MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  log2sumbnd Structured version   Visualization version   GIF version

Theorem log2sumbnd 26892
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 13878 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (1...(⌊‘𝐴)) ∈ Fin)
2 elfznn 13470 . . . . . . . . . . . 12 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
32adantl 482 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
43nnrpd 12955 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ+)
54relogcld 25978 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℝ)
65resqcld 14030 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((log‘𝑛)↑2) ∈ ℝ)
71, 6fsumrecl 15619 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) ∈ ℝ)
8 rpre 12923 . . . . . . . . 9 (𝐴 ∈ ℝ+𝐴 ∈ ℝ)
98adantr 481 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ)
10 relogcl 25931 . . . . . . . . . . 11 (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ)
1110adantr 481 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℝ)
1211resqcld 14030 . . . . . . . . 9 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((log‘𝐴)↑2) ∈ ℝ)
13 2re 12227 . . . . . . . . . 10 2 ∈ ℝ
14 remulcl 11136 . . . . . . . . . . 11 ((2 ∈ ℝ ∧ (log‘𝐴) ∈ ℝ) → (2 · (log‘𝐴)) ∈ ℝ)
1513, 11, 14sylancr 587 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (2 · (log‘𝐴)) ∈ ℝ)
16 resubcl 11465 . . . . . . . . . 10 ((2 ∈ ℝ ∧ (2 · (log‘𝐴)) ∈ ℝ) → (2 − (2 · (log‘𝐴))) ∈ ℝ)
1713, 15, 16sylancr 587 . . . . . . . . 9 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (2 − (2 · (log‘𝐴))) ∈ ℝ)
1812, 17readdcld 11184 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))) ∈ ℝ)
199, 18remulcld 11185 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))) ∈ ℝ)
207, 19resubcld 11583 . . . . . 6 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) ∈ ℝ)
2120recnd 11183 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) ∈ ℂ)
2221abscld 15321 . . . 4 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) ∈ ℝ)
23 resubcl 11465 . . . 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 586 . . 3 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))) − 2) ∈ ℝ)
25 2cn 12228 . . . . . 6 2 ∈ ℂ
2625negcli 11469 . . . . 5 -2 ∈ ℂ
27 subcl 11400 . . . . 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 586 . . . 4 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2) ∈ ℂ)
2928abscld 15321 . . 3 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2) − (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))) − -2)) ∈ ℝ)
3025absnegi 15285 . . . . . 6 (abs‘-2) = (abs‘2)
31 0le2 12255 . . . . . . 7 0 ≤ 2
32 absid 15181 . . . . . . 7 ((2 ∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2)
3313, 31, 32mp2an 690 . . . . . 6 (abs‘2) = 2
3430, 33eqtri 2764 . . . . 5 (abs‘-2) = 2
3534oveq2i 7368 . . . 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 15217 . . . . 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 586 . . . 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 5141 . . 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 6842 . . . . . . . . . . 11 (𝑥 = 𝐴 → (⌊‘𝑥) = (⌊‘𝐴))
4039oveq2d 7373 . . . . . . . . . 10 (𝑥 = 𝐴 → (1...(⌊‘𝑥)) = (1...(⌊‘𝐴)))
4140sumeq1d 15586 . . . . . . . . 9 (𝑥 = 𝐴 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = Σ𝑛 ∈ (1...(⌊‘𝐴))((log‘𝑛)↑2))
42 id 22 . . . . . . . . . 10 (𝑥 = 𝐴𝑥 = 𝐴)
43 fveq2 6842 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
4443oveq1d 7372 . . . . . . . . . . 11 (𝑥 = 𝐴 → ((log‘𝑥)↑2) = ((log‘𝐴)↑2))
