Proof of Theorem lgamgulmlem3
| Step | Hyp | Ref
| Expression |
| 1 | | lgamgulm.r |
. . . . . . . 8
⊢ (𝜑 → 𝑅 ∈ ℕ) |
| 2 | | lgamgulm.u |
. . . . . . . 8
⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))} |
| 3 | 1, 2 | lgamgulmlem1 27072 |
. . . . . . 7
⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖
ℕ))) |
| 4 | | lgamgulm.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ 𝑈) |
| 5 | 3, 4 | sseldd 3984 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖
ℕ))) |
| 6 | 5 | eldifad 3963 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 7 | | lgamgulm.n |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 8 | 7 | peano2nnd 12283 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁 + 1) ∈ ℕ) |
| 9 | 8 | nnrpd 13075 |
. . . . . . . 8
⊢ (𝜑 → (𝑁 + 1) ∈
ℝ+) |
| 10 | 7 | nnrpd 13075 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈
ℝ+) |
| 11 | 9, 10 | rpdivcld 13094 |
. . . . . . 7
⊢ (𝜑 → ((𝑁 + 1) / 𝑁) ∈
ℝ+) |
| 12 | 11 | relogcld 26665 |
. . . . . 6
⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) ∈ ℝ) |
| 13 | 12 | recnd 11289 |
. . . . 5
⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) ∈ ℂ) |
| 14 | 6, 13 | mulcld 11281 |
. . . 4
⊢ (𝜑 → (𝐴 · (log‘((𝑁 + 1) / 𝑁))) ∈ ℂ) |
| 15 | 7 | nncnd 12282 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 16 | 7 | nnne0d 12316 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ≠ 0) |
| 17 | 6, 15, 16 | divcld 12043 |
. . . . . 6
⊢ (𝜑 → (𝐴 / 𝑁) ∈ ℂ) |
| 18 | | 1cnd 11256 |
. . . . . 6
⊢ (𝜑 → 1 ∈
ℂ) |
| 19 | 17, 18 | addcld 11280 |
. . . . 5
⊢ (𝜑 → ((𝐴 / 𝑁) + 1) ∈ ℂ) |
| 20 | 5, 7 | dmgmdivn0 27071 |
. . . . 5
⊢ (𝜑 → ((𝐴 / 𝑁) + 1) ≠ 0) |
| 21 | 19, 20 | logcld 26612 |
. . . 4
⊢ (𝜑 → (log‘((𝐴 / 𝑁) + 1)) ∈ ℂ) |
| 22 | 14, 21 | subcld 11620 |
. . 3
⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1))) ∈ ℂ) |
| 23 | 22 | abscld 15475 |
. 2
⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1)))) ∈ ℝ) |
| 24 | 14, 17 | subcld 11620 |
. . . 4
⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁)) ∈ ℂ) |
| 25 | 24 | abscld 15475 |
. . 3
⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) ∈ ℝ) |
| 26 | 17, 21 | subcld 11620 |
. . . 4
⊢ (𝜑 → ((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1))) ∈ ℂ) |
| 27 | 26 | abscld 15475 |
. . 3
⊢ (𝜑 → (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1)))) ∈ ℝ) |
| 28 | 25, 27 | readdcld 11290 |
. 2
⊢ (𝜑 → ((abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) + (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1))))) ∈ ℝ) |
| 29 | 1 | nnred 12281 |
. . 3
⊢ (𝜑 → 𝑅 ∈ ℝ) |
| 30 | | 2re 12340 |
. . . . . 6
⊢ 2 ∈
ℝ |
| 31 | 30 | a1i 11 |
. . . . 5
⊢ (𝜑 → 2 ∈
ℝ) |
| 32 | | 1red 11262 |
. . . . . 6
⊢ (𝜑 → 1 ∈
ℝ) |
| 33 | 29, 32 | readdcld 11290 |
. . . . 5
⊢ (𝜑 → (𝑅 + 1) ∈ ℝ) |
| 34 | 31, 33 | remulcld 11291 |
. . . 4
⊢ (𝜑 → (2 · (𝑅 + 1)) ∈
ℝ) |
| 35 | 7 | nnsqcld 14283 |
. . . 4
⊢ (𝜑 → (𝑁↑2) ∈ ℕ) |
| 36 | 34, 35 | nndivred 12320 |
