Proof of Theorem cos2bnd
| Step | Hyp | Ref
| Expression |
| 1 | | 7cn 12360 |
. . . . . 6
⊢ 7 ∈
ℂ |
| 2 | | 9cn 12366 |
. . . . . 6
⊢ 9 ∈
ℂ |
| 3 | | 9re 12365 |
. . . . . . 7
⊢ 9 ∈
ℝ |
| 4 | | 9pos 12379 |
. . . . . . 7
⊢ 0 <
9 |
| 5 | 3, 4 | gt0ne0ii 11799 |
. . . . . 6
⊢ 9 ≠
0 |
| 6 | | divneg 11959 |
. . . . . 6
⊢ ((7
∈ ℂ ∧ 9 ∈ ℂ ∧ 9 ≠ 0) → -(7 / 9) = (-7 /
9)) |
| 7 | 1, 2, 5, 6 | mp3an 1463 |
. . . . 5
⊢ -(7 / 9)
= (-7 / 9) |
| 8 | | 2cn 12341 |
. . . . . . 7
⊢ 2 ∈
ℂ |
| 9 | 2, 5 | pm3.2i 470 |
. . . . . . 7
⊢ (9 ∈
ℂ ∧ 9 ≠ 0) |
| 10 | | divsubdir 11961 |
. . . . . . 7
⊢ ((2
∈ ℂ ∧ 9 ∈ ℂ ∧ (9 ∈ ℂ ∧ 9 ≠ 0))
→ ((2 − 9) / 9) = ((2 / 9) − (9 / 9))) |
| 11 | 8, 2, 9, 10 | mp3an 1463 |
. . . . . 6
⊢ ((2
− 9) / 9) = ((2 / 9) − (9 / 9)) |
| 12 | 2, 8 | negsubdi2i 11595 |
. . . . . . . 8
⊢ -(9
− 2) = (2 − 9) |
| 13 | | 7p2e9 12427 |
. . . . . . . . . 10
⊢ (7 + 2) =
9 |
| 14 | 2, 8, 1 | subadd2i 11597 |
. . . . . . . . . 10
⊢ ((9
− 2) = 7 ↔ (7 + 2) = 9) |
| 15 | 13, 14 | mpbir 231 |
. . . . . . . . 9
⊢ (9
− 2) = 7 |
| 16 | 15 | negeqi 11501 |
. . . . . . . 8
⊢ -(9
− 2) = -7 |
| 17 | 12, 16 | eqtr3i 2767 |
. . . . . . 7
⊢ (2
− 9) = -7 |
| 18 | 17 | oveq1i 7441 |
. . . . . 6
⊢ ((2
− 9) / 9) = (-7 / 9) |
| 19 | 11, 18 | eqtr3i 2767 |
. . . . 5
⊢ ((2 / 9)
− (9 / 9)) = (-7 / 9) |
| 20 | 2, 5 | dividi 12000 |
. . . . . 6
⊢ (9 / 9) =
1 |
| 21 | 20 | oveq2i 7442 |
. . . . 5
⊢ ((2 / 9)
− (9 / 9)) = ((2 / 9) − 1) |
| 22 | 7, 19, 21 | 3eqtr2ri 2772 |
. . . 4
⊢ ((2 / 9)
− 1) = -(7 / 9) |
| 23 | | ax-1cn 11213 |
. . . . . . . 8
⊢ 1 ∈
ℂ |
| 24 | 8, 23, 2, 5 | divassi 12023 |
. . . . . . 7
⊢ ((2
· 1) / 9) = (2 · (1 / 9)) |
| 25 | | 2t1e2 12429 |
. . . . . . . 8
⊢ (2
· 1) = 2 |
| 26 | 25 | oveq1i 7441 |
. . . . . . 7
⊢ ((2
· 1) / 9) = (2 / 9) |
| 27 | 24, 26 | eqtr3i 2767 |
. . . . . 6
⊢ (2
· (1 / 9)) = (2 / 9) |
| 28 | | 3cn 12347 |
. . . . . . . . . 10
⊢ 3 ∈
ℂ |
| 29 | | 3ne0 12372 |
. . . . . . . . . 10
⊢ 3 ≠
0 |
| 30 | 23, 28, 29 | sqdivi 14224 |
. . . . . . . . 9
⊢ ((1 /
3)↑2) = ((1↑2) / (3↑2)) |
| 31 | | sq1 14234 |
. . . . . . . . . 10
⊢
(1↑2) = 1 |
| 32 | | sq3 14237 |