4543oveq2d 7373 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (2 · (log‘𝑥)) = (2 · (log‘𝐴)))
4645oveq2d 7373 . . . . . . . . . . 11 (𝑥 = 𝐴 → (2 − (2 · (log‘𝑥))) = (2 − (2 · (log‘𝐴))))
4744, 46oveq12d 7375 . . . . . . . . . 10 (𝑥 = 𝐴 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴)))))
4842, 47oveq12d 7375 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (𝐴 · (((log‘𝐴)↑2) + (2 − (2 · (log‘𝐴))))))
4941, 48oveq12d 7375 . . . . . . . 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 2736 . . . . . . . 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 7390 . . . . . . . 8 𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) ∈ V
5249, 50, 51fvmpt3i 6953 . . . . . . 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 12919 . . . . . . 7 1 ∈ ℝ+
55 fveq2 6842 . . . . . . . . . . . . . 14 (𝑥 = 1 → (⌊‘𝑥) = (⌊‘1))
56 1z 12533 . . . . . . . . . . . . . . 15 1 ∈ ℤ
57 flid 13713 . . . . . . . . . . . . . . 15 (1 ∈ ℤ → (⌊‘1) = 1)
5856, 57ax-mp 5 . . . . . . . . . . . . . 14 (⌊‘1) = 1
5955, 58eqtrdi 2792 . . . . . . . . . . . . 13 (𝑥 = 1 → (⌊‘𝑥) = 1)
6059oveq2d 7373 . . . . . . . . . . . 12 (𝑥 = 1 → (1...(⌊‘𝑥)) = (1...1))
6160sumeq1d 15586 . . . . . . . . . . 11 (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = Σ𝑛 ∈ (1...1)((log‘𝑛)↑2))
62 0cn 11147 . . . . . . . . . . . 12 0 ∈ ℂ
63 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑛 = 1 → (log‘𝑛) = (log‘1))
64 log1 25941 . . . . . . . . . . . . . . 15 (log‘1) = 0
6563, 64eqtrdi 2792 . . . . . . . . . . . . . 14 (𝑛 = 1 → (log‘𝑛) = 0)
6665sq0id 14098 . . . . . . . . . . . . 13 (𝑛 = 1 → ((log‘𝑛)↑2) = 0)
6766fsum1 15632 . . . . . . . . . . . 12 ((1 ∈ ℤ ∧ 0 ∈ ℂ) → Σ𝑛 ∈ (1...1)((log‘𝑛)↑2) = 0)
6856, 62, 67mp2an 690 . . . . . . . . . . 11 Σ𝑛 ∈ (1...1)((log‘𝑛)↑2) = 0
6961, 68eqtrdi 2792 . . . . . . . . . 10 (𝑥 = 1 → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) = 0)
70 id 22 . . . . . . . . . . . 12 (𝑥 = 1 → 𝑥 = 1)
71 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑥 = 1 → (log‘𝑥) = (log‘1))
7271, 64eqtrdi 2792 . . . . . . . . . . . . . . 15 (𝑥 = 1 → (log‘𝑥) = 0)
7372sq0id 14098 . . . . . . . . . . . . . 14 (𝑥 = 1 → ((log‘𝑥)↑2) = 0)
7472oveq2d 7373 . . . . . . . . . . . . . . . . 17 (𝑥 = 1 → (2 · (log‘𝑥)) = (2 · 0))
75 2t0e0 12322 . . . . . . . . . . . . . . . . 17 (2 · 0) = 0
7674, 75eqtrdi 2792 . . . . . . . . . . . . . . . 16 (𝑥 = 1 → (2 · (log‘𝑥)) = 0)
7776oveq2d 7373 . . . . . . . . . . . . . . 15 (𝑥 = 1 → (2 − (2 · (log‘𝑥))) = (2 − 0))
7825subid1i 11473 . . . . . . . . . . . . . . 15 (2 − 0) = 2
7977, 78eqtrdi 2792 . . . . . . . . . . . . . 14 (𝑥 = 1 → (2 − (2 · (log‘𝑥))) = 2)
8073, 79oveq12d 7375 . . . . . . . . . . . . 13 (𝑥 = 1 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = (0 + 2))
8125addid2i 11343 . . . . . . . . . . . . 13 (0 + 2) = 2
8280, 81eqtrdi 2792 . . . . . . . . . . . 12 (𝑥 = 1 → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) = 2)
8370, 82oveq12d 7375 . . . . . . . . . . 11 (𝑥 = 1 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = (1 · 2))
8425mulid2i 11160 . . . . . . . . . . 11 (1 · 2) = 2
8583, 84eqtrdi 2792 . . . . . . . . . 10 (𝑥 = 1 → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) = 2)
8669, 85oveq12d 7375 . . . . . . . . 9 (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = (0 − 2))
87 df-neg 11388 . . . . . . . . 9 -2 = (0 − 2)
8886, 87eqtr4di 2794 . . . . . . . 8 (𝑥 = 1 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘𝑛)↑2) − (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))) = -2)
