Step |
Hyp |
Ref |
Expression |
1 |
|
tru |
⊢ ⊤ |
2 |
|
8lt9 |
⊢ 8 < 9 |
3 |
|
2z |
⊢ 2 ∈ ℤ |
4 |
|
uzid |
⊢ ( 2 ∈ ℤ → 2 ∈ ( ℤ≥ ‘ 2 ) ) |
5 |
3 4
|
ax-mp |
⊢ 2 ∈ ( ℤ≥ ‘ 2 ) |
6 |
|
8nn |
⊢ 8 ∈ ℕ |
7 |
|
nnrp |
⊢ ( 8 ∈ ℕ → 8 ∈ ℝ+ ) |
8 |
6 7
|
ax-mp |
⊢ 8 ∈ ℝ+ |
9 |
|
9nn |
⊢ 9 ∈ ℕ |
10 |
|
nnrp |
⊢ ( 9 ∈ ℕ → 9 ∈ ℝ+ ) |
11 |
9 10
|
ax-mp |
⊢ 9 ∈ ℝ+ |
12 |
5 8 11
|
3pm3.2i |
⊢ ( 2 ∈ ( ℤ≥ ‘ 2 ) ∧ 8 ∈ ℝ+ ∧ 9 ∈ ℝ+ ) |
13 |
|
logblt |
⊢ ( ( 2 ∈ ( ℤ≥ ‘ 2 ) ∧ 8 ∈ ℝ+ ∧ 9 ∈ ℝ+ ) → ( 8 < 9 ↔ ( 2 logb 8 ) < ( 2 logb 9 ) ) ) |
14 |
12 13
|
ax-mp |
⊢ ( 8 < 9 ↔ ( 2 logb 8 ) < ( 2 logb 9 ) ) |
15 |
2 14
|
mpbi |
⊢ ( 2 logb 8 ) < ( 2 logb 9 ) |
16 |
15
|
a1i |
⊢ ( ⊤ → ( 2 logb 8 ) < ( 2 logb 9 ) ) |
17 |
|
eqid |
⊢ 8 = 8 |
18 |
|
cu2 |
⊢ ( 2 ↑ 3 ) = 8 |
19 |
17 18
|
eqtr4i |
⊢ 8 = ( 2 ↑ 3 ) |
20 |
19
|
a1i |
⊢ ( ⊤ → 8 = ( 2 ↑ 3 ) ) |
21 |
20
|
oveq2d |
⊢ ( ⊤ → ( 2 logb 8 ) = ( 2 logb ( 2 ↑ 3 ) ) ) |
22 |
|
2rp |
⊢ 2 ∈ ℝ+ |
23 |
22
|
a1i |
⊢ ( ⊤ → 2 ∈ ℝ+ ) |
24 |
|
1red |
⊢ ( ⊤ → 1 ∈ ℝ ) |
25 |
|
1lt2 |
⊢ 1 < 2 |
26 |
25
|
a1i |
⊢ ( ⊤ → 1 < 2 ) |
27 |
24 26
|
ltned |
⊢ ( ⊤ → 1 ≠ 2 ) |
28 |
27
|
necomd |
⊢ ( ⊤ → 2 ≠ 1 ) |
29 |
|
3z |
⊢ 3 ∈ ℤ |
30 |
29
|
a1i |
⊢ ( ⊤ → 3 ∈ ℤ ) |
31 |
23 28 30
|
relogbexpd |
⊢ ( ⊤ → ( 2 logb ( 2 ↑ 3 ) ) = 3 ) |
32 |
21 31
|
eqtrd |
⊢ ( ⊤ → ( 2 logb 8 ) = 3 ) |
33 |
|
eqid |
⊢ 9 = 9 |
34 |
|
sq3 |
⊢ ( 3 ↑ 2 ) = 9 |
35 |
33 34
|
eqtr4i |
⊢ 9 = ( 3 ↑ 2 ) |
36 |
35
|
a1i |
⊢ ( ⊤ → 9 = ( 3 ↑ 2 ) ) |
37 |
36
|
oveq2d |
⊢ ( ⊤ → ( 2 logb 9 ) = ( 2 logb ( 3 ↑ 2 ) ) ) |
38 |
16 32 37
|
3brtr3d |
⊢ ( ⊤ → 3 < ( 2 logb ( 3 ↑ 2 ) ) ) |
39 |
|
3re |
⊢ 3 ∈ ℝ |
40 |
39
|
a1i |
⊢ ( ⊤ → 3 ∈ ℝ ) |
41 |
40
|
recnd |
⊢ ( ⊤ → 3 ∈ ℂ ) |
42 |
|
2re |
⊢ 2 ∈ ℝ |
43 |
42
|
a1i |
⊢ ( ⊤ → 2 ∈ ℝ ) |
44 |
43
|
recnd |
⊢ ( ⊤ → 2 ∈ ℂ ) |
45 |
|
2pos |
⊢ 0 < 2 |
46 |
45
|
a1i |
⊢ ( ⊤ → 0 < 2 ) |
47 |
46
|
gt0ne0d |
⊢ ( ⊤ → 2 ≠ 0 ) |
48 |