. . . . . . . . . 10
⊢
(3↑2) = 9 |
| 33 | 31, 32 | oveq12i 7443 |
. . . . . . . . 9
⊢
((1↑2) / (3↑2)) = (1 / 9) |
| 34 | 30, 33 | eqtri 2765 |
. . . . . . . 8
⊢ ((1 /
3)↑2) = (1 / 9) |
| 35 | | cos1bnd 16223 |
. . . . . . . . . 10
⊢ ((1 / 3)
< (cos‘1) ∧ (cos‘1) < (2 / 3)) |
| 36 | 35 | simpli 483 |
. . . . . . . . 9
⊢ (1 / 3)
< (cos‘1) |
| 37 | | 0le1 11786 |
. . . . . . . . . . 11
⊢ 0 ≤
1 |
| 38 | | 3pos 12371 |
. . . . . . . . . . 11
⊢ 0 <
3 |
| 39 | | 1re 11261 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℝ |
| 40 | | 3re 12346 |
. . . . . . . . . . . 12
⊢ 3 ∈
ℝ |
| 41 | 39, 40 | divge0i 12177 |
. . . . . . . . . . 11
⊢ ((0 ≤
1 ∧ 0 < 3) → 0 ≤ (1 / 3)) |
| 42 | 37, 38, 41 | mp2an 692 |
. . . . . . . . . 10
⊢ 0 ≤ (1
/ 3) |
| 43 | | 0re 11263 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ |
| 44 | | recoscl 16177 |
. . . . . . . . . . . 12
⊢ (1 ∈
ℝ → (cos‘1) ∈ ℝ) |
| 45 | 39, 44 | ax-mp 5 |
. . . . . . . . . . 11
⊢
(cos‘1) ∈ ℝ |
| 46 | 40, 29 | rereccli 12032 |
. . . . . . . . . . . . 13
⊢ (1 / 3)
∈ ℝ |
| 47 | 43, 46, 45 | lelttri 11388 |
. . . . . . . . . . . 12
⊢ ((0 ≤
(1 / 3) ∧ (1 / 3) < (cos‘1)) → 0 <
(cos‘1)) |
| 48 | 42, 36, 47 | mp2an 692 |
. . . . . . . . . . 11
⊢ 0 <
(cos‘1) |
| 49 | 43, 45, 48 | ltleii 11384 |
. . . . . . . . . 10
⊢ 0 ≤
(cos‘1) |
| 50 | 46, 45 | lt2sqi 14228 |
. . . . . . . . . 10
⊢ ((0 ≤
(1 / 3) ∧ 0 ≤ (cos‘1)) → ((1 / 3) < (cos‘1) ↔
((1 / 3)↑2) < ((cos‘1)↑2))) |
| 51 | 42, 49, 50 | mp2an 692 |
. . . . . . . . 9
⊢ ((1 / 3)
< (cos‘1) ↔ ((1 / 3)↑2) <
((cos‘1)↑2)) |
| 52 | 36, 51 | mpbi 230 |
. . . . . . . 8
⊢ ((1 /
3)↑2) < ((cos‘1)↑2) |
| 53 | 34, 52 | eqbrtrri 5166 |
. . . . . . 7
⊢ (1 / 9)
< ((cos‘1)↑2) |
| 54 | | 2pos 12369 |
. . . . . . . 8
⊢ 0 <
2 |
| 55 | 3, 5 | rereccli 12032 |
. . . . . . . . 9
⊢ (1 / 9)
∈ ℝ |
| 56 | 45 | resqcli 14225 |
. . . . . . . . 9
⊢
((cos‘1)↑2) ∈ ℝ |
| 57 | | 2re 12340 |
. . . . . . . . 9
⊢ 2 ∈
ℝ |
| 58 | 55, 56, 57 | ltmul2i 12189 |
. . . . . . . 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 230 |
. . . . . 6
⊢ (2
· (1 / 9)) < (2 · ((cos‘1)↑2)) |
| 61 | 27, 60 | eqbrtrri 5166 |
. . . . 5
