Proof of Theorem dvrelog2b
| Step | Hyp | Ref
| Expression |
| 1 | | dvrelog2b.5 |
. . . . 5
⊢ 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 logb 𝑥)) |
| 2 | 1 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 logb 𝑥))) |
| 3 | | 2cnd 12344 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ) |
| 4 | | 2ne0 12370 |
. . . . . . . . 9
⊢ 2 ≠
0 |
| 5 | 4 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0) |
| 6 | | 1red 11262 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ) |
| 7 | | 1lt2 12437 |
. . . . . . . . . . 11
⊢ 1 <
2 |
| 8 | 7 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 2) |
| 9 | 6, 8 | ltned 11397 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ≠ 2) |
| 10 | 9 | necomd 2996 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 1) |
| 11 | 5, 10 | nelprd 4657 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 2 ∈ {0,
1}) |
| 12 | 3, 11 | eldifd 3962 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ (ℂ ∖ {0,
1})) |
| 13 | | elioore 13417 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ) |
| 14 | | recn 11245 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℂ) |
| 15 | 13, 14 | syl 17 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℂ) |
| 16 | 15 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℂ) |
| 17 | | elsni 4643 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ {0} → 𝑥 = 0) |
| 18 | | dvrelog2b.3 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 0 ≤ 𝐴) |
| 19 | | 0xr 11308 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 0 ∈
ℝ* |
| 20 | 19 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 0 ∈
ℝ*) |
| 21 | | dvrelog2b.1 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| 22 | | xrlenlt 11326 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((0
∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (0 ≤
𝐴 ↔ ¬ 𝐴 < 0)) |
| 23 | 20, 21, 22 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0)) |
| 24 | 18, 23 | mpbid 232 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ¬ 𝐴 < 0) |
| 25 | 24 | orcd 874 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (¬ 𝐴 < 0 ∨ ¬ 0 < 𝐵)) |
| 26 | | ianor 984 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬
(𝐴 < 0 ∧ 0 <
𝐵) ↔ (¬ 𝐴 < 0 ∨ ¬ 0 < 𝐵)) |
| 27 | 25, 26 | sylibr 234 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ¬ (𝐴 < 0 ∧ 0 < 𝐵)) |
| 28 | | dvrelog2b.2 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
| 29 | | elioo5 13444 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 0 ∈ ℝ*) → (0 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 0 ∧ 0 < 𝐵))) |
| 30 | 21, 28, 20, 29 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (0 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 0 ∧ 0 < 𝐵))) |
| 31 | 30 | notbid 318 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (¬ 0 ∈ (𝐴(,)𝐵) ↔ ¬ (𝐴 < 0 ∧ 0 < 𝐵))) |
| 32 | 27, 31 | mpbird 257 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵)) |
| 33 | 32 | a1d 25 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (0 ∈ (𝐴(,)𝐵) → ¬ 0 ∈ (𝐴(,)𝐵))) |
| 34 | 33 | imp 406 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 0 ∈ (𝐴(,)𝐵)) → ¬ 0 ∈ (𝐴(,)𝐵)) |
| 35 | 34 | pm2.01da 799 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵)) |
| 36 | 35 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 = 0) → ¬ 0 ∈ (𝐴(,)𝐵)) |
| 37 | | eleq1 2829 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 0 → (𝑥 ∈ (𝐴(,)𝐵) ↔ 0 ∈ (𝐴(,)𝐵))) |
| 38 | 37 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 = 0) → (𝑥 ∈ (𝐴(,)𝐵) ↔ 0 ∈ (𝐴(,)𝐵))) |
| 39 | 36, 38 | mtbird 325 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 = 0) → ¬ 𝑥 ∈ (𝐴(,)𝐵)) |
| 40 | 17, 39 | sylan2 593 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ {0}) → ¬ 𝑥 ∈ (𝐴(,)𝐵)) |
| 41 | 40 | ex 412 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ {0} → ¬ 𝑥 ∈ (𝐴(,)𝐵))) |
| 42 | 41 | con2d 134 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) → ¬ 𝑥 ∈ {0})) |
