Proof of Theorem 3lexlogpow5ineq2
Step | Hyp | Ref
| Expression |
1 | | 1nn0 11935 |
. . . . . 6
⊢ 1 ∈
ℕ0 |
2 | | 1nn 11670 |
. . . . . 6
⊢ 1 ∈
ℕ |
3 | 1, 2 | decnncl 12142 |
. . . . 5
⊢ ;11 ∈ ℕ |
4 | 3 | a1i 11 |
. . . 4
⊢ (𝜑 → ;11 ∈ ℕ) |
5 | 4 | nnred 11674 |
. . 3
⊢ (𝜑 → ;11 ∈ ℝ) |
6 | | 7re 11752 |
. . . 4
⊢ 7 ∈
ℝ |
7 | 6 | a1i 11 |
. . 3
⊢ (𝜑 → 7 ∈
ℝ) |
8 | | 0red 10667 |
. . . . 5
⊢ (𝜑 → 0 ∈
ℝ) |
9 | | 7pos 11770 |
. . . . . 6
⊢ 0 <
7 |
10 | 9 | a1i 11 |
. . . . 5
⊢ (𝜑 → 0 < 7) |
11 | 8, 10 | ltned 10799 |
. . . 4
⊢ (𝜑 → 0 ≠ 7) |
12 | 11 | necomd 3004 |
. . 3
⊢ (𝜑 → 7 ≠ 0) |
13 | 5, 7, 12 | redivcld 11491 |
. 2
⊢ (𝜑 → (;11 / 7) ∈ ℝ) |
14 | | 2re 11733 |
. . . 4
⊢ 2 ∈
ℝ |
15 | 14 | a1i 11 |
. . 3
⊢ (𝜑 → 2 ∈
ℝ) |
16 | | 2pos 11762 |
. . . 4
⊢ 0 <
2 |
17 | 16 | a1i 11 |
. . 3
⊢ (𝜑 → 0 < 2) |
18 | | 3lexlogpow5ineq2.1 |
. . 3
⊢ (𝜑 → 𝑋 ∈ ℝ) |
19 | | 3re 11739 |
. . . . 5
⊢ 3 ∈
ℝ |
20 | 19 | a1i 11 |
. . . 4
⊢ (𝜑 → 3 ∈
ℝ) |
21 | | 3pos 11764 |
. . . . 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 10823 |
. . 3
⊢ (𝜑 → 0 < 𝑋) |
25 | | 1red 10665 |
. . . . 5
⊢ (𝜑 → 1 ∈
ℝ) |
26 | | 1lt2 11830 |
. . . . . 6
⊢ 1 <
2 |
27 | 26 | a1i 11 |
. . . . 5
⊢ (𝜑 → 1 < 2) |
28 | 25, 27 | ltned 10799 |
. . . 4
⊢ (𝜑 → 1 ≠ 2) |
29 | 28 | necomd 3004 |
. . 3
⊢ (𝜑 → 2 ≠ 1) |
30 | 15, 17, 18, 24, 29 | relogbcld 39524 |
. 2
⊢ (𝜑 → (2 logb 𝑋) ∈
ℝ) |
31 | | 5nn0 11939 |
. . 3
⊢ 5 ∈
ℕ0 |
32 | 31 | a1i 11 |
. 2
⊢ (𝜑 → 5 ∈
ℕ0) |
33 | | 7nn 11751 |
. . . . 5
⊢ 7 ∈
ℕ |
34 | 33 | a1i 11 |
. . . 4
⊢ (𝜑 → 7 ∈
ℕ) |
35 | 34 | nnrpd 12455 |
. . 3
⊢ (𝜑 → 7 ∈
ℝ+) |
36 | | 0nn0 11934 |
. . . . 5
⊢ 0 ∈
ℕ0 |
37 | | tru 1543 |
. . . . . 6
⊢
⊤ |
38 | | 0red 10667 |
. . . . . . 7
⊢ (⊤
→ 0 ∈ ℝ) |
39 | | 9re 11758 |
. . . . . . . 8
⊢ 9 ∈
ℝ |
40 | 39 | a1i 11 |
. . . . . . 7
⊢ (⊤
→ 9 ∈ ℝ) |
41 | | 9pos 11772 |
. . . . . . . 8
⊢ 0 <
9 |
42 | 41 | a1i 11 |
. . . . . . 7
⊢ (⊤
→ 0 < 9) |
43 | 38, 40, 42 | ltled 10811 |
. . . . . 6
⊢ (⊤
→ 0 ≤ 9) |
44 | 37, 43 | ax-mp 5 |
. . . . 5
⊢ 0 ≤
9 |
45 | 2, 1, 36, 44 | declei 12158 |
. . . 4
⊢ 0 ≤
;11 |
46 | 45 | a1i 11 |
. . 3
⊢ (𝜑 → 0 ≤ ;11) |
47 | 5, 35, 46 | divge0d 12497 |
. 2
⊢ (𝜑 → 0 ≤ (;11 / 7)) |
48 | 15, 17, 20, 22, 29 | relogbcld 39524 |
. . 3
⊢ (𝜑 → (2 logb 3)
∈ ℝ) |
49 | | 2exp11 16466 |
. . . . . . . . . . 11
⊢
(2↑;11) = ;;;2048 |
50 | 49 | eqcomi 2768 |
