Proof of Theorem chpchtlim
| Step | Hyp | Ref
| Expression |
| 1 | | 1red 11262 |
. . 3
⊢ (⊤
→ 1 ∈ ℝ) |
| 2 | | 1red 11262 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 1 ∈ ℝ) |
| 3 | | 2re 12340 |
. . . . . . . . . . 11
⊢ 2 ∈
ℝ |
| 4 | | elicopnf 13485 |
. . . . . . . . . . 11
⊢ (2 ∈
ℝ → (𝑥 ∈
(2[,)+∞) ↔ (𝑥
∈ ℝ ∧ 2 ≤ 𝑥))) |
| 5 | 3, 4 | ax-mp 5 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (2[,)+∞) ↔
(𝑥 ∈ ℝ ∧ 2
≤ 𝑥)) |
| 6 | 5 | simplbi 497 |
. . . . . . . . 9
⊢ (𝑥 ∈ (2[,)+∞) →
𝑥 ∈
ℝ) |
| 7 | 6 | adantl 481 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 𝑥 ∈ ℝ) |
| 8 | | 0red 11264 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (2[,)+∞) → 0
∈ ℝ) |
| 9 | 3 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (2[,)+∞) → 2
∈ ℝ) |
| 10 | | 2pos 12369 |
. . . . . . . . . . . . 13
⊢ 0 <
2 |
| 11 | 10 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (2[,)+∞) → 0
< 2) |
| 12 | 5 | simprbi 496 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (2[,)+∞) → 2
≤ 𝑥) |
| 13 | 8, 9, 6, 11, 12 | ltletrd 11421 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (2[,)+∞) → 0
< 𝑥) |
| 14 | 6, 13 | elrpd 13074 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (2[,)+∞) →
𝑥 ∈
ℝ+) |
| 15 | 14 | adantl 481 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 𝑥 ∈ ℝ+) |
| 16 | 15 | rpge0d 13081 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 0 ≤ 𝑥) |
| 17 | 7, 16 | resqrtcld 15456 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (√‘𝑥) ∈ ℝ) |
| 18 | 15 | relogcld 26665 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (log‘𝑥) ∈ ℝ) |
| 19 | 17, 18 | remulcld 11291 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((√‘𝑥) · (log‘𝑥)) ∈ ℝ) |
| 20 | 12 | adantl 481 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 2 ≤ 𝑥) |
| 21 | | chtrpcl 27218 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ ∧ 2 ≤
𝑥) →
(θ‘𝑥) ∈
ℝ+) |
| 22 | 7, 20, 21 | syl2anc 584 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (θ‘𝑥) ∈
ℝ+) |
| 23 | 19, 22 | rerpdivcld 13108 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)) ∈ ℝ) |
| 24 | 6 | ssriv 3987 |
. . . . . 6
⊢
(2[,)+∞) ⊆ ℝ |
| 25 | 1 | recnd 11289 |
. . . . . 6
⊢ (⊤
→ 1 ∈ ℂ) |
| 26 | | rlimconst 15580 |
. . . . . 6
⊢
(((2[,)+∞) ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ (2[,)+∞) ↦
1) ⇝𝑟 1) |
| 27 | 24, 25, 26 | sylancr 587 |
. . . . 5
⊢ (⊤
→ (𝑥 ∈
(2[,)+∞) ↦ 1) ⇝𝑟 1) |
| 28 | | ovexd 7466 |
. . . . . . . 8
⊢ (⊤
→ (2[,)+∞) ∈ V) |
| 29 | 7, 22 | rerpdivcld 13108 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (𝑥 / (θ‘𝑥)) ∈ ℝ) |
| 30 | | ovexd 7466 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (((√‘𝑥) · (log‘𝑥)) / 𝑥) ∈ V) |
| 31 | | eqidd 2738 |
. . . . . . . 8
⊢ (⊤
→ (𝑥 ∈
(2[,)+∞) ↦ (𝑥 /
(θ‘𝑥))) =
(𝑥 ∈ (2[,)+∞)
↦ (𝑥 /
(θ‘𝑥)))) |
| 32 | 7 | recnd 11289 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 𝑥 ∈ ℂ) |