8988, 50, 51fvmpt3i 6953 . . . . . . 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 7375 . . . . 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 6846 . . . 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 13342 . . . . . 6 (0(,)+∞) = ℝ+
9493eqcomi 2745 . . . . 5 + = (0(,)+∞)
95 nnuz 12806 . . . . 5 ℕ = (ℤ‘1)
9656a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ∈ ℤ)
97 1red 11156 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ)
98 pnfxr 11209 . . . . . 6 +∞ ∈ ℝ*
9998a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → +∞ ∈ ℝ*)
100 1re 11155 . . . . . . 7 1 ∈ ℝ
101 1nn0 12429 . . . . . . 7 1 ∈ ℕ0
102100, 101nn0addge1i 12461 . . . . . 6 1 ≤ (1 + 1)
103102a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ≤ (1 + 1))
104 0red 11158 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 0 ∈ ℝ)
105 rpre 12923 . . . . . . 7 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
106105adantl 482 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ)
107 simpr 485 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
108107relogcld 25978 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ)
109108resqcld 14030 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥)↑2) ∈ ℝ)
110 remulcl 11136 . . . . . . . . 9 ((2 ∈ ℝ ∧ (log‘𝑥) ∈ ℝ) → (2 · (log‘𝑥)) ∈ ℝ)
11113, 108, 110sylancr 587 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 · (log‘𝑥)) ∈ ℝ)
112 resubcl 11465 . . . . . . . 8 ((2 ∈ ℝ ∧ (2 · (log‘𝑥)) ∈ ℝ) → (2 − (2 · (log‘𝑥))) ∈ ℝ)
11313, 111, 112sylancr 587 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 − (2 · (log‘𝑥))) ∈ ℝ)
114109, 113readdcld 11184 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) ∈ ℝ)
115106, 114remulcld 11185 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))) ∈ ℝ)
116 nnrp 12926 . . . . . 6 (𝑥 ∈ ℕ → 𝑥 ∈ ℝ+)
117116, 109sylan2 593 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℕ) → ((log‘𝑥)↑2) ∈ ℝ)
118 reelprrecn 11143 . . . . . . . 8 ℝ ∈ {ℝ, ℂ}
119118a1i 11 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℝ ∈ {ℝ, ℂ})
120106recnd 11183 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ)
121 1red 11156 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 1 ∈ ℝ)
122 recn 11141 . . . . . . . . 9 (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
123122adantl 482 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
124 1red 11156 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
125119dvmptid 25321 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
126 rpssre 12922 . . . . . . . . 9 + ⊆ ℝ
127126a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℝ+ ⊆ ℝ)
128 eqid 2736 . . . . . . . . 9 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
129128tgioo2 24166 . . . . . . . 8 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
130 iooretop 24129 . . . . . . . . . 10 (0(,)+∞) ∈ (topGen‘ran (,))
13193, 130eqeltrri 2835 . . . . . . . . 9 + ∈ (topGen‘ran (,))
132131a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℝ+ ∈ (topGen‘ran (,)))
133119, 123, 124, 125, 127, 129, 128, 132dvmptres 25327 . . . . . . 7 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+𝑥)) = (𝑥 ∈ ℝ+ ↦ 1))
134114recnd 11183 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))) ∈ ℂ)
135 resubcl 11465 . . . . . . . . 9 (((2 · (log‘𝑥)) ∈ ℝ ∧ 2 ∈ ℝ) → ((2 · (log‘𝑥)) − 2) ∈ ℝ)
136111, 13, 135sylancl 586 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((2 · (log‘𝑥)) − 2) ∈ ℝ)
137136, 107rerpdivcld 12988 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) − 2) / 𝑥) ∈ ℝ)
138109recnd 11183 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥)↑2) ∈ ℂ)