. . . . . . . . . 10
⊢ ;;;2048 =
(2↑;11) |
51 | 50 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ;;;2048 = (2↑;11)) |
52 | 51 | oveq2d 7159 |
. . . . . . . 8
⊢ (𝜑 → (2 logb ;;;2048)
= (2 logb (2↑;11))) |
53 | 15, 17 | elrpd 12454 |
. . . . . . . . 9
⊢ (𝜑 → 2 ∈
ℝ+) |
54 | 4 | nnzd 12110 |
. . . . . . . . 9
⊢ (𝜑 → ;11 ∈ ℤ) |
55 | 53, 29, 54 | relogbexpd 39525 |
. . . . . . . 8
⊢ (𝜑 → (2 logb
(2↑;11)) = ;11) |
56 | 52, 55 | eqtrd 2794 |
. . . . . . 7
⊢ (𝜑 → (2 logb ;;;2048)
= ;11) |
57 | 56 | eqcomd 2765 |
. . . . . 6
⊢ (𝜑 → ;11 = (2 logb ;;;2048)) |
58 | | 2z 12038 |
. . . . . . . 8
⊢ 2 ∈
ℤ |
59 | 58 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 2 ∈
ℤ) |
60 | 15 | leidd 11229 |
. . . . . . 7
⊢ (𝜑 → 2 ≤ 2) |
61 | | 2nn0 11936 |
. . . . . . . . . . . 12
⊢ 2 ∈
ℕ0 |
62 | 61, 36 | deccl 12137 |
. . . . . . . . . . 11
⊢ ;20 ∈
ℕ0 |
63 | | 4nn0 11938 |
. . . . . . . . . . 11
⊢ 4 ∈
ℕ0 |
64 | 62, 63 | deccl 12137 |
. . . . . . . . . 10
⊢ ;;204 ∈ ℕ0 |
65 | | 8nn 11754 |
. . . . . . . . . 10
⊢ 8 ∈
ℕ |
66 | 64, 65 | decnncl 12142 |
. . . . . . . . 9
⊢ ;;;2048
∈ ℕ |
67 | 66 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ;;;2048 ∈ ℕ) |
68 | 67 | nnred 11674 |
. . . . . . 7
⊢ (𝜑 → ;;;2048 ∈ ℝ) |
69 | | 4nn 11742 |
. . . . . . . . . 10
⊢ 4 ∈
ℕ |
70 | 62, 69 | decnncl 12142 |
. . . . . . . . 9
⊢ ;;204 ∈ ℕ |
71 | | 8nn0 11942 |
. . . . . . . . 9
⊢ 8 ∈
ℕ0 |
72 | 70, 71, 36, 44 | decltdi 12161 |
. . . . . . . 8
⊢ 0 <
;;;2048 |
73 | 72 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 0 < ;;;2048) |
74 | 61, 1 | deccl 12137 |
. . . . . . . . . . 11
⊢ ;21 ∈
ℕ0 |
75 | 74, 71 | deccl 12137 |
. . . . . . . . . 10
⊢ ;;218 ∈ ℕ0 |
76 | 75, 33 | decnncl 12142 |
. . . . . . . . 9
⊢ ;;;2187
∈ ℕ |
77 | 76 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ;;;2187 ∈ ℕ) |
78 | 77 | nnred 11674 |
. . . . . . 7
⊢ (𝜑 → ;;;2187 ∈ ℝ) |
79 | 74, 65 | decnncl 12142 |
. . . . . . . . 9
⊢ ;;218 ∈ ℕ |
80 | | 7nn0 11941 |
. . . . . . . . 9
⊢ 7 ∈
ℕ0 |
81 | 79, 80, 36, 44 | decltdi 12161 |
. . . . . . . 8
⊢ 0 <
;;;2187 |
82 | 81 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 0 < ;;;2187) |
83 | | 8re 11755 |
. . . . . . . . . . 11
⊢ 8 ∈
ℝ |
84 | 83, 1 | nn0addge1i 11967 |
. . . . . . . . . 10
⊢ 8 ≤ (8
+ 1) |
85 | | 8p1e9 11809 |
. . . . . . . . . 10
⊢ (8 + 1) =
