Proof of Theorem 3lexlogpow5ineq2
| Step | Hyp | Ref
| Expression |
| 1 | | 1nn0 12542 |
. . . . . 6
⊢ 1 ∈
ℕ0 |
| 2 | | 1nn 12277 |
. . . . . 6
⊢ 1 ∈
ℕ |
| 3 | 1, 2 | decnncl 12753 |
. . . . 5
⊢ ;11 ∈ ℕ |
| 4 | 3 | a1i 11 |
. . . 4
⊢ (𝜑 → ;11 ∈ ℕ) |
| 5 | 4 | nnred 12281 |
. . 3
⊢ (𝜑 → ;11 ∈ ℝ) |
| 6 | | 7re 12359 |
. . . 4
⊢ 7 ∈
ℝ |
| 7 | 6 | a1i 11 |
. . 3
⊢ (𝜑 → 7 ∈
ℝ) |
| 8 | | 0red 11264 |
. . . . 5
⊢ (𝜑 → 0 ∈
ℝ) |
| 9 | | 7pos 12377 |
. . . . . 6
⊢ 0 <
7 |
| 10 | 9 | a1i 11 |
. . . . 5
⊢ (𝜑 → 0 < 7) |
| 11 | 8, 10 | ltned 11397 |
. . . 4
⊢ (𝜑 → 0 ≠ 7) |
| 12 | 11 | necomd 2996 |
. . 3
⊢ (𝜑 → 7 ≠ 0) |
| 13 | 5, 7, 12 | redivcld 12095 |
. 2
⊢ (𝜑 → (;11 / 7) ∈ ℝ) |
| 14 | | 2re 12340 |
. . . 4
⊢ 2 ∈
ℝ |
| 15 | 14 | a1i 11 |
. . 3
⊢ (𝜑 → 2 ∈
ℝ) |
| 16 | | 2pos 12369 |
. . . 4
⊢ 0 <
2 |
| 17 | 16 | a1i 11 |
. . 3
⊢ (𝜑 → 0 < 2) |
| 18 | | 3lexlogpow5ineq2.1 |
. . 3
⊢ (𝜑 → 𝑋 ∈ ℝ) |
| 19 | | 3re 12346 |
. . . . 5
⊢ 3 ∈
ℝ |
| 20 | 19 | a1i 11 |
. . . 4
⊢ (𝜑 → 3 ∈
ℝ) |
| 21 | | 3pos 12371 |
. . . . 5
⊢ 0 <
3 |
| 22 | 21 | a1i 11 |
. . . 4
⊢ (𝜑 → 0 < 3) |
| 23 | | 3lexlogpow5ineq2.2 |
. . . 4
⊢ (𝜑 → 3 ≤ 𝑋) |
| 24 | 8, 20, 18, 22, 23 | ltletrd 11421 |
. . 3
⊢ (𝜑 → 0 < 𝑋) |
| 25 | | 1red 11262 |
. . . . 5
⊢ (𝜑 → 1 ∈
ℝ) |
| 26 | | 1lt2 12437 |
. . . . . 6
⊢ 1 <
2 |
| 27 | 26 | a1i 11 |
. . . . 5
⊢ (𝜑 → 1 < 2) |
| 28 | 25, 27 | ltned 11397 |
. . . 4
⊢ (𝜑 → 1 ≠ 2) |
| 29 | 28 | necomd 2996 |
. . 3
⊢ (𝜑 → 2 ≠ 1) |
| 30 | 15, 17, 18, 24, 29 | relogbcld 41974 |
. 2
⊢ (𝜑 → (2 logb 𝑋) ∈
ℝ) |
| 31 | | 5nn0 12546 |
. . 3
⊢ 5 ∈
ℕ0 |
| 32 | 31 | a1i 11 |
. 2
⊢ (𝜑 → 5 ∈
ℕ0) |
| 33 | | 7nn 12358 |
. . . . 5
⊢ 7 ∈
ℕ |
| 34 | 33 | a1i 11 |
. . . 4
⊢ (𝜑 → 7 ∈
ℕ) |
| 35 | 34 | nnrpd 13075 |
. . 3
⊢ (𝜑 → 7 ∈
ℝ+) |
| 36 | | 0nn0 12541 |
. . . . 5
⊢ 0 ∈
ℕ0 |
| 37 | | tru 1544 |
. . . . . 6
⊢
⊤ |
| 38 | | 0red 11264 |
. . . . . . 7
⊢ (⊤
→ 0 ∈ ℝ) |
| 39 | | 9re 12365 |
. . . . . . . 8
⊢ 9 ∈
ℝ |
| 40 | 39 | a1i 11 |
. . . . . . 7
⊢ (⊤
→ 9 ∈ ℝ) |
| 41 | | 9pos 12379 |
. . . . . . . 8
⊢ 0 <
9 |
| 42 | 41 | a1i 11 |
. . . . . . 7
⊢ (⊤
→ 0 < 9) |
| 43 | 38, 40, 42 | ltled 11409 |
. . . . . 6
⊢ (⊤
→ 0 ≤ 9) |
| 44 | 37, 43 | ax-mp 5 |
. . . . 5
⊢ 0 ≤
9 |
| 45 | 2, 1, 36, 44 | declei 12769 |
. . . 4
⊢ 0 ≤
;11 |
| 46 | 45 | a1i 11 |
. . 3
⊢ (𝜑 → 0 ≤ ;11) |
| 47 | 5, 35, 46 | divge0d 13117 |
. 2
⊢ (𝜑 → 0 ≤ (;11 / 7)) |
| 48 | 15, 17, 20, 22, 29 | relogbcld 41974 |
. . 3
⊢ (𝜑 → (2 logb 3)
∈ ℝ) |
| 49 | | 2exp11 17127 |