139111recnd 11183 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 · (log‘𝑥)) ∈ ℂ)
140107rpreccld 12967 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ+)
141140rpcnd 12959 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℂ)
142139, 141mulcld 11175 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((2 · (log‘𝑥)) · (1 / 𝑥)) ∈ ℂ)
143 cnelprrecn 11144 . . . . . . . . . . 11 ℂ ∈ {ℝ, ℂ}
144143a1i 11 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ℂ ∈ {ℝ, ℂ})
145108recnd 11183 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ)
146 sqcl 14023 . . . . . . . . . . 11 (𝑦 ∈ ℂ → (𝑦↑2) ∈ ℂ)
147146adantl 482 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → (𝑦↑2) ∈ ℂ)
148 simpr 485 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
149 mulcl 11135 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (2 · 𝑦) ∈ ℂ)
15025, 148, 149sylancr 587 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑦 ∈ ℂ) → (2 · 𝑦) ∈ ℂ)
151 relogf1o 25922 . . . . . . . . . . . . . . 15 (log ↾ ℝ+):ℝ+1-1-onto→ℝ
152 f1of 6784 . . . . . . . . . . . . . . 15 ((log ↾ ℝ+):ℝ+1-1-onto→ℝ → (log ↾ ℝ+):ℝ+⟶ℝ)
153151, 152mp1i 13 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log ↾ ℝ+):ℝ+⟶ℝ)
154153feqmptd 6910 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ ((log ↾ ℝ+)‘𝑥)))
155 fvres 6861 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+ → ((log ↾ ℝ+)‘𝑥) = (log‘𝑥))
156155mpteq2ia 5208 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ ↦ ((log ↾ ℝ+)‘𝑥)) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))
157154, 156eqtrdi 2792 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥)))
158157oveq2d 7373 . . . . . . . . . . 11 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (log ↾ ℝ+)) = (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))))
159 dvrelog 25992 . . . . . . . . . . 11 (ℝ D (log ↾ ℝ+)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))
160158, 159eqtr3di 2791 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)))
161 2nn 12226 . . . . . . . . . . . 12 2 ∈ ℕ
162 dvexp 25317 . . . . . . . . . . . 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 12279 . . . . . . . . . . . . . . 15 (2 − 1) = 1
165164oveq2i 7368 . . . . . . . . . . . . . 14 (𝑦↑(2 − 1)) = (𝑦↑1)
166 exp1 13973 . . . . . . . . . . . . . 14 (𝑦 ∈ ℂ → (𝑦↑1) = 𝑦)
167165, 166eqtrid 2788 . . . . . . . . . . . . 13 (𝑦 ∈ ℂ → (𝑦↑(2 − 1)) = 𝑦)
168167oveq2d 7373 . . . . . . . . . . . 12 (𝑦 ∈ ℂ → (2 · (𝑦↑(2 − 1))) = (2 · 𝑦))
169168mpteq2ia 5208 . . . . . . . . . . 11 (𝑦 ∈ ℂ ↦ (2 · (𝑦↑(2 − 1)))) = (𝑦 ∈ ℂ ↦ (2 · 𝑦))
170163, 169eqtrdi 2792 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑2))) = (𝑦 ∈ ℂ ↦ (2 · 𝑦)))
171 oveq1 7364 . . . . . . . . . 10 (𝑦 = (log‘𝑥) → (𝑦↑2) = ((log‘𝑥)↑2))
172 oveq2 7365 . . . . . . . . . 10 (𝑦 = (log‘𝑥) → (2 · 𝑦) = (2 · (log‘𝑥)))
173119, 144, 145, 140, 147, 150, 160, 170, 171, 172dvmptco 25336 . . . . . . . . 9 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ ((log‘𝑥)↑2))) = (𝑥 ∈ ℝ+ ↦ ((2 · (log‘𝑥)) · (1 / 𝑥))))
174113recnd 11183 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 − (2 · (log‘𝑥))) ∈ ℂ)
175 ovexd 7392 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (0 − (2 · (1 / 𝑥))) ∈ V)
176 2cnd 12231 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 2 ∈ ℂ)
177 0red 11158 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 0 ∈ ℝ)
178 2cnd 12231 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 2 ∈ ℂ)
179 0red 11158 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
180 2cnd 12231 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 2 ∈ ℂ)
