Proof of Theorem chto1ub
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | rpssre 13043 | . . . 4
⊢
ℝ+ ⊆ ℝ | 
| 2 | 1 | a1i 11 | . . 3
⊢ (⊤
→ ℝ+ ⊆ ℝ) | 
| 3 |  | rpre 13044 | . . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℝ) | 
| 4 |  | chtcl 27153 | . . . . . . 7
⊢ (𝑥 ∈ ℝ →
(θ‘𝑥) ∈
ℝ) | 
| 5 | 3, 4 | syl 17 | . . . . . 6
⊢ (𝑥 ∈ ℝ+
→ (θ‘𝑥)
∈ ℝ) | 
| 6 |  | rerpdivcl 13066 | . . . . . 6
⊢
(((θ‘𝑥)
∈ ℝ ∧ 𝑥
∈ ℝ+) → ((θ‘𝑥) / 𝑥) ∈ ℝ) | 
| 7 | 5, 6 | mpancom 688 | . . . . 5
⊢ (𝑥 ∈ ℝ+
→ ((θ‘𝑥) /
𝑥) ∈
ℝ) | 
| 8 | 7 | recnd 11290 | . . . 4
⊢ (𝑥 ∈ ℝ+
→ ((θ‘𝑥) /
𝑥) ∈
ℂ) | 
| 9 | 8 | adantl 481 | . . 3
⊢
((⊤ ∧ 𝑥
∈ ℝ+) → ((θ‘𝑥) / 𝑥) ∈ ℂ) | 
| 10 |  | 3re 12347 | . . . 4
⊢ 3 ∈
ℝ | 
| 11 | 10 | a1i 11 | . . 3
⊢ (⊤
→ 3 ∈ ℝ) | 
| 12 |  | 2rp 13040 | . . . . . 6
⊢ 2 ∈
ℝ+ | 
| 13 |  | relogcl 26618 | . . . . . 6
⊢ (2 ∈
ℝ+ → (log‘2) ∈ ℝ) | 
| 14 | 12, 13 | ax-mp 5 | . . . . 5
⊢
(log‘2) ∈ ℝ | 
| 15 |  | 2re 12341 | . . . . 5
⊢ 2 ∈
ℝ | 
| 16 | 14, 15 | remulcli 11278 | . . . 4
⊢
((log‘2) · 2) ∈ ℝ | 
| 17 | 16 | a1i 11 | . . 3
⊢ (⊤
→ ((log‘2) · 2) ∈ ℝ) | 
| 18 |  | chtge0 27156 | . . . . . . . . 9
⊢ (𝑥 ∈ ℝ → 0 ≤
(θ‘𝑥)) | 
| 19 | 3, 18 | syl 17 | . . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ 0 ≤ (θ‘𝑥)) | 
| 20 |  | rpregt0 13050 | . . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℝ
∧ 0 < 𝑥)) | 
| 21 |  | divge0 12138 | . . . . . . . 8
⊢
((((θ‘𝑥)
∈ ℝ ∧ 0 ≤ (θ‘𝑥)) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → 0 ≤
((θ‘𝑥) / 𝑥)) | 
| 22 | 5, 19, 20, 21 | syl21anc 837 | . . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ 0 ≤ ((θ‘𝑥) / 𝑥)) | 
| 23 | 7, 22 | absidd 15462 | . . . . . 6
⊢ (𝑥 ∈ ℝ+
→ (abs‘((θ‘𝑥) / 𝑥)) = ((θ‘𝑥) / 𝑥)) | 
| 24 | 23 | adantr 480 | . . . . 5
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
(abs‘((θ‘𝑥) / 𝑥)) = ((θ‘𝑥) / 𝑥)) | 
| 25 | 7 | adantr 480 | . . . . . 6
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
((θ‘𝑥) / 𝑥) ∈
ℝ) | 
| 26 | 16 | a1i 11 | . . . . . 6
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
((log‘2) · 2) ∈ ℝ) | 
| 27 | 5 | adantr 480 | . . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
(θ‘𝑥) ∈
ℝ) | 
| 28 | 3 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
𝑥 ∈
ℝ) | 
| 29 |  | remulcl 11241 | . . . . . . . . . . . 12
⊢ ((2
∈ ℝ ∧ 𝑥
∈ ℝ) → (2 · 𝑥) ∈ ℝ) | 
| 30 | 15, 28, 29 | sylancr 587 | . . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) → (2
· 𝑥) ∈
