Proof of Theorem cos2bnd
Step | Hyp | Ref
| Expression |
1 | | 7cn 8917 |
. . . . . 6
⊢ 7 ∈
ℂ |
2 | | 9cn 8921 |
. . . . . 6
⊢ 9 ∈
ℂ |
3 | | 9re 8920 |
. . . . . . 7
⊢ 9 ∈
ℝ |
4 | | 9pos 8937 |
. . . . . . 7
⊢ 0 <
9 |
5 | 3, 4 | gt0ap0ii 8503 |
. . . . . 6
⊢ 9 #
0 |
6 | | divnegap 8579 |
. . . . . 6
⊢ ((7
∈ ℂ ∧ 9 ∈ ℂ ∧ 9 # 0) → -(7 / 9) = (-7 /
9)) |
7 | 1, 2, 5, 6 | mp3an 1319 |
. . . . 5
⊢ -(7 / 9)
= (-7 / 9) |
8 | | 2cn 8904 |
. . . . . . 7
⊢ 2 ∈
ℂ |
9 | 2, 5 | pm3.2i 270 |
. . . . . . 7
⊢ (9 ∈
ℂ ∧ 9 # 0) |
10 | | divsubdirap 8581 |
. . . . . . 7
⊢ ((2
∈ ℂ ∧ 9 ∈ ℂ ∧ (9 ∈ ℂ ∧ 9 # 0))
→ ((2 − 9) / 9) = ((2 / 9) − (9 / 9))) |
11 | 8, 2, 9, 10 | mp3an 1319 |
. . . . . 6
⊢ ((2
− 9) / 9) = ((2 / 9) − (9 / 9)) |
12 | 2, 8 | negsubdi2i 8161 |
. . . . . . . 8
⊢ -(9
− 2) = (2 − 9) |
13 | | 7p2e9 8984 |
. . . . . . . . . 10
⊢ (7 + 2) =
9 |
14 | 2, 8, 1 | subadd2i 8163 |
. . . . . . . . . 10
⊢ ((9
− 2) = 7 ↔ (7 + 2) = 9) |
15 | 13, 14 | mpbir 145 |
. . . . . . . . 9
⊢ (9
− 2) = 7 |
16 | 15 | negeqi 8069 |
. . . . . . . 8
⊢ -(9
− 2) = -7 |
17 | 12, 16 | eqtr3i 2180 |
. . . . . . 7
⊢ (2
− 9) = -7 |
18 | 17 | oveq1i 5834 |
. . . . . 6
⊢ ((2
− 9) / 9) = (-7 / 9) |
19 | 11, 18 | eqtr3i 2180 |
. . . . 5
⊢ ((2 / 9)
− (9 / 9)) = (-7 / 9) |
20 | 2, 5 | dividapi 8618 |
. . . . . 6
⊢ (9 / 9) =
1 |
21 | 20 | oveq2i 5835 |
. . . . 5
⊢ ((2 / 9)
− (9 / 9)) = ((2 / 9) − 1) |
22 | 7, 19, 21 | 3eqtr2ri 2185 |
. . . 4
⊢ ((2 / 9)
− 1) = -(7 / 9) |
23 | | ax-1cn 7825 |
. . . . . . . 8
⊢ 1 ∈
ℂ |
24 | 8, 23, 2, 5 | divassapi 8641 |
. . . . . . 7
⊢ ((2
· 1) / 9) = (2 · (1 / 9)) |
25 | | 2t1e2 8986 |
. . . . . . . 8
⊢ (2
· 1) = 2 |
26 | 25 | oveq1i 5834 |
. . . . . . 7
⊢ ((2
· 1) / 9) = (2 / 9) |
27 | 24, 26 | eqtr3i 2180 |
. . . . . 6
⊢ (2
· (1 / 9)) = (2 / 9) |
28 | | 3cn 8908 |
. . . . . . . . . 10
⊢ 3 ∈
ℂ |
29 | | 3ap0 8929 |
. . . . . . . . . 10
⊢ 3 #
0 |
30 | 23, 28, 29 | sqdivapi 10502 |
. . . . . . . . 9
⊢ ((1 /
3)↑2) = ((1↑2) / (3↑2)) |
31 | | sq1 10512 |
. . . . . . . . . 10
⊢
(1↑2) = 1 |
32 | | sq3 10515 |
. . . . . . . . . 10
⊢
(3↑2) = 9 |
33 | 31, 32 | oveq12i 5836 |
. . . . . . . . 9
⊢
((1↑2) / (3↑2)) = (1 / 9) |
34 | 30, 33 | eqtri 2178 |
. . . . . . . 8
⊢ ((1 /
