Proof of Theorem chpchtlim
Step | Hyp | Ref
| Expression |
1 | | 1red 9934 |
. . 3
⊢ (⊤
→ 1 ∈ ℝ) |
2 | | 1red 9934 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 1 ∈ ℝ) |
3 | | 2re 10967 |
. . . . . . . . . . 11
⊢ 2 ∈
ℝ |
4 | | elicopnf 12140 |
. . . . . . . . . . 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 475 |
. . . . . . . . 9
⊢ (𝑥 ∈ (2[,)+∞) →
𝑥 ∈
ℝ) |
7 | 6 | adantl 481 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 𝑥 ∈ ℝ) |
8 | | 0red 9920 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (2[,)+∞) → 0
∈ ℝ) |
9 | 3 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (2[,)+∞) → 2
∈ ℝ) |
10 | | 2pos 10989 |
. . . . . . . . . . . . 13
⊢ 0 <
2 |
11 | 10 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (2[,)+∞) → 0
< 2) |
12 | 5 | simprbi 479 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (2[,)+∞) → 2
≤ 𝑥) |
13 | 8, 9, 6, 11, 12 | ltletrd 10076 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (2[,)+∞) → 0
< 𝑥) |
14 | 6, 13 | elrpd 11745 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (2[,)+∞) →
𝑥 ∈
ℝ+) |
15 | 14 | adantl 481 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 𝑥 ∈ ℝ+) |
16 | 15 | rpge0d 11752 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 0 ≤ 𝑥) |
17 | 7, 16 | resqrtcld 14004 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (√‘𝑥) ∈ ℝ) |
18 | 15 | relogcld 24173 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (log‘𝑥) ∈ ℝ) |
19 | 17, 18 | remulcld 9949 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((√‘𝑥) · (log‘𝑥)) ∈ ℝ) |
20 | 12 | adantl 481 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 2 ≤ 𝑥) |
21 | | chtrpcl 24701 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ ∧ 2 ≤
𝑥) →
(θ‘𝑥) ∈
ℝ+) |
22 | 7, 20, 21 | syl2anc 691 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (θ‘𝑥) ∈
ℝ+) |
23 | 19, 22 | rerpdivcld 11779 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)) ∈ ℝ) |
24 | 6 | ssriv 3572 |
. . . . . 6
⊢
(2[,)+∞) ⊆ ℝ |
25 | 1 | recnd 9947 |
. . . . . 6
⊢ (⊤
→ 1 ∈ ℂ) |
26 | | rlimconst 14123 |
. . . . . 6
⊢
(((2[,)+∞) ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ (2[,)+∞) ↦
1) ⇝𝑟 1) |
27 | 24, 25, 26 | sylancr 694 |
. . . . 5
⊢ (⊤
→ (𝑥 ∈
(2[,)+∞) ↦ 1) ⇝𝑟 1) |
28 | | ovex 6577 |
. . . . . . . . 9
⊢
(2[,)+∞) ∈ V |
29 | 28 | a1i 11 |
. . . . . . . 8
⊢ (⊤
→ (2[,)+∞) ∈ V) |
30 | 7, 22 | rerpdivcld 11779 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (𝑥 / (θ‘𝑥)) ∈ ℝ) |
31 | | ovex 6577 |
. . . . . . . . 9
⊢
(((√‘𝑥)
· (log‘𝑥)) /
𝑥) ∈
V |
32 | 31 | a1i 11 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (((√‘𝑥) · (log‘𝑥)) / 𝑥) ∈ V) |
33 | | eqidd 2611 |
. . . . . . . 8
⊢ (⊤
→ (𝑥 ∈
(2[,)+∞) ↦ (𝑥 /
(θ‘𝑥))) =
(𝑥 ∈ (2[,)+∞)
↦ (𝑥 /