. . . . . . . . . . 11
⊢
(2↑;11) = ;;;2048 |
| 50 | 49 | eqcomi 2746 |
. . . . . . . . . 10
⊢ ;;;2048 =
(2↑;11) |
| 51 | 50 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ;;;2048 = (2↑;11)) |
| 52 | 51 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝜑 → (2 logb ;;;2048)
= (2 logb (2↑;11))) |
| 53 | 15, 17 | elrpd 13074 |
. . . . . . . . 9
⊢ (𝜑 → 2 ∈
ℝ+) |
| 54 | 4 | nnzd 12640 |
. . . . . . . . 9
⊢ (𝜑 → ;11 ∈ ℤ) |
| 55 | 53, 29, 54 | relogbexpd 41975 |
. . . . . . . 8
⊢ (𝜑 → (2 logb
(2↑;11)) = ;11) |
| 56 | 52, 55 | eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → (2 logb ;;;2048)
= ;11) |
| 57 | 56 | eqcomd 2743 |
. . . . . 6
⊢ (𝜑 → ;11 = (2 logb ;;;2048)) |
| 58 | | 2z 12649 |
. . . . . . . 8
⊢ 2 ∈
ℤ |
| 59 | 58 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 2 ∈
ℤ) |
| 60 | 15 | leidd 11829 |
. . . . . . 7
⊢ (𝜑 → 2 ≤ 2) |
| 61 | | 2nn0 12543 |
. . . . . . . . . . . 12
⊢ 2 ∈
ℕ0 |
| 62 | 61, 36 | deccl 12748 |
. . . . . . . . . . 11
⊢ ;20 ∈
ℕ0 |
| 63 | | 4nn0 12545 |
. . . . . . . . . . 11
⊢ 4 ∈
ℕ0 |
| 64 | 62, 63 | deccl 12748 |
. . . . . . . . . 10
⊢ ;;204 ∈ ℕ0 |
| 65 | | 8nn 12361 |
. . . . . . . . . 10
⊢ 8 ∈
ℕ |
| 66 | 64, 65 | decnncl 12753 |
. . . . . . . . 9
⊢ ;;;2048
∈ ℕ |
| 67 | 66 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ;;;2048 ∈ ℕ) |
| 68 | 67 | nnred 12281 |
. . . . . . 7
⊢ (𝜑 → ;;;2048 ∈ ℝ) |
| 69 | | 4nn 12349 |
. . . . . . . . . 10
⊢ 4 ∈
ℕ |
| 70 | 62, 69 | decnncl 12753 |
. . . . . . . . 9
⊢ ;;204 ∈ ℕ |
| 71 | | 8nn0 12549 |
. . . . . . . . 9
⊢ 8 ∈
ℕ0 |
| 72 | 70, 71, 36, 44 | decltdi 12772 |
. . . . . . . 8
⊢ 0 <
;;;2048 |
| 73 | 72 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 0 < ;;;2048) |
| 74 | 61, 1 | deccl 12748 |
. . . . . . . . . . 11
⊢ ;21 ∈
ℕ0 |
| 75 | 74, 71 | deccl 12748 |
. . . . . . . . . 10
⊢ ;;218 ∈ ℕ0 |
| 76 | 75, 33 | decnncl 12753 |
. . . . . . . . 9
⊢ ;;;2187
∈ ℕ |
| 77 | 76 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ;;;2187 ∈ ℕ) |
| 78 | 77 | nnred 12281 |
. . . . . . 7
⊢ (𝜑 → ;;;2187 ∈ ℝ) |
| 79 | 74, 65 | decnncl 12753 |
. . . . . . . . 9
⊢ ;;218 ∈ ℕ |
| 80 | | 7nn0 12548 |
. . . . . . . . 9
⊢ 7 ∈
ℕ0 |
| 81 | 79, 80, 36, 44 | decltdi 12772 |
. . . . . . . 8
⊢ 0 <
;;;2187 |
| 82 | 81 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 0 < ;;;2187) |
| 83 | | 8re 12362 |
. . . . . . . . . . 11
⊢ 8 ∈
ℝ |
| 84 | 83, 1 | nn0addge1i 12574 |
. . . . . . . . . 10
⊢ 8 ≤ (8