9 |
86 | 84, 85 | breqtri 5050 |
. . . . . . . . 9
⊢ 8 ≤
9 |
87 | | 4lt10 12258 |
. . . . . . . . . 10
⊢ 4 <
;10 |
88 | | 0lt1 11185 |
. . . . . . . . . . 11
⊢ 0 <
1 |
89 | 61, 36, 2, 88 | declt 12150 |
. . . . . . . . . 10
⊢ ;20 < ;21 |
90 | 62, 74, 63, 71, 87, 89 | decltc 12151 |
. . . . . . . . 9
⊢ ;;204 < ;;218 |
91 | 64, 75, 71, 80, 86, 90 | decleh 12157 |
. . . . . . . 8
⊢ ;;;2048
≤ ;;;2187 |
92 | 91 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → ;;;2048 ≤ ;;;2187) |
93 | 59, 60, 68, 73, 78, 82, 92 | logblebd 39527 |
. . . . . 6
⊢ (𝜑 → (2 logb ;;;2048)
≤ (2 logb ;;;2187)) |
94 | 57, 93 | eqbrtrd 5047 |
. . . . 5
⊢ (𝜑 → ;11 ≤ (2 logb ;;;2187)) |
95 | 5 | recnd 10692 |
. . . . . . 7
⊢ (𝜑 → ;11 ∈ ℂ) |
96 | 7 | recnd 10692 |
. . . . . . 7
⊢ (𝜑 → 7 ∈
ℂ) |
97 | 95, 96, 12 | divcan1d 11440 |
. . . . . 6
⊢ (𝜑 → ((;11 / 7) · 7) = ;11) |
98 | 97 | eqcomd 2765 |
. . . . 5
⊢ (𝜑 → ;11 = ((;11 / 7) · 7)) |
99 | | 3exp7 39605 |
. . . . . . . . . 10
⊢
(3↑7) = ;;;2187 |
100 | 99 | eqcomi 2768 |
. . . . . . . . 9
⊢ ;;;2187 =
(3↑7) |
101 | 100 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ;;;2187 = (3↑7)) |
102 | 101 | oveq2d 7159 |
. . . . . . 7
⊢ (𝜑 → (2 logb ;;;2187)
= (2 logb (3↑7))) |
103 | 20, 22 | elrpd 12454 |
. . . . . . . 8
⊢ (𝜑 → 3 ∈
ℝ+) |
104 | 34 | nnzd 12110 |
. . . . . . . 8
⊢ (𝜑 → 7 ∈
ℤ) |
105 | 53, 29, 103, 104 | relogbzexpd 39526 |
. . . . . . 7
⊢ (𝜑 → (2 logb
(3↑7)) = (7 · (2 logb 3))) |
106 | 102, 105 | eqtrd 2794 |
. . . . . 6
⊢ (𝜑 → (2 logb ;;;2187)
= (7 · (2 logb 3))) |
107 | 48 | recnd 10692 |
. . . . . . 7
⊢ (𝜑 → (2 logb 3)
∈ ℂ) |
108 | 96, 107 | mulcomd 10685 |
. . . . . 6
⊢ (𝜑 → (7 · (2
logb 3)) = ((2 logb 3) · 7)) |
109 | 106, 108 | eqtrd 2794 |
. . . . 5
⊢ (𝜑 → (2 logb ;;;2187)
= ((2 logb 3) · 7)) |
110 | 94, 98, 109 | 3brtr3d 5056 |
. . . 4
⊢ (𝜑 → ((;11 / 7) · 7) ≤ ((2 logb 3)
· 7)) |
111 | 13, 48, 35 | lemul1d 12500 |
. . . 4
⊢ (𝜑 → ((;11 / 7) ≤ (2 logb 3) ↔ ((;11 / 7) · 7) ≤ ((2
logb 3) · 7))) |
112 | 110, 111 | mpbird 260 |
. . 3
⊢ (𝜑 → (;11 / 7) ≤ (2 logb 3)) |
113 | 59, 60, 20, 22, 18, 24, 23 | logblebd 39527 |
. . 3
⊢ (𝜑 → (2 logb 3) ≤
(2 logb 𝑋)) |
114 | 13, 48, 30, 112, 113 | letrd 10820 |
. 2
⊢ (𝜑 → (;11 / 7) ≤ (2 logb 𝑋)) |
115 | 13, 30, 32, 47, 114 | leexp1ad 39523 |
1
⊢ (𝜑 → ((;11 / 7)↑5) ≤ ((2 logb 𝑋)↑5)) |