| 43 | 42 | imp 406 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 ∈ {0}) |
| 44 | 16, 43 | eldifd 3962 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (ℂ ∖
{0})) |
| 45 | | logbval 26809 |
. . . . . 6
⊢ ((2
∈ (ℂ ∖ {0, 1}) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (2
logb 𝑥) =
((log‘𝑥) /
(log‘2))) |
| 46 | 12, 44, 45 | syl2anc 584 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 logb 𝑥) = ((log‘𝑥) /
(log‘2))) |
| 47 | 46 | mpteq2dva 5242 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 logb 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))) |
| 48 | 2, 47 | eqtrd 2777 |
. . 3
⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))) |
| 49 | 48 | oveq2d 7447 |
. 2
⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2))))) |
| 50 | | reelprrecn 11247 |
. . . . 5
⊢ ℝ
∈ {ℝ, ℂ} |
| 51 | 50 | a1i 11 |
. . . 4
⊢ (𝜑 → ℝ ∈ {ℝ,
ℂ}) |
| 52 | 39 | ex 412 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 = 0 → ¬ 𝑥 ∈ (𝐴(,)𝐵))) |
| 53 | 52 | con2d 134 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) → ¬ 𝑥 = 0)) |
| 54 | | biidd 262 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝑥 = 0 ↔ 𝑥 = 0)) |
| 55 | 54 | necon3bbid 2978 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝐴(,)𝐵) → (¬ 𝑥 = 0 ↔ 𝑥 ≠ 0)) |
| 56 | 55 | pm5.74i 271 |
. . . . . . 7
⊢ ((𝑥 ∈ (𝐴(,)𝐵) → ¬ 𝑥 = 0) ↔ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ≠ 0)) |
| 57 | 53, 56 | sylib 218 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ≠ 0)) |
| 58 | 57 | imp 406 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0) |
| 59 | 16, 58 | logcld 26612 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘𝑥) ∈ ℂ) |
| 60 | 13 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ) |
| 61 | 6, 60, 58 | redivcld 12095 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / 𝑥) ∈ ℝ) |
| 62 | | dvrelog2b.4 |
. . . . 5
⊢ (𝜑 → 𝐴 ≤ 𝐵) |
| 63 | | eqid 2737 |
. . . . 5
⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥)) |
| 64 | | eqid 2737 |
. . . . 5
⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) |
| 65 | 21, 28, 18, 62, 63, 64 | dvrelog3 42066 |
. . . 4
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))) |
| 66 | | 2cnd 12344 |
. . . . 5
⊢ (𝜑 → 2 ∈
ℂ) |
| 67 | 4 | a1i 11 |
. . . . 5
⊢ (𝜑 → 2 ≠ 0) |
| 68 | 66, 67 | logcld 26612 |
. . . 4
⊢ (𝜑 → (log‘2) ∈
ℂ) |
| 69 | | 0red 11264 |
. . . . . 6
⊢ (𝜑 → 0 ∈
ℝ) |
| 70 | | 2rp 13039 |
. . . . . . . . 9
⊢ 2 ∈
ℝ+ |
| 71 | | loggt0b 26674 |
. . . . . . . . 9
⊢ (2 ∈
ℝ+ → (0 < (log‘2) ↔ 1 <
2)) |
| 72 | 70, 71 | ax-mp 5 |
. . . . . . . 8
⊢ (0 <
(log‘2) ↔ 1 < 2) |
| 73 | 7, 72 | mpbir 231 |
. . . . . . 7
⊢ 0 <
(log‘2) |
| 74 | 73 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 0 <
(log‘2)) |
| 75 | 69, 74 | ltned 11397 |
. . . . 5
⊢ (𝜑 → 0 ≠
(log‘2)) |
| 76 | 75 | necomd 2996 |
. . . 4
⊢ (𝜑 → (log‘2) ≠
0) |
| 77 | 51, 59, 61, 65, 68, 76 | dvmptdivc 26003 |
. . 3
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((1 / 𝑥) / (log‘2)))) |
| 78 | 3, 5 | logcld 26612 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈
ℂ) |
| 79 | 76 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ≠
0) |
| 80 | 16, 78, 58, 79 | recdiv2d 12061 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((1 / 𝑥) / (log‘2)) = (1 / (𝑥 · (log‘2)))) |
| 81 | 80 | mpteq2dva 5242 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((1 / 𝑥) / (log‘2))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2))))) |
| 82 | | dvrelog2b.6 |
. . . . . 6
⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))) |
| 83 | 82 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2))))) |
| 84 | 83 | eqcomd 2743 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))) = 𝐺) |
| 85 | 81, 84 | eqtrd 2777 |
. . 3
⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((1 / 𝑥) / (log‘2))) = 𝐺) |
| 86 | 77, 85 | eqtrd 2777 |
. 2
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))) = 𝐺) |
| 87 | 49, 86 | eqtrd 2777 |
1
⊢ (𝜑 → (ℝ D 𝐹) = 𝐺) |