. . 3
⊢ (𝜑 → ((2 · (𝑅 + 1)) / (𝑁↑2)) ∈ ℝ) |
| 37 | 29, 36 | remulcld 11291 |
. 2
⊢ (𝜑 → (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))) ∈ ℝ) |
| 38 | 14, 21, 17 | abs3difd 15499 |
. 2
⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1)))) ≤ ((abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) + (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1)))))) |
| 39 | 7 | nnrecred 12317 |
. . . . . 6
⊢ (𝜑 → (1 / 𝑁) ∈ ℝ) |
| 40 | 8 | nnrecred 12317 |
. . . . . 6
⊢ (𝜑 → (1 / (𝑁 + 1)) ∈ ℝ) |
| 41 | 39, 40 | resubcld 11691 |
. . . . 5
⊢ (𝜑 → ((1 / 𝑁) − (1 / (𝑁 + 1))) ∈ ℝ) |
| 42 | 29, 41 | remulcld 11291 |
. . . 4
⊢ (𝜑 → (𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) ∈ ℝ) |
| 43 | 31, 29 | remulcld 11291 |
. . . . . . . . 9
⊢ (𝜑 → (2 · 𝑅) ∈
ℝ) |
| 44 | 7 | nnred 12281 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ ℝ) |
| 45 | 1 | nnrpd 13075 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑅 ∈
ℝ+) |
| 46 | 29, 45 | ltaddrpd 13110 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑅 < (𝑅 + 𝑅)) |
| 47 | 1 | nncnd 12282 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑅 ∈ ℂ) |
| 48 | 47 | 2timesd 12509 |
. . . . . . . . . 10
⊢ (𝜑 → (2 · 𝑅) = (𝑅 + 𝑅)) |
| 49 | 46, 48 | breqtrrd 5171 |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 < (2 · 𝑅)) |
| 50 | | lgamgulm.l |
. . . . . . . . 9
⊢ (𝜑 → (2 · 𝑅) ≤ 𝑁) |
| 51 | 29, 43, 44, 49, 50 | ltletrd 11421 |
. . . . . . . 8
⊢ (𝜑 → 𝑅 < 𝑁) |
| 52 | | difrp 13073 |
. . . . . . . . 9
⊢ ((𝑅 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑅 < 𝑁 ↔ (𝑁 − 𝑅) ∈
ℝ+)) |
| 53 | 29, 44, 52 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → (𝑅 < 𝑁 ↔ (𝑁 − 𝑅) ∈
ℝ+)) |
| 54 | 51, 53 | mpbid 232 |
. . . . . . 7
⊢ (𝜑 → (𝑁 − 𝑅) ∈
ℝ+) |
| 55 | 54 | rprecred 13088 |
. . . . . 6
⊢ (𝜑 → (1 / (𝑁 − 𝑅)) ∈ ℝ) |
| 56 | 55, 39 | resubcld 11691 |
. . . . 5
⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)) ∈ ℝ) |
| 57 | 29, 56 | remulcld 11291 |
. . . 4
⊢ (𝜑 → (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))) ∈ ℝ) |
| 58 | 42, 57 | readdcld 11290 |
. . 3
⊢ (𝜑 → ((𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) + (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))) ∈ ℝ) |
| 59 | 6, 15, 16 | divrecd 12046 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴 / 𝑁) = (𝐴 · (1 / 𝑁))) |
| 60 | 59 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁)) = ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 · (1 / 𝑁)))) |
| 61 | 39 | recnd 11289 |
. . . . . . . . 9
⊢ (𝜑 → (1 / 𝑁) ∈ ℂ) |
| 62 | 6, 13, 61 | subdid 11719 |
. . . . . . . 8
⊢ (𝜑 → (𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) = ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 · (1 / 𝑁)))) |
| 63 | 60, 62 | eqtr4d 2780 |
. . . . . . 7
⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁)) = (𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))) |
| 64 | 63 | fveq2d 6910 |