ℝ) | 
| 31 |  | resubcl 11574 | . . . . . . . . . . 11
⊢ (((2
· 𝑥) ∈ ℝ
∧ 3 ∈ ℝ) → ((2 · 𝑥) − 3) ∈ ℝ) | 
| 32 | 30, 10, 31 | sylancl 586 | . . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) → ((2
· 𝑥) − 3)
∈ ℝ) | 
| 33 |  | remulcl 11241 | . . . . . . . . . 10
⊢
(((log‘2) ∈ ℝ ∧ ((2 · 𝑥) − 3) ∈ ℝ) →
((log‘2) · ((2 · 𝑥) − 3)) ∈
ℝ) | 
| 34 | 14, 32, 33 | sylancr 587 | . . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
((log‘2) · ((2 · 𝑥) − 3)) ∈
ℝ) | 
| 35 |  | remulcl 11241 | . . . . . . . . . 10
⊢
(((log‘2) ∈ ℝ ∧ (2 · 𝑥) ∈ ℝ) → ((log‘2)
· (2 · 𝑥))
∈ ℝ) | 
| 36 | 14, 30, 35 | sylancr 587 | . . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
((log‘2) · (2 · 𝑥)) ∈ ℝ) | 
| 37 | 15 | a1i 11 | . . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) → 2
∈ ℝ) | 
| 38 | 10 | a1i 11 | . . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) → 3
∈ ℝ) | 
| 39 |  | 2lt3 12439 | . . . . . . . . . . . 12
⊢ 2 <
3 | 
| 40 | 39 | a1i 11 | . . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) → 2
< 3) | 
| 41 |  | simpr 484 | . . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) → 3
≤ 𝑥) | 
| 42 | 37, 38, 28, 40, 41 | ltletrd 11422 | . . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) → 2
< 𝑥) | 
| 43 |  | chtub 27257 | . . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ 2 <
𝑥) →
(θ‘𝑥) <
((log‘2) · ((2 · 𝑥) − 3))) | 
| 44 | 28, 42, 43 | syl2anc 584 | . . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
(θ‘𝑥) <
((log‘2) · ((2 · 𝑥) − 3))) | 
| 45 |  | 3rp 13041 | . . . . . . . . . . 11
⊢ 3 ∈
ℝ+ | 
| 46 |  | ltsubrp 13072 | . . . . . . . . . . 11
⊢ (((2
· 𝑥) ∈ ℝ
∧ 3 ∈ ℝ+) → ((2 · 𝑥) − 3) < (2 · 𝑥)) | 
| 47 | 30, 45, 46 | sylancl 586 | . . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) → ((2
· 𝑥) − 3) <
(2 · 𝑥)) | 
| 48 |  | 1lt2 12438 | . . . . . . . . . . . . . 14
⊢ 1 <
2 | 
| 49 |  | rplogcl 26647 | . . . . . . . . . . . . . 14
⊢ ((2
∈ ℝ ∧ 1 < 2) → (log‘2) ∈
ℝ+) | 
| 50 | 15, 48, 49 | mp2an 692 | . . . . . . . . . . . . 13
⊢
(log‘2) ∈ ℝ+ | 
| 51 |  | elrp 13037 | . . . . . . . . . . . . 13
⊢
((log‘2) ∈ ℝ+ ↔ ((log‘2) ∈
ℝ ∧ 0 < (log‘2))) | 
| 52 | 50, 51 | mpbi 230 | . . . . . . . . . . . 12
⊢
((log‘2) ∈ ℝ ∧ 0 <
(log‘2)) | 
| 53 | 52 | a1i 11 | . . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
((log‘2) ∈ ℝ ∧ 0 < (log‘2))) | 
| 54 |  | ltmul2 12119 | . . . . . . . . . . 11
⊢ ((((2
· 𝑥) − 3)