(θ‘𝑥)))) |
34 | 7 | recnd 9947 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 𝑥 ∈ ℂ) |
35 | | cxpsqrt 24249 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℂ → (𝑥↑𝑐(1 /
2)) = (√‘𝑥)) |
36 | 34, 35 | syl 17 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (𝑥↑𝑐(1 / 2)) =
(√‘𝑥)) |
37 | 36 | oveq2d 6565 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((log‘𝑥) / (𝑥↑𝑐(1 / 2))) =
((log‘𝑥) /
(√‘𝑥))) |
38 | 18 | recnd 9947 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (log‘𝑥) ∈ ℂ) |
39 | 15 | rpsqrtcld 13998 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (√‘𝑥) ∈
ℝ+) |
40 | 39 | rpcnne0d 11757 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((√‘𝑥) ∈ ℂ ∧ (√‘𝑥) ≠ 0)) |
41 | | divcan5 10606 |
. . . . . . . . . . 11
⊢
(((log‘𝑥)
∈ ℂ ∧ ((√‘𝑥) ∈ ℂ ∧ (√‘𝑥) ≠ 0) ∧
((√‘𝑥) ∈
ℂ ∧ (√‘𝑥) ≠ 0)) → (((√‘𝑥) · (log‘𝑥)) / ((√‘𝑥) · (√‘𝑥))) = ((log‘𝑥) / (√‘𝑥))) |
42 | 38, 40, 40, 41 | syl3anc 1318 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (((√‘𝑥) · (log‘𝑥)) / ((√‘𝑥) · (√‘𝑥))) = ((log‘𝑥) / (√‘𝑥))) |
43 | | remsqsqrt 13845 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ ∧ 0 ≤
𝑥) →
((√‘𝑥) ·
(√‘𝑥)) = 𝑥) |
44 | 7, 16, 43 | syl2anc 691 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((√‘𝑥) · (√‘𝑥)) = 𝑥) |
45 | 44 | oveq2d 6565 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (((√‘𝑥) · (log‘𝑥)) / ((√‘𝑥) · (√‘𝑥))) = (((√‘𝑥) · (log‘𝑥)) / 𝑥)) |
46 | 37, 42, 45 | 3eqtr2d 2650 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((log‘𝑥) / (𝑥↑𝑐(1 / 2))) =
(((√‘𝑥)
· (log‘𝑥)) /
𝑥)) |
47 | 46 | mpteq2dva 4672 |
. . . . . . . 8
⊢ (⊤
→ (𝑥 ∈
(2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑𝑐(1 / 2)))) = (𝑥 ∈ (2[,)+∞) ↦
(((√‘𝑥)
· (log‘𝑥)) /
𝑥))) |
48 | 29, 30, 32, 33, 47 | offval2 6812 |
. . . . . . 7
⊢ (⊤
→ ((𝑥 ∈
(2[,)+∞) ↦ (𝑥 /
(θ‘𝑥)))
∘𝑓 · (𝑥 ∈ (2[,)+∞) ↦
((log‘𝑥) / (𝑥↑𝑐(1 /
2))))) = (𝑥 ∈
(2[,)+∞) ↦ ((𝑥
/ (θ‘𝑥))
· (((√‘𝑥) · (log‘𝑥)) / 𝑥)))) |
49 | 15 | rpne0d 11753 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 𝑥 ≠ 0) |
50 | 22 | rpcnne0d 11757 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((θ‘𝑥) ∈ ℂ ∧ (θ‘𝑥) ≠ 0)) |
51 | 19 | recnd 9947 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((√‘𝑥) · (log‘𝑥)) ∈ ℂ) |
52 | | dmdcan 10614 |
. . . . . . . . 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))) |
53 | 34, 49, 50, 51, 52 | syl211anc 1324 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((𝑥 / (θ‘𝑥)) · (((√‘𝑥) · (log‘𝑥)) / 𝑥)) = (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥))) |
54 | 53 | mpteq2dva 4672 |
. . . . . . 7
⊢ (⊤
→ (𝑥 ∈
(2[,)+∞) ↦ ((𝑥
/ (θ‘𝑥))
· (((√‘𝑥) · (log‘𝑥)) / 𝑥))) = (𝑥 ∈ (2[,)+∞) ↦
(((√‘𝑥)
· (log‘𝑥)) /
(θ‘𝑥)))) |
55 | 48, 54 | eqtrd 2644 |
. . . . . 6
⊢ (⊤
→ ((𝑥 ∈