181119, 180dvmptc 25322 . . . . . . . . . . 11 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 2)) = (𝑥 ∈ ℝ ↦ 0))
182119, 178, 179, 181, 127, 129, 128, 132dvmptres 25327 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ 2)) = (𝑥 ∈ ℝ+ ↦ 0))
183 mulcl 11135 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ (1 / 𝑥) ∈ ℂ) → (2 · (1 / 𝑥)) ∈ ℂ)
18425, 141, 183sylancr 587 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (2 · (1 / 𝑥)) ∈ ℂ)
185119, 145, 140, 160, 180dvmptcmul 25328 . . . . . . . . . 10 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (2 · (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ (2 · (1 / 𝑥))))
186119, 176, 177, 182, 139, 184, 185dvmptsub 25331 . . . . . . . . 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 25324 . . . . . . . 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 11612 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) − 2) · (1 / 𝑥)) = (((2 · (log‘𝑥)) · (1 / 𝑥)) − (2 · (1 / 𝑥))))
189136recnd 11183 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((2 · (log‘𝑥)) − 2) ∈ ℂ)
190 rpne0 12931 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+𝑥 ≠ 0)
191190adantl 482 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0)
192189, 120, 191divrecd 11934 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) − 2) / 𝑥) = (((2 · (log‘𝑥)) − 2) · (1 / 𝑥)))
193 df-neg 11388 . . . . . . . . . . . 12 -(2 · (1 / 𝑥)) = (0 − (2 · (1 / 𝑥)))
194193oveq2i 7368 . . . . . . . . . . 11 (((2 · (log‘𝑥)) · (1 / 𝑥)) + -(2 · (1 / 𝑥))) = (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥))))
195142, 184negsubd 11518 . . . . . . . . . . 11 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) · (1 / 𝑥)) + -(2 · (1 / 𝑥))) = (((2 · (log‘𝑥)) · (1 / 𝑥)) − (2 · (1 / 𝑥))))
196194, 195eqtr3id 2790 . . . . . . . . . 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 2787 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((2 · (log‘𝑥)) · (1 / 𝑥)) + (0 − (2 · (1 / 𝑥)))) = (((2 · (log‘𝑥)) − 2) / 𝑥))
198197mpteq2dva 5205 . . . . . . . 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 2776 . . . . . . 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 25325 . . . . . 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 11173 . . . . . . . . . 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 11541 . . . . . . . . . 10 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑2) − ((2 · (log‘𝑥)) − 2)) = (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥)))))
203201, 202eqtr4d 2779 . . . . . . . . 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 11932 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((((2 · (log‘𝑥)) − 2) / 𝑥) · 𝑥) = ((2 · (log‘𝑥)) − 2))
205203, 204oveq12d 7375 . . . . . . . 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 11516 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ 𝑥 ∈ ℝ+) → ((((log‘𝑥)↑2) − ((2 · (log‘𝑥)) − 2)) + ((2 · (log‘𝑥)) − 2)) = ((log‘𝑥)↑2))
207205, 206eqtrd 2776 . . . . . . 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 5205 . . . . . 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 2776 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · (((log‘𝑥)↑2) + (2 − (2 · (log‘𝑥))))))) = (𝑥 ∈ ℝ+ ↦ ((log‘𝑥)↑2)))
210 fveq2 6842 . . . . . 6 (𝑥 = 𝑛 → (log‘𝑥) = (log‘𝑛))
211210oveq1d 7372 . . . . 5 (𝑥 = 𝑛 → ((log‘𝑥)↑2) = ((log‘𝑛)↑2))
212 simp32 1210 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑥𝑛)
213 simp2l 1199 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑥 ∈ ℝ+)