∈ ℝ ∧ (2 · 𝑥) ∈ ℝ ∧ ((log‘2) ∈
ℝ ∧ 0 < (log‘2))) → (((2 · 𝑥) − 3) < (2 · 𝑥) ↔ ((log‘2) ·
((2 · 𝑥) − 3))
< ((log‘2) · (2 · 𝑥)))) | 
| 55 | 32, 30, 53, 54 | syl3anc 1372 | . . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) → (((2
· 𝑥) − 3) <
(2 · 𝑥) ↔
((log‘2) · ((2 · 𝑥) − 3)) < ((log‘2) · (2
· 𝑥)))) | 
| 56 | 47, 55 | mpbid 232 | . . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
((log‘2) · ((2 · 𝑥) − 3)) < ((log‘2) · (2
· 𝑥))) | 
| 57 | 27, 34, 36, 44, 56 | lttrd 11423 | . . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
(θ‘𝑥) <
((log‘2) · (2 · 𝑥))) | 
| 58 | 14 | recni 11276 | . . . . . . . . . 10
⊢
(log‘2) ∈ ℂ | 
| 59 | 58 | a1i 11 | . . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
(log‘2) ∈ ℂ) | 
| 60 |  | 2cnd 12345 | . . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) → 2
∈ ℂ) | 
| 61 | 3 | recnd 11290 | . . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℂ) | 
| 62 | 61 | adantr 480 | . . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
𝑥 ∈
ℂ) | 
| 63 | 59, 60, 62 | mulassd 11285 | . . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
(((log‘2) · 2) · 𝑥) = ((log‘2) · (2 · 𝑥))) | 
| 64 | 57, 63 | breqtrrd 5170 | . . . . . . 7
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
(θ‘𝑥) <
(((log‘2) · 2) · 𝑥)) | 
| 65 | 20 | adantr 480 | . . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
(𝑥 ∈ ℝ ∧ 0
< 𝑥)) | 
| 66 |  | ltdivmul2 12146 | . . . . . . . 8
⊢
(((θ‘𝑥)
∈ ℝ ∧ ((log‘2) · 2) ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 <
𝑥)) →
(((θ‘𝑥) / 𝑥) < ((log‘2) ·
2) ↔ (θ‘𝑥)
< (((log‘2) · 2) · 𝑥))) | 
| 67 | 27, 26, 65, 66 | syl3anc 1372 | . . . . . . 7
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
(((θ‘𝑥) / 𝑥) < ((log‘2) ·
2) ↔ (θ‘𝑥)
< (((log‘2) · 2) · 𝑥))) | 
| 68 | 64, 67 | mpbird 257 | . . . . . 6
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
((θ‘𝑥) / 𝑥) < ((log‘2) ·
2)) | 
| 69 | 25, 26, 68 | ltled 11410 | . . . . 5
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
((θ‘𝑥) / 𝑥) ≤ ((log‘2) ·
2)) | 
| 70 | 24, 69 | eqbrtrd 5164 | . . . 4
⊢ ((𝑥 ∈ ℝ+
∧ 3 ≤ 𝑥) →
(abs‘((θ‘𝑥) / 𝑥)) ≤ ((log‘2) ·
2)) | 
| 71 | 70 | adantl 481 | . . 3
⊢
((⊤ ∧ (𝑥
∈ ℝ+ ∧ 3 ≤ 𝑥)) → (abs‘((θ‘𝑥) / 𝑥)) ≤ ((log‘2) ·
2)) | 
| 72 | 2, 9, 11, 17, 71 | elo1d 15573 | . 2
⊢ (⊤
→ (𝑥 ∈
ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1)) | 
| 73 | 72 | mptru 1546 | 1
⊢ (𝑥 ∈ ℝ+
↦ ((θ‘𝑥)
/ 𝑥)) ∈
𝑂(1) |