(2[,)+∞) ↦ (𝑥 /
(θ‘𝑥)))
∘𝑓 · (𝑥 ∈ (2[,)+∞) ↦
((log‘𝑥) / (𝑥↑𝑐(1 /
2))))) = (𝑥 ∈
(2[,)+∞) ↦ (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)))) |
56 | | chto1lb 24967 |
. . . . . . 7
⊢ (𝑥 ∈ (2[,)+∞) ↦
(𝑥 / (θ‘𝑥))) ∈
𝑂(1) |
57 | 14 | ssriv 3572 |
. . . . . . . . 9
⊢
(2[,)+∞) ⊆ ℝ+ |
58 | 57 | a1i 11 |
. . . . . . . 8
⊢ (⊤
→ (2[,)+∞) ⊆ ℝ+) |
59 | | 1rp 11712 |
. . . . . . . . . . 11
⊢ 1 ∈
ℝ+ |
60 | | rphalfcl 11734 |
. . . . . . . . . . 11
⊢ (1 ∈
ℝ+ → (1 / 2) ∈ ℝ+) |
61 | 59, 60 | ax-mp 5 |
. . . . . . . . . 10
⊢ (1 / 2)
∈ ℝ+ |
62 | | cxploglim 24504 |
. . . . . . . . . 10
⊢ ((1 / 2)
∈ ℝ+ → (𝑥 ∈ ℝ+ ↦
((log‘𝑥) / (𝑥↑𝑐(1 /
2)))) ⇝𝑟 0) |
63 | 61, 62 | ax-mp 5 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
↦ ((log‘𝑥) /
(𝑥↑𝑐(1 / 2))))
⇝𝑟 0 |
64 | 63 | a1i 11 |
. . . . . . . 8
⊢ (⊤
→ (𝑥 ∈
ℝ+ ↦ ((log‘𝑥) / (𝑥↑𝑐(1 / 2))))
⇝𝑟 0) |
65 | 58, 64 | rlimres2 14140 |
. . . . . . 7
⊢ (⊤
→ (𝑥 ∈
(2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑𝑐(1 / 2))))
⇝𝑟 0) |
66 | | o1rlimmul 14197 |
. . . . . . 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) |
67 | 56, 65, 66 | sylancr 694 |
. . . . . 6
⊢ (⊤
→ ((𝑥 ∈
(2[,)+∞) ↦ (𝑥 /
(θ‘𝑥)))
∘𝑓 · (𝑥 ∈ (2[,)+∞) ↦
((log‘𝑥) / (𝑥↑𝑐(1 /
2))))) ⇝𝑟 0) |
68 | 55, 67 | eqbrtrrd 4607 |
. . . . 5
⊢ (⊤
→ (𝑥 ∈
(2[,)+∞) ↦ (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥))) ⇝𝑟
0) |
69 | 2, 23, 27, 68 | rlimadd 14221 |
. . . 4
⊢ (⊤
→ (𝑥 ∈
(2[,)+∞) ↦ (1 + (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)))) ⇝𝑟 (1 +
0)) |
70 | | 1p0e1 11010 |
. . . 4
⊢ (1 + 0) =
1 |
71 | 69, 70 | syl6breq 4624 |
. . 3
⊢ (⊤
→ (𝑥 ∈
(2[,)+∞) ↦ (1 + (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)))) ⇝𝑟
1) |
72 | | 1re 9918 |
. . . 4
⊢ 1 ∈
ℝ |
73 | | readdcl 9898 |
. . . 4
⊢ ((1
∈ ℝ ∧ (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)) ∈ ℝ) → (1 +
(((√‘𝑥)
· (log‘𝑥)) /
(θ‘𝑥))) ∈
ℝ) |
74 | 72, 23, 73 | sylancr 694 |
. . 3
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (1 + (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥))) ∈ ℝ) |
75 | | chpcl 24650 |
. . . . 5
⊢ (𝑥 ∈ ℝ →
(ψ‘𝑥) ∈
ℝ) |
76 | 7, 75 | syl 17 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (ψ‘𝑥) ∈ ℝ) |
77 | 76, 22 | rerpdivcld 11779 |
. . 3
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ∈ ℝ) |
78 | | chtcl 24635 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ →
(θ‘𝑥) ∈
ℝ) |
79 | 7, 78 | syl 17 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (θ‘𝑥) ∈ ℝ) |
80 | 79, 19 | readdcld 9948 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((θ‘𝑥) + ((√‘𝑥) · (log‘𝑥))) ∈ ℝ) |
81 | 3 | a1i 11 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 2 ∈ ℝ) |
82 | | 1le2 11118 |
. . . . . . . . 9
⊢ 1 ≤
2 |
83 | 82 | a1i 11 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 1 ≤ 2) |