. . . . . 6
⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) = (abs‘(𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))))) |
| 65 | 13, 61 | subcld 11620 |
. . . . . . 7
⊢ (𝜑 → ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)) ∈ ℂ) |
| 66 | 6, 65 | absmuld 15493 |
. . . . . 6
⊢ (𝜑 → (abs‘(𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))) = ((abs‘𝐴) · (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))))) |
| 67 | 64, 66 | eqtrd 2777 |
. . . . 5
⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) = ((abs‘𝐴) · (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))))) |
| 68 | 6 | abscld 15475 |
. . . . . 6
⊢ (𝜑 → (abs‘𝐴) ∈
ℝ) |
| 69 | 65 | abscld 15475 |
. . . . . 6
⊢ (𝜑 →
(abs‘((log‘((𝑁
+ 1) / 𝑁)) − (1 /
𝑁))) ∈
ℝ) |
| 70 | 6 | absge0d 15483 |
. . . . . 6
⊢ (𝜑 → 0 ≤ (abs‘𝐴)) |
| 71 | 65 | absge0d 15483 |
. . . . . 6
⊢ (𝜑 → 0 ≤
(abs‘((log‘((𝑁
+ 1) / 𝑁)) − (1 /
𝑁)))) |
| 72 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝐴 → (abs‘𝑥) = (abs‘𝐴)) |
| 73 | 72 | breq1d 5153 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝐴 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝐴) ≤ 𝑅)) |
| 74 | | fvoveq1 7454 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝐴 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝐴 + 𝑘))) |
| 75 | 74 | breq2d 5155 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝐴 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘)))) |
| 76 | 75 | ralbidv 3178 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝐴 → (∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘)))) |
| 77 | 73, 76 | anbi12d 632 |
. . . . . . . . . 10
⊢ (𝑥 = 𝐴 → (((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘))) ↔ ((abs‘𝐴) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘))))) |
| 78 | 77, 2 | elrab2 3695 |
. . . . . . . . 9
⊢ (𝐴 ∈ 𝑈 ↔ (𝐴 ∈ ℂ ∧ ((abs‘𝐴) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘))))) |
| 79 | 78 | simprbi 496 |
. . . . . . . 8
⊢ (𝐴 ∈ 𝑈 → ((abs‘𝐴) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘)))) |
| 80 | 4, 79 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ((abs‘𝐴) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘)))) |
| 81 | 80 | simpld 494 |
. . . . . 6
⊢ (𝜑 → (abs‘𝐴) ≤ 𝑅) |
| 82 | 9, 10 | relogdivd 26668 |
. . . . . . . . 9
⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) = ((log‘(𝑁 + 1)) − (log‘𝑁))) |
| 83 | | logdifbnd 27037 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℝ+
→ ((log‘(𝑁 + 1))
− (log‘𝑁)) ≤
(1 / 𝑁)) |
| 84 | 10, 83 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((log‘(𝑁 + 1)) − (log‘𝑁)) ≤ (1 / 𝑁)) |
| 85 | 82, 84 | eqbrtrd 5165 |
. . . . . . . 8
⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) ≤ (1 / 𝑁)) |
| 86 | 12, 39, 85 | abssuble0d 15471 |
. . . . . . 7
⊢ (𝜑 →
(abs‘((log‘((𝑁
+ 1) / 𝑁)) − (1 /
𝑁))) = ((1 / 𝑁) − (log‘((𝑁 + 1) / 𝑁)))) |
| 87 | | logdiflbnd 27038 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℝ+