+ 1) |
| 85 | | 8p1e9 12416 |
. . . . . . . . . 10
⊢ (8 + 1) =
9 |
| 86 | 84, 85 | breqtri 5168 |
. . . . . . . . 9
⊢ 8 ≤
9 |
| 87 | | 4lt10 12869 |
. . . . . . . . . 10
⊢ 4 <
;10 |
| 88 | | 0lt1 11785 |
. . . . . . . . . . 11
⊢ 0 <
1 |
| 89 | 61, 36, 2, 88 | declt 12761 |
. . . . . . . . . 10
⊢ ;20 < ;21 |
| 90 | 62, 74, 63, 71, 87, 89 | decltc 12762 |
. . . . . . . . 9
⊢ ;;204 < ;;218 |
| 91 | 64, 75, 71, 80, 86, 90 | decleh 12768 |
. . . . . . . 8
⊢ ;;;2048
≤ ;;;2187 |
| 92 | 91 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → ;;;2048 ≤ ;;;2187) |
| 93 | 59, 60, 68, 73, 78, 82, 92 | logblebd 41977 |
. . . . . 6
⊢ (𝜑 → (2 logb ;;;2048)
≤ (2 logb ;;;2187)) |
| 94 | 57, 93 | eqbrtrd 5165 |
. . . . 5
⊢ (𝜑 → ;11 ≤ (2 logb ;;;2187)) |
| 95 | 5 | recnd 11289 |
. . . . . . 7
⊢ (𝜑 → ;11 ∈ ℂ) |
| 96 | 7 | recnd 11289 |
. . . . . . 7
⊢ (𝜑 → 7 ∈
ℂ) |
| 97 | 95, 96, 12 | divcan1d 12044 |
. . . . . 6
⊢ (𝜑 → ((;11 / 7) · 7) = ;11) |
| 98 | 97 | eqcomd 2743 |
. . . . 5
⊢ (𝜑 → ;11 = ((;11 / 7) · 7)) |
| 99 | | 3exp7 42054 |
. . . . . . . . . 10
⊢
(3↑7) = ;;;2187 |
| 100 | 99 | eqcomi 2746 |
. . . . . . . . 9
⊢ ;;;2187 =
(3↑7) |
| 101 | 100 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ;;;2187 = (3↑7)) |
| 102 | 101 | oveq2d 7447 |
. . . . . . 7
⊢ (𝜑 → (2 logb ;;;2187)
= (2 logb (3↑7))) |
| 103 | 20, 22 | elrpd 13074 |
. . . . . . . 8
⊢ (𝜑 → 3 ∈
ℝ+) |
| 104 | 34 | nnzd 12640 |
. . . . . . . 8
⊢ (𝜑 → 7 ∈
ℤ) |
| 105 | 53, 29, 103, 104 | relogbzexpd 41976 |
. . . . . . 7
⊢ (𝜑 → (2 logb
(3↑7)) = (7 · (2 logb 3))) |
| 106 | 102, 105 | eqtrd 2777 |
. . . . . 6
⊢ (𝜑 → (2 logb ;;;2187)
= (7 · (2 logb 3))) |
| 107 | 48 | recnd 11289 |
. . . . . . 7
⊢ (𝜑 → (2 logb 3)
∈ ℂ) |
| 108 | 96, 107 | mulcomd 11282 |
. . . . . 6
⊢ (𝜑 → (7 · (2
logb 3)) = ((2 logb 3) · 7)) |
| 109 | 106, 108 | eqtrd 2777 |
. . . . 5
⊢ (𝜑 → (2 logb ;;;2187)
= ((2 logb 3) · 7)) |
| 110 | 94, 98, 109 | 3brtr3d 5174 |
. . . 4
⊢ (𝜑 → ((;11 / 7) · 7) ≤ ((2 logb 3)
· 7)) |
| 111 | 13, 48, 35 | lemul1d 13120 |
. . . 4
⊢ (𝜑 → ((;11 / 7) ≤ (2 logb 3) ↔ ((;11 / 7) · 7) ≤ ((2
logb 3) · 7))) |
| 112 | 110, 111 | mpbird 257 |
. . 3
⊢ (𝜑 → (;11 / 7) ≤ (2 logb 3)) |
| 113 | 59, 60, 20, 22, 18, 24, 23 | logblebd 41977 |
. . 3
⊢ (𝜑 → (2 logb 3) ≤
(2 logb 𝑋)) |
| 114 | 13, 48, 30, 112, 113 | letrd 11418 |
. 2
⊢ (𝜑 → (;11 / 7) ≤ (2 logb 𝑋)) |
| 115 | 13, 30, 32, 47, 114 | leexp1ad 41973 |
1
⊢ (𝜑 → ((;11 / 7)↑5) ≤ ((2 logb 𝑋)↑5)) |