41 44 47
|
divcan1d |
⊢ ( ⊤ → ( ( 3 / 2 ) · 2 ) = 3 ) |
49 |
48
|
eqcomd |
⊢ ( ⊤ → 3 = ( ( 3 / 2 ) · 2 ) ) |
50 |
|
3pos |
⊢ 0 < 3 |
51 |
50
|
a1i |
⊢ ( ⊤ → 0 < 3 ) |
52 |
40 51
|
elrpd |
⊢ ( ⊤ → 3 ∈ ℝ+ ) |
53 |
3
|
a1i |
⊢ ( ⊤ → 2 ∈ ℤ ) |
54 |
23 28 52 53
|
relogbzexpd |
⊢ ( ⊤ → ( 2 logb ( 3 ↑ 2 ) ) = ( 2 · ( 2 logb 3 ) ) ) |
55 |
43 46 40 51 28
|
relogbcld |
⊢ ( ⊤ → ( 2 logb 3 ) ∈ ℝ ) |
56 |
55
|
recnd |
⊢ ( ⊤ → ( 2 logb 3 ) ∈ ℂ ) |
57 |
44 56
|
mulcomd |
⊢ ( ⊤ → ( 2 · ( 2 logb 3 ) ) = ( ( 2 logb 3 ) · 2 ) ) |
58 |
54 57
|
eqtrd |
⊢ ( ⊤ → ( 2 logb ( 3 ↑ 2 ) ) = ( ( 2 logb 3 ) · 2 ) ) |
59 |
38 49 58
|
3brtr3d |
⊢ ( ⊤ → ( ( 3 / 2 ) · 2 ) < ( ( 2 logb 3 ) · 2 ) ) |
60 |
40
|
rehalfcld |
⊢ ( ⊤ → ( 3 / 2 ) ∈ ℝ ) |
61 |
60 55 23
|
ltmul1d |
⊢ ( ⊤ → ( ( 3 / 2 ) < ( 2 logb 3 ) ↔ ( ( 3 / 2 ) · 2 ) < ( ( 2 logb 3 ) · 2 ) ) ) |
62 |
59 61
|
mpbird |
⊢ ( ⊤ → ( 3 / 2 ) < ( 2 logb 3 ) ) |
63 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
64 |
|
3nn0 |
⊢ 3 ∈ ℕ0 |
65 |
|
7nn0 |
⊢ 7 ∈ ℕ0 |
66 |
|
7lt10 |
⊢ 7 < ; 1 0 |
67 |
|
2lt3 |
⊢ 2 < 3 |
68 |
63 64 65 63 66 67
|
decltc |
⊢ ; 2 7 < ; 3 2 |
69 |
|
7nn |
⊢ 7 ∈ ℕ |
70 |
63 69
|
decnncl |
⊢ ; 2 7 ∈ ℕ |
71 |
|
nnrp |
⊢ ( ; 2 7 ∈ ℕ → ; 2 7 ∈ ℝ+ ) |
72 |
70 71
|
ax-mp |
⊢ ; 2 7 ∈ ℝ+ |
73 |
|
2nn |
⊢ 2 ∈ ℕ |
74 |
64 73
|
decnncl |
⊢ ; 3 2 ∈ ℕ |
75 |
|
nnrp |
⊢ ( ; 3 2 ∈ ℕ → ; 3 2 ∈ ℝ+ ) |
76 |
74 75
|
ax-mp |
⊢ ; 3 2 ∈ ℝ+ |
77 |
5 72 76
|
3pm3.2i |
⊢ ( 2 ∈ ( ℤ≥ ‘ 2 ) ∧ ; 2 7 ∈ ℝ+ ∧ ; 3 2 ∈ ℝ+ ) |
78 |
|
logblt |
⊢ ( ( 2 ∈ ( ℤ≥ ‘ 2 ) ∧ ; 2 7 ∈ ℝ+ ∧ ; 3 2 ∈ ℝ+ ) → ( ; 2 7 < ; 3 2 ↔ ( 2 logb ; 2 7 ) < ( 2 logb ; 3 2 ) ) ) |
79 |
77 78
|
ax-mp |
⊢ ( ; 2 7 < ; 3 2 ↔ ( 2 logb ; 2 7 ) < ( 2 logb ; 3 2 ) ) |
80 |
68 79
|
mpbi |
⊢ ( 2 logb ; 2 7 ) < ( 2 logb ; 3 2 ) |
81 |
80
|
a1i |
⊢ ( ⊤ → ( 2 logb ; 2 7 ) < ( 2 logb ; 3 2 ) ) |
82 |
|
eqid |
⊢ ; 3 2 = ; 3 2 |
83 |
|
2exp5 |