| 33 | | cxpsqrt 26745 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℂ → (𝑥↑𝑐(1 /
2)) = (√‘𝑥)) |
| 34 | 32, 33 | syl 17 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (𝑥↑𝑐(1 / 2)) =
(√‘𝑥)) |
| 35 | 34 | oveq2d 7447 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((log‘𝑥) / (𝑥↑𝑐(1 / 2))) =
((log‘𝑥) /
(√‘𝑥))) |
| 36 | 18 | recnd 11289 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (log‘𝑥) ∈ ℂ) |
| 37 | 15 | rpsqrtcld 15450 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (√‘𝑥) ∈
ℝ+) |
| 38 | 37 | rpcnne0d 13086 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((√‘𝑥) ∈ ℂ ∧ (√‘𝑥) ≠ 0)) |
| 39 | | divcan5 11969 |
. . . . . . . . . . 11
⊢
(((log‘𝑥)
∈ ℂ ∧ ((√‘𝑥) ∈ ℂ ∧ (√‘𝑥) ≠ 0) ∧
((√‘𝑥) ∈
ℂ ∧ (√‘𝑥) ≠ 0)) → (((√‘𝑥) · (log‘𝑥)) / ((√‘𝑥) · (√‘𝑥))) = ((log‘𝑥) / (√‘𝑥))) |
| 40 | 36, 38, 38, 39 | syl3anc 1373 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (((√‘𝑥) · (log‘𝑥)) / ((√‘𝑥) · (√‘𝑥))) = ((log‘𝑥) / (√‘𝑥))) |
| 41 | | remsqsqrt 15295 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ ∧ 0 ≤
𝑥) →
((√‘𝑥) ·
(√‘𝑥)) = 𝑥) |
| 42 | 7, 16, 41 | syl2anc 584 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((√‘𝑥) · (√‘𝑥)) = 𝑥) |
| 43 | 42 | oveq2d 7447 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (((√‘𝑥) · (log‘𝑥)) / ((√‘𝑥) · (√‘𝑥))) = (((√‘𝑥) · (log‘𝑥)) / 𝑥)) |
| 44 | 35, 40, 43 | 3eqtr2d 2783 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((log‘𝑥) / (𝑥↑𝑐(1 / 2))) =
(((√‘𝑥)
· (log‘𝑥)) /
𝑥)) |
| 45 | 44 | mpteq2dva 5242 |
. . . . . . . 8
⊢ (⊤
→ (𝑥 ∈
(2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑𝑐(1 / 2)))) = (𝑥 ∈ (2[,)+∞) ↦
(((√‘𝑥)
· (log‘𝑥)) /
𝑥))) |
| 46 | 28, 29, 30, 31, 45 | offval2 7717 |
. . . . . . 7
⊢ (⊤
→ ((𝑥 ∈
(2[,)+∞) ↦ (𝑥 /
(θ‘𝑥)))
∘f · (𝑥 ∈ (2[,)+∞) ↦
((log‘𝑥) / (𝑥↑𝑐(1 /
2))))) = (𝑥 ∈
(2[,)+∞) ↦ ((𝑥
/ (θ‘𝑥))
· (((√‘𝑥) · (log‘𝑥)) / 𝑥)))) |
| 47 | 15 | rpne0d 13082 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 𝑥 ≠ 0) |
| 48 | 22 | rpcnne0d 13086 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((θ‘𝑥) ∈ ℂ ∧ (θ‘𝑥) ≠ 0)) |
| 49 | 19 | recnd 11289 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((√‘𝑥) · (log‘𝑥)) ∈ ℂ) |
| 50 | | dmdcan 11977 |
. . . . . . . . 9
⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧
((θ‘𝑥) ∈
ℂ ∧ (θ‘𝑥) ≠ 0) ∧ ((√‘𝑥) · (log‘𝑥)) ∈ ℂ) →
((𝑥 / (θ‘𝑥)) ·
(((√‘𝑥)
· (log‘𝑥)) /
𝑥)) =
(((√‘𝑥)
· (log‘𝑥)) /
(θ‘𝑥))) |
| 51 | 32, 47, 48, 49, 50 | syl211anc 1378 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((𝑥 / (θ‘𝑥)) · (((√‘𝑥) · (log‘𝑥)) / 𝑥)) = (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥))) |
| 52 | 51 | mpteq2dva 5242 |
. . . . . . 7
⊢ (⊤
→ (𝑥 ∈
(2[,)+∞) ↦ ((𝑥
/ (θ‘𝑥))
· (((√‘𝑥) · (log‘𝑥)) / 𝑥))) = (𝑥 ∈ (2[,)+∞) ↦
(((√‘𝑥)
· (log‘𝑥)) /
(θ‘𝑥)))) |
| 53 | 46, 52 | eqtrd 2777 |
. . . . . 6
⊢ (⊤
→ ((𝑥 ∈
(2[,)+∞) ↦ (𝑥 /
(θ‘𝑥)))
∘f · (𝑥 ∈ (2[,)+∞) ↦
((log‘𝑥) / (𝑥↑𝑐(1 /