214 simp2r 1200 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑛 ∈ ℝ+)
215213, 214logled 25982 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (𝑥𝑛 ↔ (log‘𝑥) ≤ (log‘𝑛)))
216212, 215mpbid 231 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (log‘𝑥) ≤ (log‘𝑛))
217213relogcld 25978 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (log‘𝑥) ∈ ℝ)
218214relogcld 25978 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (log‘𝑛) ∈ ℝ)
219 simp31 1209 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 1 ≤ 𝑥)
220 logleb 25958 . . . . . . . . . 10 ((1 ∈ ℝ+𝑥 ∈ ℝ+) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
22154, 213, 220sylancr 587 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
222219, 221mpbid 231 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → (log‘1) ≤ (log‘𝑥))
22364, 222eqbrtrrid 5141 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 0 ≤ (log‘𝑥))
224214rpred 12957 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑛 ∈ ℝ)
225 1red 11156 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 1 ∈ ℝ)
226213rpred 12957 . . . . . . . . 9 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 𝑥 ∈ ℝ)
227225, 226, 224, 219, 212letrd 11312 . . . . . . . 8 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 1 ≤ 𝑛)
228224, 227logge0d 25985 . . . . . . 7 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → 0 ≤ (log‘𝑛))
229217, 218, 223, 228le2sqd 14160 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → ((log‘𝑥) ≤ (log‘𝑛) ↔ ((log‘𝑥)↑2) ≤ ((log‘𝑛)↑2)))
230216, 229mpbid 231 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑥𝑥𝑛𝑛 ≤ +∞)) → ((log‘𝑥)↑2) ≤ ((log‘𝑛)↑2))
231 relogcl 25931 . . . . . . 7 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
232231ad2antrl 726 . . . . . 6 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
233232sqge0d 14042 . . . . 5 (((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘𝑥)↑2))
23454a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ+)
235 simpl 483 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ+)
236 1le1 11783 . . . . . 6 1 ≤ 1
237236a1i 11 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ≤ 1)
238 simpr 485 . . . . 5 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ≤ 𝐴)
2399rexrd 11205 . . . . . 6 ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ*)
240 pnfge 13051 . . . . . 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 25398 . . . 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 5129 . . 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 11312 . 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 11752 . 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 231 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  Vcvv 3445  wss 3910  {cpr 4588   class class class wbr 5105  cmpt 5188  ran crn 5634  cres 5635  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  +∞cpnf 11186  *cxr 11188  cle 11190  cmin 11385  -cneg 11386   / cdiv 11812  cn 12153  2c2 12208  cz 12499  +crp 12915  (,)cioo 13264  ...cfz 13424  cfl 13695  cexp 13967  abscabs 15119  Σcsu 15570  TopOpenctopn 17303  topGenctg 17319  fldccnfld 20796   D cdv 25227  logclog 25910
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-ef 15950  df-sin 15952  df-cos 15953  df-pi 15955  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-haus 22666  df-cmp 22738  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-xms 23673  df-ms 23674  df-tms 23675  df-cncf 24241  df-limc 25230  df-dv 25231  df-log 25912
This theorem is referenced by:  selberglem2  26894
  Copyright terms: Public domain W3C validator