→ (1 / (𝑁 + 1)) ≤
((log‘(𝑁 + 1))
− (log‘𝑁))) |
| 88 | 10, 87 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (1 / (𝑁 + 1)) ≤ ((log‘(𝑁 + 1)) − (log‘𝑁))) |
| 89 | 88, 82 | breqtrrd 5171 |
. . . . . . . 8
⊢ (𝜑 → (1 / (𝑁 + 1)) ≤ (log‘((𝑁 + 1) / 𝑁))) |
| 90 | 40, 12, 39, 89 | lesub2dd 11880 |
. . . . . . 7
⊢ (𝜑 → ((1 / 𝑁) − (log‘((𝑁 + 1) / 𝑁))) ≤ ((1 / 𝑁) − (1 / (𝑁 + 1)))) |
| 91 | 86, 90 | eqbrtrd 5165 |
. . . . . 6
⊢ (𝜑 →
(abs‘((log‘((𝑁
+ 1) / 𝑁)) − (1 /
𝑁))) ≤ ((1 / 𝑁) − (1 / (𝑁 + 1)))) |
| 92 | 68, 29, 69, 41, 70, 71, 81, 91 | lemul12ad 12210 |
. . . . 5
⊢ (𝜑 → ((abs‘𝐴) ·
(abs‘((log‘((𝑁
+ 1) / 𝑁)) − (1 /
𝑁)))) ≤ (𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1))))) |
| 93 | 67, 92 | eqbrtrd 5165 |
. . . 4
⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) ≤ (𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1))))) |
| 94 | 1, 2, 7, 4, 50 | lgamgulmlem2 27073 |
. . . 4
⊢ (𝜑 → (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1)))) ≤ (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))) |
| 95 | 25, 27, 42, 57, 93, 94 | le2addd 11882 |
. . 3
⊢ (𝜑 → ((abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) + (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1))))) ≤ ((𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) + (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))))) |
| 96 | 15, 47 | subcld 11620 |
. . . . . . . 8
⊢ (𝜑 → (𝑁 − 𝑅) ∈ ℂ) |
| 97 | 15, 18 | addcld 11280 |
. . . . . . . 8
⊢ (𝜑 → (𝑁 + 1) ∈ ℂ) |
| 98 | 29, 51 | gtned 11396 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ≠ 𝑅) |
| 99 | 15, 47, 98 | subne0d 11629 |
. . . . . . . 8
⊢ (𝜑 → (𝑁 − 𝑅) ≠ 0) |
| 100 | 8 | nnne0d 12316 |
. . . . . . . 8
⊢ (𝜑 → (𝑁 + 1) ≠ 0) |
| 101 | 96, 97, 99, 100 | subrecd 12098 |
. . . . . . 7
⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))) = (((𝑁 + 1) − (𝑁 − 𝑅)) / ((𝑁 − 𝑅) · (𝑁 + 1)))) |
| 102 | 15, 18, 47 | pnncand 11659 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑁 + 1) − (𝑁 − 𝑅)) = (1 + 𝑅)) |
| 103 | 18, 47, 102 | comraddd 11475 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁 + 1) − (𝑁 − 𝑅)) = (𝑅 + 1)) |
| 104 | 103 | oveq1d 7446 |
. . . . . . 7
⊢ (𝜑 → (((𝑁 + 1) − (𝑁 − 𝑅)) / ((𝑁 − 𝑅) · (𝑁 + 1))) = ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1)))) |
| 105 | 101, 104 | eqtr2d 2778 |
. . . . . 6
⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) = ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1)))) |
| 106 | 105 | oveq2d 7447 |
. . . . 5
⊢ (𝜑 → (𝑅 · ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1)))) = (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))))) |
| 107 | 97, 100 | reccld 12036 |
. . . . . . . 8
⊢ (𝜑 → (1 / (𝑁 + 1)) ∈ ℂ) |
| 108 | 96, 99 | reccld 12036 |
. . . . . . . 8
⊢ (𝜑 → (1 / (𝑁 − 𝑅)) ∈ ℂ) |
| 109 | 61, 107, 108 | npncan3d 11656 |
. . . . . . 7
⊢ (𝜑 → (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))) = ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1)))) |