3)↑2) = (1 / 9) |
35 | | cos1bnd 11656 |
. . . . . . . . . 10
⊢ ((1 / 3)
< (cos‘1) ∧ (cos‘1) < (2 / 3)) |
36 | 35 | simpli 110 |
. . . . . . . . 9
⊢ (1 / 3)
< (cos‘1) |
37 | | 0le1 8356 |
. . . . . . . . . . 11
⊢ 0 ≤
1 |
38 | | 3pos 8927 |
. . . . . . . . . . 11
⊢ 0 <
3 |
39 | | 1re 7877 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℝ |
40 | | 3re 8907 |
. . . . . . . . . . . 12
⊢ 3 ∈
ℝ |
41 | 39, 40 | divge0i 8782 |
. . . . . . . . . . 11
⊢ ((0 ≤
1 ∧ 0 < 3) → 0 ≤ (1 / 3)) |
42 | 37, 38, 41 | mp2an 423 |
. . . . . . . . . 10
⊢ 0 ≤ (1
/ 3) |
43 | | 0re 7878 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ |
44 | | recoscl 11618 |
. . . . . . . . . . . 12
⊢ (1 ∈
ℝ → (cos‘1) ∈ ℝ) |
45 | 39, 44 | ax-mp 5 |
. . . . . . . . . . 11
⊢
(cos‘1) ∈ ℝ |
46 | 40, 29 | rerecclapi 8650 |
. . . . . . . . . . . . 13
⊢ (1 / 3)
∈ ℝ |
47 | 43, 46, 45 | lelttri 7982 |
. . . . . . . . . . . 12
⊢ ((0 ≤
(1 / 3) ∧ (1 / 3) < (cos‘1)) → 0 <
(cos‘1)) |
48 | 42, 36, 47 | mp2an 423 |
. . . . . . . . . . 11
⊢ 0 <
(cos‘1) |
49 | 43, 45, 48 | ltleii 7979 |
. . . . . . . . . 10
⊢ 0 ≤
(cos‘1) |
50 | 46, 45 | lt2sqi 10506 |
. . . . . . . . . 10
⊢ ((0 ≤
(1 / 3) ∧ 0 ≤ (cos‘1)) → ((1 / 3) < (cos‘1) ↔
((1 / 3)↑2) < ((cos‘1)↑2))) |
51 | 42, 49, 50 | mp2an 423 |
. . . . . . . . 9
⊢ ((1 / 3)
< (cos‘1) ↔ ((1 / 3)↑2) <
((cos‘1)↑2)) |
52 | 36, 51 | mpbi 144 |
. . . . . . . 8
⊢ ((1 /
3)↑2) < ((cos‘1)↑2) |
53 | 34, 52 | eqbrtrri 3987 |
. . . . . . 7
⊢ (1 / 9)
< ((cos‘1)↑2) |
54 | | 2pos 8924 |
. . . . . . . 8
⊢ 0 <
2 |
55 | 3, 5 | rerecclapi 8650 |
. . . . . . . . 9
⊢ (1 / 9)
∈ ℝ |
56 | 45 | resqcli 10503 |
. . . . . . . . 9
⊢
((cos‘1)↑2) ∈ ℝ |
57 | | 2re 8903 |
. . . . . . . . 9
⊢ 2 ∈
ℝ |
58 | 55, 56, 57 | ltmul2i 8794 |
. . . . . . . 8
⊢ (0 < 2
→ ((1 / 9) < ((cos‘1)↑2) ↔ (2 · (1 / 9)) < (2
· ((cos‘1)↑2)))) |
59 | 54, 58 | ax-mp 5 |
. . . . . . 7
⊢ ((1 / 9)
< ((cos‘1)↑2) ↔ (2 · (1 / 9)) < (2 ·
((cos‘1)↑2))) |
60 | 53, 59 | mpbi 144 |
. . . . . 6
⊢ (2
· (1 / 9)) < (2 · ((cos‘1)↑2)) |
61 | 27, 60 | eqbrtrri 3987 |
. . . . 5
⊢ (2 / 9)
< (2 · ((cos‘1)↑2)) |
62 | 57, 3, 5 | redivclapi 8652 |
. . . . . 6
⊢ (2 / 9)
∈ ℝ |
63 | 57, 56 | remulcli 7892 |
. . . . . 6
⊢ (2
· ((cos‘1)↑2)) ∈ ℝ |