2))))) = (𝑥 ∈
(2[,)+∞) ↦ (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)))) |
| 54 | | chto1lb 27522 |
. . . . . . 7
⊢ (𝑥 ∈ (2[,)+∞) ↦
(𝑥 / (θ‘𝑥))) ∈
𝑂(1) |
| 55 | 14 | ssriv 3987 |
. . . . . . . . 9
⊢
(2[,)+∞) ⊆ ℝ+ |
| 56 | 55 | a1i 11 |
. . . . . . . 8
⊢ (⊤
→ (2[,)+∞) ⊆ ℝ+) |
| 57 | | 1rp 13038 |
. . . . . . . . . . 11
⊢ 1 ∈
ℝ+ |
| 58 | | rphalfcl 13062 |
. . . . . . . . . . 11
⊢ (1 ∈
ℝ+ → (1 / 2) ∈ ℝ+) |
| 59 | 57, 58 | ax-mp 5 |
. . . . . . . . . 10
⊢ (1 / 2)
∈ ℝ+ |
| 60 | | cxploglim 27021 |
. . . . . . . . . 10
⊢ ((1 / 2)
∈ ℝ+ → (𝑥 ∈ ℝ+ ↦
((log‘𝑥) / (𝑥↑𝑐(1 /
2)))) ⇝𝑟 0) |
| 61 | 59, 60 | ax-mp 5 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
↦ ((log‘𝑥) /
(𝑥↑𝑐(1 / 2))))
⇝𝑟 0 |
| 62 | 61 | a1i 11 |
. . . . . . . 8
⊢ (⊤
→ (𝑥 ∈
ℝ+ ↦ ((log‘𝑥) / (𝑥↑𝑐(1 / 2))))
⇝𝑟 0) |
| 63 | 56, 62 | rlimres2 15597 |
. . . . . . 7
⊢ (⊤
→ (𝑥 ∈
(2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑𝑐(1 / 2))))
⇝𝑟 0) |
| 64 | | o1rlimmul 15655 |
. . . . . . 7
⊢ (((𝑥 ∈ (2[,)+∞) ↦
(𝑥 / (θ‘𝑥))) ∈ 𝑂(1) ∧
(𝑥 ∈ (2[,)+∞)
↦ ((log‘𝑥) /
(𝑥↑𝑐(1 / 2))))
⇝𝑟 0) → ((𝑥 ∈ (2[,)+∞) ↦ (𝑥 / (θ‘𝑥))) ∘f ·
(𝑥 ∈ (2[,)+∞)
↦ ((log‘𝑥) /
(𝑥↑𝑐(1 / 2)))))
⇝𝑟 0) |
| 65 | 54, 63, 64 | sylancr 587 |
. . . . . 6
⊢ (⊤
→ ((𝑥 ∈
(2[,)+∞) ↦ (𝑥 /
(θ‘𝑥)))
∘f · (𝑥 ∈ (2[,)+∞) ↦
((log‘𝑥) / (𝑥↑𝑐(1 /
2))))) ⇝𝑟 0) |
| 66 | 53, 65 | eqbrtrrd 5167 |
. . . . 5
⊢ (⊤
→ (𝑥 ∈
(2[,)+∞) ↦ (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥))) ⇝𝑟
0) |
| 67 | 2, 23, 27, 66 | rlimadd 15679 |
. . . 4
⊢ (⊤
→ (𝑥 ∈
(2[,)+∞) ↦ (1 + (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)))) ⇝𝑟 (1 +
0)) |
| 68 | | 1p0e1 12390 |
. . . 4
⊢ (1 + 0) =
1 |
| 69 | 67, 68 | breqtrdi 5184 |
. . 3
⊢ (⊤
→ (𝑥 ∈
(2[,)+∞) ↦ (1 + (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)))) ⇝𝑟
1) |
| 70 | | 1re 11261 |
. . . 4
⊢ 1 ∈
ℝ |
| 71 | | readdcl 11238 |
. . . 4
⊢ ((1
∈ ℝ ∧ (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)) ∈ ℝ) → (1 +
(((√‘𝑥)
· (log‘𝑥)) /
(θ‘𝑥))) ∈
ℝ) |
| 72 | 70, 23, 71 | sylancr 587 |
. . 3
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (1 + (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥))) ∈ ℝ) |
| 73 | | chpcl 27167 |
. . . . 5
⊢ (𝑥 ∈ ℝ →
(ψ‘𝑥) ∈
ℝ) |
| 74 | 7, 73 | syl 17 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (ψ‘𝑥) ∈ ℝ) |
| 75 | 74, 22 | rerpdivcld 13108 |
. . 3
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ∈ ℝ) |
| 76 | | chtcl 27152 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ →
(θ‘𝑥) ∈
ℝ) |
| 77 | 7, 76 | syl 17 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (θ‘𝑥) ∈ ℝ) |
| 78 | 77, 19 | readdcld 11290 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((θ‘𝑥) + ((√‘𝑥) · (log‘𝑥))) ∈ ℝ) |
| 79 | 3 | a1i 11 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 2 ∈ ℝ) |
| 80 | | 1le2 12475 |
. . . . . . . . 9
⊢ 1 ≤
2 |
| 81 | 80 | a1i 11 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 1 ≤ 2) |