| 110 | 109 | eqcomd 2743 |
. . . . . 6
⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))) = (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))) |
| 111 | 110 | oveq2d 7447 |
. . . . 5
⊢ (𝜑 → (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1)))) = (𝑅 · (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))))) |
| 112 | 41 | recnd 11289 |
. . . . . 6
⊢ (𝜑 → ((1 / 𝑁) − (1 / (𝑁 + 1))) ∈ ℂ) |
| 113 | 56 | recnd 11289 |
. . . . . 6
⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)) ∈ ℂ) |
| 114 | 47, 112, 113 | adddid 11285 |
. . . . 5
⊢ (𝜑 → (𝑅 · (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))) = ((𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) + (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))))) |
| 115 | 106, 111,
114 | 3eqtrd 2781 |
. . . 4
⊢ (𝜑 → (𝑅 · ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1)))) = ((𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) + (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))))) |
| 116 | 54, 9 | rpmulcld 13093 |
. . . . . 6
⊢ (𝜑 → ((𝑁 − 𝑅) · (𝑁 + 1)) ∈
ℝ+) |
| 117 | 33, 116 | rerpdivcld 13108 |
. . . . 5
⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) ∈ ℝ) |
| 118 | 45 | rpge0d 13081 |
. . . . 5
⊢ (𝜑 → 0 ≤ 𝑅) |
| 119 | | 2z 12649 |
. . . . . . . . . 10
⊢ 2 ∈
ℤ |
| 120 | 119 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 2 ∈
ℤ) |
| 121 | 10, 120 | rpexpcld 14286 |
. . . . . . . 8
⊢ (𝜑 → (𝑁↑2) ∈
ℝ+) |
| 122 | 121 | rphalfcld 13089 |
. . . . . . 7
⊢ (𝜑 → ((𝑁↑2) / 2) ∈
ℝ+) |
| 123 | | 0le1 11786 |
. . . . . . . . 9
⊢ 0 ≤
1 |
| 124 | 123 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 0 ≤ 1) |
| 125 | 29, 32, 118, 124 | addge0d 11839 |
. . . . . . 7
⊢ (𝜑 → 0 ≤ (𝑅 + 1)) |
| 126 | 15 | sqvald 14183 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁↑2) = (𝑁 · 𝑁)) |
| 127 | 126 | oveq1d 7446 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑁↑2) / 2) = ((𝑁 · 𝑁) / 2)) |
| 128 | 31 | recnd 11289 |
. . . . . . . . . 10
⊢ (𝜑 → 2 ∈
ℂ) |
| 129 | | 2ne0 12370 |
. . . . . . . . . . 11
⊢ 2 ≠
0 |
| 130 | 129 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 2 ≠ 0) |
| 131 | 15, 15, 128, 130 | div23d 12080 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑁 · 𝑁) / 2) = ((𝑁 / 2) · 𝑁)) |
| 132 | 127, 131 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁↑2) / 2) = ((𝑁 / 2) · 𝑁)) |
| 133 | 44 | rehalfcld 12513 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁 / 2) ∈ ℝ) |
| 134 | 44, 29 | resubcld 11691 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁 − 𝑅) ∈ ℝ) |
| 135 | 44, 32 | readdcld 11290 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁 + 1) ∈ ℝ) |
| 136 | | 2rp 13039 |
. . . . . . . . . . 11
⊢ 2 ∈
ℝ+ |
| 137 | 136 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 2 ∈
ℝ+) |
| 138 | 10 | rpge0d 13081 |
. . . . . . . . . 10
⊢ (𝜑 → 0 ≤ 𝑁) |