64 | | ltsub1 8333 |
. . . . . 6
⊢ (((2 / 9)
∈ ℝ ∧ (2 · ((cos‘1)↑2)) ∈ ℝ ∧ 1
∈ ℝ) → ((2 / 9) < (2 · ((cos‘1)↑2)) ↔
((2 / 9) − 1) < ((2 · ((cos‘1)↑2)) −
1))) |
65 | 62, 63, 39, 64 | mp3an 1319 |
. . . . 5
⊢ ((2 / 9)
< (2 · ((cos‘1)↑2)) ↔ ((2 / 9) − 1) < ((2
· ((cos‘1)↑2)) − 1)) |
66 | 61, 65 | mpbi 144 |
. . . 4
⊢ ((2 / 9)
− 1) < ((2 · ((cos‘1)↑2)) −
1) |
67 | 22, 66 | eqbrtrri 3987 |
. . 3
⊢ -(7 / 9)
< ((2 · ((cos‘1)↑2)) − 1) |
68 | 25 | fveq2i 5471 |
. . . 4
⊢
(cos‘(2 · 1)) = (cos‘2) |
69 | | cos2t 11647 |
. . . . 5
⊢ (1 ∈
ℂ → (cos‘(2 · 1)) = ((2 ·
((cos‘1)↑2)) − 1)) |
70 | 23, 69 | ax-mp 5 |
. . . 4
⊢
(cos‘(2 · 1)) = ((2 · ((cos‘1)↑2))
− 1) |
71 | 68, 70 | eqtr3i 2180 |
. . 3
⊢
(cos‘2) = ((2 · ((cos‘1)↑2)) −
1) |
72 | 67, 71 | breqtrri 3991 |
. 2
⊢ -(7 / 9)
< (cos‘2) |
73 | 35 | simpri 112 |
. . . . . . . . 9
⊢
(cos‘1) < (2 / 3) |
74 | | 0le2 8923 |
. . . . . . . . . . 11
⊢ 0 ≤
2 |
75 | 57, 40 | divge0i 8782 |
. . . . . . . . . . 11
⊢ ((0 ≤
2 ∧ 0 < 3) → 0 ≤ (2 / 3)) |
76 | 74, 38, 75 | mp2an 423 |
. . . . . . . . . 10
⊢ 0 ≤ (2
/ 3) |
77 | 57, 40, 29 | redivclapi 8652 |
. . . . . . . . . . 11
⊢ (2 / 3)
∈ ℝ |
78 | 45, 77 | lt2sqi 10506 |
. . . . . . . . . 10
⊢ ((0 ≤
(cos‘1) ∧ 0 ≤ (2 / 3)) → ((cos‘1) < (2 / 3) ↔
((cos‘1)↑2) < ((2 / 3)↑2))) |
79 | 49, 76, 78 | mp2an 423 |
. . . . . . . . 9
⊢
((cos‘1) < (2 / 3) ↔ ((cos‘1)↑2) < ((2 /
3)↑2)) |
80 | 73, 79 | mpbi 144 |
. . . . . . . 8
⊢
((cos‘1)↑2) < ((2 / 3)↑2) |
81 | 8, 28, 29 | sqdivapi 10502 |
. . . . . . . . 9
⊢ ((2 /
3)↑2) = ((2↑2) / (3↑2)) |
82 | | sq2 10514 |
. . . . . . . . . 10
⊢
(2↑2) = 4 |
83 | 82, 32 | oveq12i 5836 |
. . . . . . . . 9
⊢
((2↑2) / (3↑2)) = (4 / 9) |
84 | 81, 83 | eqtri 2178 |
. . . . . . . 8
⊢ ((2 /
3)↑2) = (4 / 9) |
85 | 80, 84 | breqtri 3989 |
. . . . . . 7
⊢
((cos‘1)↑2) < (4 / 9) |
86 | | 4re 8910 |
. . . . . . . . . 10
⊢ 4 ∈
ℝ |
87 | 86, 3, 5 | redivclapi 8652 |
. . . . . . . . 9
⊢ (4 / 9)
∈ ℝ |
88 | 56, 87, 57 | ltmul2i 8794 |
. . . . . . . 8
⊢ (0 < 2
→ (((cos‘1)↑2) < (4 / 9) ↔ (2 ·
((cos‘1)↑2)) < (2 · (4 / 9)))) |
89 | 54, 88 | ax-mp 5 |
. . . . . . 7
⊢
(((cos‘1)↑2) < (4 / 9) ↔ (2 ·
((cos‘1)↑2)) < (2 · (4 / 9))) |
90 | 85, 89 | mpbi 144 |
. . . . . 6
⊢ (2
