld 10045 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((2 · (log‘𝑥)) · (1 / 𝑥)) ∈ ℂ)
143 cnelprrecn 10014 . . . . . . . . . . 11 ℂ ∈ {ℝ, ℂ}
144143a1i 11 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℂ ∈ {ℝ, ℂ})
145108recnd 10053 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ)
146 sqcl 12908 . . . . . . . . . . 11 (𝑦 ∈ ℂ → (𝑦↑2) ∈ ℂ)
147146adantl 482 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → (𝑦↑2) ∈ ℂ)
148 simpr 477 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
149 mulcl 10005 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (2 · 𝑦) ∈ ℂ)
15025, 148, 149sylancr 694 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → (2 · 𝑦) ∈ ℂ)
151 dvrelog 24364 . . . . . . . . . . 11 (ℝ D (log ↾ ℝ+)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))
152 relogf1o 24294 . . . . . . . . . . . . . . 15 (log ↾ ℝ+):ℝ+1-1-onto→ℝ
153 f1of 6124 . . . . . . . . . . . . . . 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 6236 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ ((log ↾ ℝ+)‘𝑥)))
156 fvres 6194 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+ → ((log ↾ ℝ+)‘𝑥) = (log‘𝑥))
157156mpteq2ia 4731 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ ↦ ((log ↾ ℝ+)‘𝑥)) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))
158155, 157syl6eq 2670 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥)))
159158oveq2d 6651 . . . . . . . . . . 11 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (log ↾ ℝ+)) = (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))))
160151, 159syl5reqr 2669 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)))
161 2nn 11170 . . . . . . . . . . . 12 2 ∈ ℕ
162 dvexp 23697 . . . . . . . . . . . 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 11120 . . . . . . . . . . . . . . 15 (2 − 1) = 1
165164oveq2i 6646 . . . . . . . . . . . . . 14 (𝑦↑(2 − 1)) = (𝑦↑1)
166 exp1 12849 . . . . . . . . . . . . . 14 (𝑦 ∈ ℂ → (𝑦↑1) = 𝑦)
167165, 166syl5eq 2666 . . . . . . . . . . . . 13 (𝑦 ∈ ℂ → (𝑦↑(2 − 1)) = 𝑦)
168167oveq2d 6651 . . . . . . . . . . . 12 (𝑦 ∈ ℂ → (2 · (𝑦↑(2 − 1))) = (2 · 𝑦))
169168mpteq2ia 4731 . . . . . . . . . . 11 (𝑦 ∈ ℂ ↦ (2 · (𝑦↑(2 − 1)))) = (𝑦 ∈ ℂ ↦ (2 · 𝑦))
170163, 169syl6eq 2670 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑2))) = (𝑦 ∈ ℂ ↦ (2 · 𝑦)))
171 oveq1 6642 . . . . . . . . . 10 (𝑦 = (log‘𝑥) → (𝑦↑2) = ((log‘𝑥)↑2))
172 oveq2 6643 . . . . . . . . . 10 (𝑦 = (log‘𝑥) → (2 · 𝑦) = (2 · (log‘𝑥)))
173119, 144, 145, 140, 147, 150, 160, 170, 171, 172dvmptco 23716 . . . . . . . . 9 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ ((log‘𝑥)↑2))) = (𝑥 ∈ ℝ+ ↦ ((2 · (log‘𝑥)) · (1 / 𝑥))))
174113recnd 10053 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 − (2 · (log‘𝑥))) ∈ ℂ)
175 ovexd 6665 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (0 − (2 · (1 / 𝑥))) ∈ V)
176 2cnd 11078 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 2 ∈ ℂ)
177 0red 10026 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 0 ∈ ℝ)
178 2cnd 11078 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 2 ∈ ℂ)
179 0red 10026 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
180 2cnd 11078 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 2 ∈ ℂ)
181119, 180dvmptc 23702 . . . . . . . . . . 11 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 2)) = (𝑥 ∈ ℝ ↦ 0))
182119, 178, 179, 181, 127, 129, 128, 132dvmptres 23707 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ 2)) = (𝑥 ∈ ℝ+ ↦ 0))
183 mulcl 10005 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ (1 / 𝑥) ∈ ℂ) → (2 · (1 / 𝑥)) ∈ ℂ)
18425, 141, 183sylancr 694 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 · (1 / 𝑥)) ∈ ℂ)
185119, 145, 140, 160, 180dvmptcmul 23708 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (2 · (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ (2 · (1 / 𝑥))))
186119, 176, 177, 182, 139, 184, 185dvmptsub 23711 . . . . . . . . 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 23704 . . . . . . . 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 10472 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) − 2) · (1 / 𝑥)) = (((2 · (log‘𝑥)) · (1 / 𝑥)) − (2 · (1 / 𝑥))))
189136recnd 10053 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((2 · (log‘𝑥)) − 2) ∈ ℂ)
190 rpne0 11833 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+𝑥 ≠ 0)
191190adantl 482 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0)
192189, 120, 191divrecd 10789 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) − 2) / 𝑥) = (((2 · (log‘𝑥)) − 2) · (1 / 𝑥)))
193 df-neg 10254 . . . . . . . . . . . 12 -(2 · (1 / 𝑥)) = (0 − (2 · (1 / 𝑥)))
194193oveq2i 6646 . . . . . . . . . . 11 (((2 · (log‘𝑥)) · (1 / 𝑥)) + -(2 · (1 / 𝑥))) = (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥))))
195142, 184negsubd 10383 . . . . . . . . . . 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 2668 . . . . . . . . . 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 2665 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥)))) = (((2 · (log‘𝑥)) − 2) / 𝑥))
198197mpteq2dva 4735 . . . . . . . 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 2654 . . . . . . 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 23705 . . . . . 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 10043 . . . . . . . . . 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 10406 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑2) − ((2 · (log‘𝑥)) − 2)) = (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))
203201, 202eqtr4d 2657 . . . . . . . . 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 10787 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((((2 · (log‘𝑥)) − 2) / 𝑥) · 𝑥) = ((2 · (log‘𝑥)) − 2))