| 139 | 44, 137, 138 | divge0d 13117 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ (𝑁 / 2)) |
| 140 | 29, 44, 137 | lemuldiv2d 13127 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((2 · 𝑅) ≤ 𝑁 ↔ 𝑅 ≤ (𝑁 / 2))) |
| 141 | 50, 140 | mpbid 232 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑅 ≤ (𝑁 / 2)) |
| 142 | 15 | 2halvesd 12512 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑁 / 2) + (𝑁 / 2)) = 𝑁) |
| 143 | 133 | recnd 11289 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 / 2) ∈ ℂ) |
| 144 | 15, 143, 143 | subaddd 11638 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑁 − (𝑁 / 2)) = (𝑁 / 2) ↔ ((𝑁 / 2) + (𝑁 / 2)) = 𝑁)) |
| 145 | 142, 144 | mpbird 257 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑁 − (𝑁 / 2)) = (𝑁 / 2)) |
| 146 | 141, 145 | breqtrrd 5171 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑅 ≤ (𝑁 − (𝑁 / 2))) |
| 147 | 29, 44, 133, 146 | lesubd 11867 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁 / 2) ≤ (𝑁 − 𝑅)) |
| 148 | 44 | lep1d 12199 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ≤ (𝑁 + 1)) |
| 149 | 133, 134,
44, 135, 139, 138, 147, 148 | lemul12ad 12210 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁 / 2) · 𝑁) ≤ ((𝑁 − 𝑅) · (𝑁 + 1))) |
| 150 | 132, 149 | eqbrtrd 5165 |
. . . . . . 7
⊢ (𝜑 → ((𝑁↑2) / 2) ≤ ((𝑁 − 𝑅) · (𝑁 + 1))) |
| 151 | 122, 116,
33, 125, 150 | lediv2ad 13099 |
. . . . . 6
⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) ≤ ((𝑅 + 1) / ((𝑁↑2) / 2))) |
| 152 | 1 | peano2nnd 12283 |
. . . . . . . . 9
⊢ (𝜑 → (𝑅 + 1) ∈ ℕ) |
| 153 | 152 | nncnd 12282 |
. . . . . . . 8
⊢ (𝜑 → (𝑅 + 1) ∈ ℂ) |
| 154 | 35 | nncnd 12282 |
. . . . . . . 8
⊢ (𝜑 → (𝑁↑2) ∈ ℂ) |
| 155 | 35 | nnne0d 12316 |
. . . . . . . 8
⊢ (𝜑 → (𝑁↑2) ≠ 0) |
| 156 | 153, 154,
128, 155, 130 | divdiv2d 12075 |
. . . . . . 7
⊢ (𝜑 → ((𝑅 + 1) / ((𝑁↑2) / 2)) = (((𝑅 + 1) · 2) / (𝑁↑2))) |
| 157 | 153, 128 | mulcomd 11282 |
. . . . . . . 8
⊢ (𝜑 → ((𝑅 + 1) · 2) = (2 · (𝑅 + 1))) |
| 158 | 157 | oveq1d 7446 |
. . . . . . 7
⊢ (𝜑 → (((𝑅 + 1) · 2) / (𝑁↑2)) = ((2 · (𝑅 + 1)) / (𝑁↑2))) |
| 159 | 156, 158 | eqtr2d 2778 |
. . . . . 6
⊢ (𝜑 → ((2 · (𝑅 + 1)) / (𝑁↑2)) = ((𝑅 + 1) / ((𝑁↑2) / 2))) |
| 160 | 151, 159 | breqtrrd 5171 |
. . . . 5
⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) ≤ ((2 · (𝑅 + 1)) / (𝑁↑2))) |
| 161 | 117, 36, 29, 118, 160 | lemul2ad 12208 |
. . . 4
⊢ (𝜑 → (𝑅 · ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2)))) |
| 162 | 115, 161 | eqbrtrrd 5167 |
. . 3
⊢ (𝜑 → ((𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) + (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2)))) |
| 163 | 28, 58, 37, 95, 162 | letrd 11418 |
. 2
⊢ (𝜑 → ((abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) + (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1))))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2)))) |
| 164 | 23, 28, 37, 38, 163 | letrd 11418 |
1
⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2)))) |