84 | 2, 81, 7, 83, 20 | letrd 10073 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 1 ≤ 𝑥) |
85 | | chpub 24745 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ ∧ 1 ≤
𝑥) →
(ψ‘𝑥) ≤
((θ‘𝑥) +
((√‘𝑥) ·
(log‘𝑥)))) |
86 | 7, 84, 85 | syl2anc 691 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (ψ‘𝑥) ≤ ((θ‘𝑥) + ((√‘𝑥) · (log‘𝑥)))) |
87 | 76, 80, 22, 86 | lediv1dd 11806 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ≤ (((θ‘𝑥) + ((√‘𝑥) · (log‘𝑥))) / (θ‘𝑥))) |
88 | 22 | rpcnd 11750 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (θ‘𝑥) ∈ ℂ) |
89 | | divdir 10589 |
. . . . . . 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)))) |
90 | 88, 51, 50, 89 | syl3anc 1318 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (((θ‘𝑥) + ((√‘𝑥) · (log‘𝑥))) / (θ‘𝑥)) = (((θ‘𝑥) / (θ‘𝑥)) + (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)))) |
91 | | divid 10593 |
. . . . . . . 8
⊢
(((θ‘𝑥)
∈ ℂ ∧ (θ‘𝑥) ≠ 0) → ((θ‘𝑥) / (θ‘𝑥)) = 1) |
92 | 50, 91 | syl 17 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((θ‘𝑥) / (θ‘𝑥)) = 1) |
93 | 92 | oveq1d 6564 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (((θ‘𝑥) / (θ‘𝑥)) + (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥))) = (1 + (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)))) |
94 | 90, 93 | eqtrd 2644 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (((θ‘𝑥) + ((√‘𝑥) · (log‘𝑥))) / (θ‘𝑥)) = (1 + (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)))) |
95 | 87, 94 | breqtrd 4609 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ≤ (1 + (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)))) |
96 | 95 | adantrr 749 |
. . 3
⊢
((⊤ ∧ (𝑥
∈ (2[,)+∞) ∧ 1 ≤ 𝑥)) → ((ψ‘𝑥) / (θ‘𝑥)) ≤ (1 + (((√‘𝑥) · (log‘𝑥)) / (θ‘𝑥)))) |
97 | 88 | mulid2d 9937 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (1 · (θ‘𝑥)) = (θ‘𝑥)) |
98 | | chtlepsi 24731 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ →
(θ‘𝑥) ≤
(ψ‘𝑥)) |
99 | 7, 98 | syl 17 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (θ‘𝑥) ≤ (ψ‘𝑥)) |
100 | 97, 99 | eqbrtrd 4605 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → (1 · (θ‘𝑥)) ≤ (ψ‘𝑥)) |
101 | 2, 76, 22 | lemuldivd 11797 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → ((1 · (θ‘𝑥)) ≤ (ψ‘𝑥) ↔ 1 ≤ ((ψ‘𝑥) / (θ‘𝑥)))) |
102 | 100, 101 | mpbid 221 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ (2[,)+∞)) → 1 ≤ ((ψ‘𝑥) / (θ‘𝑥))) |
103 | 102 | adantrr 749 |
. . 3
⊢
((⊤ ∧ (𝑥
∈ (2[,)+∞) ∧ 1 ≤ 𝑥)) → 1 ≤ ((ψ‘𝑥) / (θ‘𝑥))) |
104 | 1, 1, 71, 74, 77, 96, 103 | rlimsqz2 14229 |
. 2
⊢ (⊤
→ (𝑥 ∈
(2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) ⇝𝑟
1) |
105 | 104 | trud 1484 |
1
⊢ (𝑥 ∈ (2[,)+∞) ↦
((ψ‘𝑥) /
(θ‘𝑥)))
⇝𝑟 1 |