⊢ (2 / 9)
< (2 · ((cos‘1)↑2)) |
| 62 | 57, 3, 5 | redivcli 12034 |
. . . . . 6
⊢ (2 / 9)
∈ ℝ |
| 63 | 57, 56 | remulcli 11277 |
. . . . . 6
⊢ (2
· ((cos‘1)↑2)) ∈ ℝ |
| 64 | | ltsub1 11759 |
. . . . . 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 1463 |
. . . . 5
⊢ ((2 / 9)
< (2 · ((cos‘1)↑2)) ↔ ((2 / 9) − 1) < ((2
· ((cos‘1)↑2)) − 1)) |
| 66 | 61, 65 | mpbi 230 |
. . . 4
⊢ ((2 / 9)
− 1) < ((2 · ((cos‘1)↑2)) −
1) |
| 67 | 22, 66 | eqbrtrri 5166 |
. . 3
⊢ -(7 / 9)
< ((2 · ((cos‘1)↑2)) − 1) |
| 68 | 25 | fveq2i 6909 |
. . . 4
⊢
(cos‘(2 · 1)) = (cos‘2) |
| 69 | | cos2t 16214 |
. . . . 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 2767 |
. . 3
⊢
(cos‘2) = ((2 · ((cos‘1)↑2)) −
1) |
| 72 | 67, 71 | breqtrri 5170 |
. 2
⊢ -(7 / 9)
< (cos‘2) |
| 73 | 35 | simpri 485 |
. . . . . . . . 9
⊢
(cos‘1) < (2 / 3) |
| 74 | | 0le2 12368 |
. . . . . . . . . . 11
⊢ 0 ≤
2 |
| 75 | 57, 40 | divge0i 12177 |
. . . . . . . . . . 11
⊢ ((0 ≤
2 ∧ 0 < 3) → 0 ≤ (2 / 3)) |
| 76 | 74, 38, 75 | mp2an 692 |
. . . . . . . . . 10
⊢ 0 ≤ (2
/ 3) |
| 77 | 57, 40, 29 | redivcli 12034 |
. . . . . . . . . . 11
⊢ (2 / 3)
∈ ℝ |
| 78 | 45, 77 | lt2sqi 14228 |
. . . . . . . . . 10
⊢ ((0 ≤
(cos‘1) ∧ 0 ≤ (2 / 3)) → ((cos‘1) < (2 / 3) ↔
((cos‘1)↑2) < ((2 / 3)↑2))) |
| 79 | 49, 76, 78 | mp2an 692 |
. . . . . . . . 9
⊢
((cos‘1) < (2 / 3) ↔ ((cos‘1)↑2) < ((2 /
3)↑2)) |
| 80 | 73, 79 | mpbi 230 |
. . . . . . . 8
⊢
((cos‘1)↑2) < ((2 / 3)↑2) |
| 81 | 8, 28, 29 | sqdivi 14224 |
. . . . . . . . 9
⊢ ((2 /
3)↑2) = ((2↑2) / (3↑2)) |
| 82 | | sq2 14236 |
. . . . . . . . . 10
⊢
(2↑2) = 4 |
| 83 | 82, 32 | oveq12i 7443 |
. . . . . . . . 9
⊢
((2↑2) / (3↑2)) = (4 / 9) |
| 84 | 81, 83 | eqtri 2765 |
. . . . . . . 8
⊢ ((2 /
3)↑2) = (4 / 9) |
| 85 | 80, 84 | breqtri 5168 |
. . . . . . 7
⊢
((cos‘1)↑2) < (4 / 9) |
| 86 | | 4re 12350 |
. . . . . . . . . 10
⊢ 4 ∈
ℝ |
| 87 | 86, 3, 5 | redivcli 12034 |
. . . . . . . . 9
⊢ (4 / 9)
∈ ℝ |
| 88 | 56, 87, 57 | ltmul2i 12189 |
. . . . . . . 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 230 |
. . . . . 6
⊢ (2
· ((cos‘1)↑2)) < (2 · (4 / 9)) |
| 91 | | 4cn 12351 |