| 82 | 2, 79, 7, 81, 20 | letrd 11418 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 1 ≤ 𝑥) |
| 83 | | chpub 27264 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ ∧ 1 ≤
𝑥) →
(ψ‘𝑥) ≤
((θ‘𝑥) +
((√‘𝑥) ·
(log‘𝑥)))) |
| 84 | 7, 82, 83 | syl2anc 584 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (ψ‘𝑥) ≤ ((θ‘𝑥) + ((√‘𝑥) · (log‘𝑥)))) |
| 85 | 74, 78, 22, 84 | lediv1dd 13135 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ≤ (((θ‘𝑥) + ((√‘𝑥) · (log‘𝑥))) / (θ‘𝑥))) |
| 86 | 22 | rpcnd 13079 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (θ‘𝑥) ∈ ℂ) |
| 87 | | divdir 11947 |
. . . . . . 7
⊢
(((θ‘𝑥)
∈ ℂ ∧ ((√‘𝑥) · (log‘𝑥)) ∈ ℂ ∧ ((θ‘𝑥) ∈ ℂ ∧
(θ‘𝑥) ≠ 0))
→ (((θ‘𝑥)
+ ((√‘𝑥)
· (log‘𝑥))) /
(θ‘𝑥)) =
(((θ‘𝑥) /
(θ‘𝑥)) +
(((√‘𝑥)
· (log‘𝑥)) /
(θ‘𝑥)))) |
| 88 | 86, 49, 48, 87 | syl3anc 1373 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (((θ‘𝑥) + ((√‘𝑥) · (log‘𝑥))) / (θ‘𝑥)) = (((θ‘𝑥) / (θ‘𝑥)) + (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)))) |
| 89 | | divid 11953 |
. . . . . . . 8
⊢
(((θ‘𝑥)
∈ ℂ ∧ (θ‘𝑥) ≠ 0) → ((θ‘𝑥) / (θ‘𝑥)) = 1) |
| 90 | 48, 89 | syl 17 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((θ‘𝑥) / (θ‘𝑥)) = 1) |
| 91 | 90 | oveq1d 7446 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (((θ‘𝑥) / (θ‘𝑥)) + (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥))) = (1 + (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)))) |
| 92 | 88, 91 | eqtrd 2777 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (((θ‘𝑥) + ((√‘𝑥) · (log‘𝑥))) / (θ‘𝑥)) = (1 + (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)))) |
| 93 | 85, 92 | breqtrd 5169 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ≤ (1 + (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)))) |
| 94 | 93 | adantrr 717 |
. . 3
⊢
((⊤ ∧ (𝑥
∈ (2[,)+∞) ∧ 1 ≤ 𝑥)) → ((ψ‘𝑥) / (θ‘𝑥)) ≤ (1 + (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)))) |
| 95 | 86 | mullidd 11279 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (1 · (θ‘𝑥)) = (θ‘𝑥)) |
| 96 | | chtlepsi 27250 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ →
(θ‘𝑥) ≤
(ψ‘𝑥)) |
| 97 | 7, 96 | syl 17 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (θ‘𝑥) ≤ (ψ‘𝑥)) |
| 98 | 95, 97 | eqbrtrd 5165 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (1 · (θ‘𝑥)) ≤ (ψ‘𝑥)) |
| 99 | 2, 74, 22 | lemuldivd 13126 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((1 · (θ‘𝑥)) ≤ (ψ‘𝑥) ↔ 1 ≤ ((ψ‘𝑥) / (θ‘𝑥)))) |
| 100 | 98, 99 | mpbid 232 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 1 ≤ ((ψ‘𝑥) / (θ‘𝑥))) |
| 101 | 100 | adantrr 717 |
. . 3
⊢
((⊤ ∧ (𝑥
∈ (2[,)+∞) ∧ 1 ≤ 𝑥)) → 1 ≤ ((ψ‘𝑥) / (θ‘𝑥))) |
| 102 | 1, 1, 69, 72, 75, 94, 101 | rlimsqz2 15687 |
. 2
⊢ (⊤
→ (𝑥 ∈
(2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) ⇝𝑟
1) |
| 103 | 102 | mptru 1547 |
1
⊢ (𝑥 ∈ (2[,)+∞) ↦
((ψ‘𝑥) /
(θ‘𝑥)))
⇝𝑟 1 |