⊢ ( 2 ↑ 5 ) = ; 3 2 |
84 |
82 83
|
eqtr4i |
⊢ ; 3 2 = ( 2 ↑ 5 ) |
85 |
84
|
a1i |
⊢ ( ⊤ → ; 3 2 = ( 2 ↑ 5 ) ) |
86 |
85
|
oveq2d |
⊢ ( ⊤ → ( 2 logb ; 3 2 ) = ( 2 logb ( 2 ↑ 5 ) ) ) |
87 |
81 86
|
breqtrd |
⊢ ( ⊤ → ( 2 logb ; 2 7 ) < ( 2 logb ( 2 ↑ 5 ) ) ) |
88 |
|
eqid |
⊢ ; 2 7 = ; 2 7 |
89 |
|
3exp3 |
⊢ ( 3 ↑ 3 ) = ; 2 7 |
90 |
88 89
|
eqtr4i |
⊢ ; 2 7 = ( 3 ↑ 3 ) |
91 |
90
|
a1i |
⊢ ( ⊤ → ; 2 7 = ( 3 ↑ 3 ) ) |
92 |
91
|
oveq2d |
⊢ ( ⊤ → ( 2 logb ; 2 7 ) = ( 2 logb ( 3 ↑ 3 ) ) ) |
93 |
23 28 52 30
|
relogbzexpd |
⊢ ( ⊤ → ( 2 logb ( 3 ↑ 3 ) ) = ( 3 · ( 2 logb 3 ) ) ) |
94 |
92 93
|
eqtrd |
⊢ ( ⊤ → ( 2 logb ; 2 7 ) = ( 3 · ( 2 logb 3 ) ) ) |
95 |
41 56
|
mulcomd |
⊢ ( ⊤ → ( 3 · ( 2 logb 3 ) ) = ( ( 2 logb 3 ) · 3 ) ) |
96 |
94 95
|
eqtrd |
⊢ ( ⊤ → ( 2 logb ; 2 7 ) = ( ( 2 logb 3 ) · 3 ) ) |
97 |
|
5re |
⊢ 5 ∈ ℝ |
98 |
97
|
a1i |
⊢ ( ⊤ → 5 ∈ ℝ ) |
99 |
98
|
recnd |
⊢ ( ⊤ → 5 ∈ ℂ ) |
100 |
51
|
gt0ne0d |
⊢ ( ⊤ → 3 ≠ 0 ) |
101 |
99 41 100
|
divcan1d |
⊢ ( ⊤ → ( ( 5 / 3 ) · 3 ) = 5 ) |
102 |
|
5nn |
⊢ 5 ∈ ℕ |
103 |
102
|
a1i |
⊢ ( ⊤ → 5 ∈ ℕ ) |
104 |
103
|
nnzd |
⊢ ( ⊤ → 5 ∈ ℤ ) |
105 |
23 28 104
|
relogbexpd |
⊢ ( ⊤ → ( 2 logb ( 2 ↑ 5 ) ) = 5 ) |
106 |
105
|
eqcomd |
⊢ ( ⊤ → 5 = ( 2 logb ( 2 ↑ 5 ) ) ) |
107 |
101 106
|
eqtrd |
⊢ ( ⊤ → ( ( 5 / 3 ) · 3 ) = ( 2 logb ( 2 ↑ 5 ) ) ) |
108 |
107
|
eqcomd |
⊢ ( ⊤ → ( 2 logb ( 2 ↑ 5 ) ) = ( ( 5 / 3 ) · 3 ) ) |
109 |
87 96 108
|
3brtr3d |
⊢ ( ⊤ → ( ( 2 logb 3 ) · 3 ) < ( ( 5 / 3 ) · 3 ) ) |
110 |
98 40 100
|
redivcld |
⊢ ( ⊤ → ( 5 / 3 ) ∈ ℝ ) |
111 |
55 110 52
|
ltmul1d |
⊢ ( ⊤ → ( ( 2 logb 3 ) < ( 5 / 3 ) ↔ ( ( 2 logb 3 ) · 3 ) < ( ( 5 / 3 ) · 3 ) ) ) |
112 |
109 111
|
mpbird |
⊢ ( ⊤ → ( 2 logb 3 ) < ( 5 / 3 ) ) |
113 |
62 112
|
jca |
⊢ ( ⊤ → ( ( 3 / 2 ) < ( 2 logb 3 ) ∧ ( 2 logb 3 ) < ( 5 / 3 ) ) ) |
114 |
1 113
|
ax-mp |
⊢ ( ( 3 / 2 ) < ( 2 logb 3 ) ∧ ( 2 logb 3 ) < ( 5 / 3 ) ) |