205203, 204oveq12d 6653 . . . . . . . 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 10381 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((((log‘𝑥)↑2) − ((2 · (log‘𝑥)) − 2)) + ((2 · (log‘𝑥)) − 2)) = ((log‘𝑥)↑2))
207205, 206eqtrd 2654 . . . . . . 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 4735 . . . . . 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 2654 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))))) = (𝑥 ∈ ℝ+ ↦ ((log‘𝑥)↑2)))
210 fveq2 6178 . . . . . 6 (𝑥 = 𝑛 → (log‘𝑥) = (log‘𝑛))
211210oveq1d 6650 . . . . 5 (𝑥 = 𝑛 → ((log‘𝑥)↑2) = ((log‘𝑛)↑2))
212 simp32 1096 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑥𝑛)
213 simp2l 1085 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑥 ∈ ℝ+)
214 simp2r 1086 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑛 ∈ ℝ+)
215213, 214logled 24354 . . . . . . 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 24350 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (log‘𝑥) ∈ ℝ)
218214relogcld 24350 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (log‘𝑛) ∈ ℝ)
219 simp31 1095 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 1 ≤ 𝑥)
220 logleb 24330 . . . . . . . . . 10 ((1 ∈ ℝ+𝑥 ∈ ℝ+) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
22154, 213, 220sylancr 694 . . . . . . . . 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 4680 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 0 ≤ (log‘𝑥))
224214rpred 11857 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑛 ∈ ℝ)
225 1red 10040 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 1 ∈ ℝ)
226213rpred 11857 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑥 ∈ ℝ)
227225, 226, 224, 219, 212letrd 10179 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 1 ≤ 𝑛)
228224, 227logge0d 24357 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 0 ≤ (log‘𝑛))
229217, 218, 223, 228le2sqd 13027 . . . . . 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 24303 . . . . . . 7 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
232231ad2antrl 763 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
233232sqge0d 13019 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘𝑥)↑2))
23454a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ+)
235 simpl 473 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ+)
236 1le1 10640 . . . . . 6 1 ≤ 1
237236a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ≤ 1)
238 simpr 477 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ≤ 𝐴)
2399rexrd 10074 . . . . . 6 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ*)
240 pnfge 11949 . . . . . 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 23778 . . . 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 4668 . . 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 10179 . 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 10609 . 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 384  w3a 1036   = wceq 1481  wcel 1988  wne 2791  Vcvv 3195  wss 3567  {cpr 4170   class class class wbr 4644  cmpt 4720  ran crn 5105  cres 5106  wf 5872  1-1-ontowf1o 5875  cfv 5876  (class class class)co 6635  cc 9919  cr 9920  0cc0 9921  1c1 9922   + caddc 9924   · cmul 9926  +∞cpnf 10056  *cxr 10058  cle 10060  cmin 10251  -cneg 10252   / cdiv 10669  cn 11005  2c2 11055  cz 11362  +crp 11817  (,)cioo 12160  ...cfz 12311  cfl 12574  cexp 12843  abscabs 13955  Σcsu 14397  TopOpenctopn 16063  topGenctg 16079  fldccnfld 19727   D cdv 23608  logclog 24282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999  ax-addf 10000  ax-mulf 10001
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-of 6882  df-om 7051  df-1st 7153  df-2nd 7154  df-supp 7281  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-pm 7845  df-ixp 7894  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fsupp 8261  df-fi 8302  df-sup 8333  df-inf 8334  df-oi 8400  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-uz 11673  df-q 11774  df-rp 11818  df-xneg 11931  df-xadd 11932  df-xmul 11933  df-ioo 12164  df-ioc 12165  df-ico 12166  df-icc 12167  df-fz 12312  df-fzo 12450  df-fl 12576  df-mod 12652  df-seq 12785  df-exp 12844  df-fac 13044  df-bc 13073  df-hash 13101  df-shft 13788  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-limsup 14183  df-clim 14200  df-rlim 14201  df-sum 14398  df-ef 14779  df-sin 14781  df-cos 14782  df-pi 14784  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-ress 15846  df-plusg 15935  df-mulr 15936  df-starv 15937  df-sca 15938  df-vsca 15939  df-ip 15940  df-tset 15941  df-ple 15942  df-ds 15945  df-unif 15946  df-hom 15947  df-cco 15948  df-rest 16064  df-topn 16065  df-0g 16083  df-gsum 16084  df-topgen 16085  df-pt 16086  df-prds 16089  df-xrs 16143  df-qtop 16148  df-imas 16149  df-xps 16151  df-mre 16227  df-mrc 16228  df-acs 16230  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-submnd 17317  df-mulg 17522  df-cntz 17731  df-cmn 18176  df-psmet 19719  df-xmet 19720  df-met 19721  df-bl 19722  df-mopn 19723  df-fbas 19724  df-fg 19725  df-cnfld 19728  df-top 20680  df-topon 20697  df-topsp 20718  df-bases 20731  df-cld 20804  df-ntr 20805  df-cls 20806  df-nei 20883  df-lp 20921  df-perf 20922  df-cn 21012  df-cnp 21013  df-haus 21100  df-cmp 21171  df-tx 21346  df-hmeo 21539  df-fil 21631  df-fm 21723  df-flim 21724  df-flf 21725  df-xms 22106  df-ms 22107  df-tms 22108  df-cncf 22662  df-limc 23611  df-dv 23612  df-log 24284
This theorem is referenced by:  selberglem2  25216
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