· ((cos‘1)↑2)) < (2 · (4 / 9)) |
91 | | 4cn 8911 |
. . . . . . . 8
⊢ 4 ∈
ℂ |
92 | 8, 91, 2, 5 | divassapi 8641 |
. . . . . . 7
⊢ ((2
· 4) / 9) = (2 · (4 / 9)) |
93 | | 4t2e8 8991 |
. . . . . . . . 9
⊢ (4
· 2) = 8 |
94 | 91, 8, 93 | mulcomli 7885 |
. . . . . . . 8
⊢ (2
· 4) = 8 |
95 | 94 | oveq1i 5834 |
. . . . . . 7
⊢ ((2
· 4) / 9) = (8 / 9) |
96 | 92, 95 | eqtr3i 2180 |
. . . . . 6
⊢ (2
· (4 / 9)) = (8 / 9) |
97 | 90, 96 | breqtri 3989 |
. . . . 5
⊢ (2
· ((cos‘1)↑2)) < (8 / 9) |
98 | | 8re 8918 |
. . . . . . 7
⊢ 8 ∈
ℝ |
99 | 98, 3, 5 | redivclapi 8652 |
. . . . . 6
⊢ (8 / 9)
∈ ℝ |
100 | | ltsub1 8333 |
. . . . . 6
⊢ (((2
· ((cos‘1)↑2)) ∈ ℝ ∧ (8 / 9) ∈ ℝ
∧ 1 ∈ ℝ) → ((2 · ((cos‘1)↑2)) < (8 /
9) ↔ ((2 · ((cos‘1)↑2)) − 1) < ((8 / 9) −
1))) |
101 | 63, 99, 39, 100 | mp3an 1319 |
. . . . 5
⊢ ((2
· ((cos‘1)↑2)) < (8 / 9) ↔ ((2 ·
((cos‘1)↑2)) − 1) < ((8 / 9) − 1)) |
102 | 97, 101 | mpbi 144 |
. . . 4
⊢ ((2
· ((cos‘1)↑2)) − 1) < ((8 / 9) −
1) |
103 | 20 | oveq2i 5835 |
. . . . 5
⊢ ((8 / 9)
− (9 / 9)) = ((8 / 9) − 1) |
104 | | divnegap 8579 |
. . . . . . 7
⊢ ((1
∈ ℂ ∧ 9 ∈ ℂ ∧ 9 # 0) → -(1 / 9) = (-1 /
9)) |
105 | 23, 2, 5, 104 | mp3an 1319 |
. . . . . 6
⊢ -(1 / 9)
= (-1 / 9) |
106 | | 8cn 8919 |
. . . . . . . . 9
⊢ 8 ∈
ℂ |
107 | 2, 106 | negsubdi2i 8161 |
. . . . . . . 8
⊢ -(9
− 8) = (8 − 9) |
108 | | 8p1e9 8973 |
. . . . . . . . . 10
⊢ (8 + 1) =
9 |
109 | 2, 106, 23, 108 | subaddrii 8164 |
. . . . . . . . 9
⊢ (9
− 8) = 1 |
110 | 109 | negeqi 8069 |
. . . . . . . 8
⊢ -(9
− 8) = -1 |
111 | 107, 110 | eqtr3i 2180 |
. . . . . . 7
⊢ (8
− 9) = -1 |
112 | 111 | oveq1i 5834 |
. . . . . 6
⊢ ((8
− 9) / 9) = (-1 / 9) |
113 | | divsubdirap 8581 |
. . . . . . 7
⊢ ((8
∈ ℂ ∧ 9 ∈ ℂ ∧ (9 ∈ ℂ ∧ 9 # 0))
→ ((8 − 9) / 9) = ((8 / 9) − (9 / 9))) |
114 | 106, 2, 9, 113 | mp3an 1319 |
. . . . . 6
⊢ ((8
− 9) / 9) = ((8 / 9) − (9 / 9)) |
115 | 105, 112,
114 | 3eqtr2ri 2185 |
. . . . 5
⊢ ((8 / 9)
− (9 / 9)) = -(1 / 9) |
116 | 103, 115 | eqtr3i 2180 |
. . . 4
⊢ ((8 / 9)
− 1) = -(1 / 9) |
117 | 102, 116 | breqtri 3989 |
. . 3
⊢ ((2
· ((cos‘1)↑2)) − 1) < -(1 / 9) |
118 | 71, 117 | eqbrtri 3985 |
. 2
⊢
(cos‘2) < -(1 / 9) |
119 | 72, 118 | pm3.2i 270 |
1
⊢ (-(7 / 9)
< (cos‘2) ∧ (cos‘2) < -(1 / 9)) |