. . . . . . . 8
⊢ 4 ∈
ℂ |
| 92 | 8, 91, 2, 5 | divassi 12023 |
. . . . . . 7
⊢ ((2
· 4) / 9) = (2 · (4 / 9)) |
| 93 | | 4t2e8 12434 |
. . . . . . . . 9
⊢ (4
· 2) = 8 |
| 94 | 91, 8, 93 | mulcomli 11270 |
. . . . . . . 8
⊢ (2
· 4) = 8 |
| 95 | 94 | oveq1i 7441 |
. . . . . . 7
⊢ ((2
· 4) / 9) = (8 / 9) |
| 96 | 92, 95 | eqtr3i 2767 |
. . . . . 6
⊢ (2
· (4 / 9)) = (8 / 9) |
| 97 | 90, 96 | breqtri 5168 |
. . . . 5
⊢ (2
· ((cos‘1)↑2)) < (8 / 9) |
| 98 | | 8re 12362 |
. . . . . . 7
⊢ 8 ∈
ℝ |
| 99 | 98, 3, 5 | redivcli 12034 |
. . . . . 6
⊢ (8 / 9)
∈ ℝ |
| 100 | | ltsub1 11759 |
. . . . . 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 1463 |
. . . . 5
⊢ ((2
· ((cos‘1)↑2)) < (8 / 9) ↔ ((2 ·
((cos‘1)↑2)) − 1) < ((8 / 9) − 1)) |
| 102 | 97, 101 | mpbi 230 |
. . . 4
⊢ ((2
· ((cos‘1)↑2)) − 1) < ((8 / 9) −
1) |
| 103 | 20 | oveq2i 7442 |
. . . . 5
⊢ ((8 / 9)
− (9 / 9)) = ((8 / 9) − 1) |
| 104 | | divneg 11959 |
. . . . . . 7
⊢ ((1
∈ ℂ ∧ 9 ∈ ℂ ∧ 9 ≠ 0) → -(1 / 9) = (-1 /
9)) |
| 105 | 23, 2, 5, 104 | mp3an 1463 |
. . . . . 6
⊢ -(1 / 9)
= (-1 / 9) |
| 106 | | 8cn 12363 |
. . . . . . . . 9
⊢ 8 ∈
ℂ |
| 107 | 2, 106 | negsubdi2i 11595 |
. . . . . . . 8
⊢ -(9
− 8) = (8 − 9) |
| 108 | | 8p1e9 12416 |
. . . . . . . . . 10
⊢ (8 + 1) =
9 |
| 109 | 2, 106, 23, 108 | subaddrii 11598 |
. . . . . . . . 9
⊢ (9
− 8) = 1 |
| 110 | 109 | negeqi 11501 |
. . . . . . . 8
⊢ -(9
− 8) = -1 |
| 111 | 107, 110 | eqtr3i 2767 |
. . . . . . 7
⊢ (8
− 9) = -1 |
| 112 | 111 | oveq1i 7441 |
. . . . . 6
⊢ ((8
− 9) / 9) = (-1 / 9) |
| 113 | | divsubdir 11961 |
. . . . . . 7
⊢ ((8
∈ ℂ ∧ 9 ∈ ℂ ∧ (9 ∈ ℂ ∧ 9 ≠ 0))
→ ((8 − 9) / 9) = ((8 / 9) − (9 / 9))) |
| 114 | 106, 2, 9, 113 | mp3an 1463 |
. . . . . 6
⊢ ((8
− 9) / 9) = ((8 / 9) − (9 / 9)) |
| 115 | 105, 112,
114 | 3eqtr2ri 2772 |
. . . . 5
⊢ ((8 / 9)
− (9 / 9)) = -(1 / 9) |
| 116 | 103, 115 | eqtr3i 2767 |
. . . 4
⊢ ((8 / 9)
− 1) = -(1 / 9) |
| 117 | 102, 116 | breqtri 5168 |
. . 3
⊢ ((2
· ((cos‘1)↑2)) − 1) < -(1 / 9) |
| 118 | 71, 117 | eqbrtri 5164 |
. 2
⊢
(cos‘2) < -(1 / 9) |
| 119 | 72, 118 | pm3.2i 470 |
1
⊢ (-(7 / 9)
< (cos‘2) ∧ (cos‘2) < -(1 / 9)) |