Proof of Theorem tanhlt1
Step | Hyp | Ref
| Expression |
1 | | ax-icn 10939 |
. . . . . . 7
⊢ i ∈
ℂ |
2 | | recn 10970 |
. . . . . . 7
⊢ (𝐴 ∈ ℝ → 𝐴 ∈
ℂ) |
3 | | mulcl 10964 |
. . . . . . 7
⊢ ((i
∈ ℂ ∧ 𝐴
∈ ℂ) → (i · 𝐴) ∈ ℂ) |
4 | 1, 2, 3 | sylancr 587 |
. . . . . 6
⊢ (𝐴 ∈ ℝ → (i
· 𝐴) ∈
ℂ) |
5 | | rpcoshcl 15875 |
. . . . . . 7
⊢ (𝐴 ∈ ℝ →
(cos‘(i · 𝐴))
∈ ℝ+) |
6 | 5 | rpne0d 12786 |
. . . . . 6
⊢ (𝐴 ∈ ℝ →
(cos‘(i · 𝐴))
≠ 0) |
7 | | tanval 15846 |
. . . . . 6
⊢ (((i
· 𝐴) ∈ ℂ
∧ (cos‘(i · 𝐴)) ≠ 0) → (tan‘(i ·
𝐴)) = ((sin‘(i
· 𝐴)) /
(cos‘(i · 𝐴)))) |
8 | 4, 6, 7 | syl2anc 584 |
. . . . 5
⊢ (𝐴 ∈ ℝ →
(tan‘(i · 𝐴))
= ((sin‘(i · 𝐴)) / (cos‘(i · 𝐴)))) |
9 | 8 | oveq1d 7299 |
. . . 4
⊢ (𝐴 ∈ ℝ →
((tan‘(i · 𝐴))
/ i) = (((sin‘(i · 𝐴)) / (cos‘(i · 𝐴))) / i)) |
10 | 4 | sincld 15848 |
. . . . 5
⊢ (𝐴 ∈ ℝ →
(sin‘(i · 𝐴))
∈ ℂ) |
11 | | recoshcl 15876 |
. . . . . 6
⊢ (𝐴 ∈ ℝ →
(cos‘(i · 𝐴))
∈ ℝ) |
12 | 11 | recnd 11012 |
. . . . 5
⊢ (𝐴 ∈ ℝ →
(cos‘(i · 𝐴))
∈ ℂ) |
13 | 1 | a1i 11 |
. . . . 5
⊢ (𝐴 ∈ ℝ → i ∈
ℂ) |
14 | | ine0 11419 |
. . . . . 6
⊢ i ≠
0 |
15 | 14 | a1i 11 |
. . . . 5
⊢ (𝐴 ∈ ℝ → i ≠
0) |
16 | 10, 12, 13, 6, 15 | divdiv32d 11785 |
. . . 4
⊢ (𝐴 ∈ ℝ →
(((sin‘(i · 𝐴)) / (cos‘(i · 𝐴))) / i) = (((sin‘(i
· 𝐴)) / i) /
(cos‘(i · 𝐴)))) |
17 | | sinhval 15872 |
. . . . . 6
⊢ (𝐴 ∈ ℂ →
((sin‘(i · 𝐴))
/ i) = (((exp‘𝐴)
− (exp‘-𝐴)) /
2)) |
18 | 2, 17 | syl 17 |
. . . . 5
⊢ (𝐴 ∈ ℝ →
((sin‘(i · 𝐴))
/ i) = (((exp‘𝐴)
− (exp‘-𝐴)) /
2)) |
19 | | coshval 15873 |
. . . . . 6
⊢ (𝐴 ∈ ℂ →
(cos‘(i · 𝐴))
= (((exp‘𝐴) +
(exp‘-𝐴)) /
2)) |
20 | 2, 19 | syl 17 |
. . . . 5
⊢ (𝐴 ∈ ℝ →
(cos‘(i · 𝐴))
= (((exp‘𝐴) +
(exp‘-𝐴)) /
2)) |
21 | 18, 20 | oveq12d 7302 |
. . . 4
⊢ (𝐴 ∈ ℝ →
(((sin‘(i · 𝐴)) / i) / (cos‘(i · 𝐴))) = ((((exp‘𝐴) − (exp‘-𝐴)) / 2) / (((exp‘𝐴) + (exp‘-𝐴)) / 2))) |
22 | 9, 16, 21 | 3eqtrd 2783 |
. . 3
⊢ (𝐴 ∈ ℝ →
((tan‘(i · 𝐴))
/ i) = ((((exp‘𝐴)
− (exp‘-𝐴)) /
2) / (((exp‘𝐴) +
(exp‘-𝐴)) /
2))) |
23 | | reefcl 15805 |
. . . . . 6
⊢ (𝐴 ∈ ℝ →
(exp‘𝐴) ∈
ℝ) |
24 | | renegcl 11293 |
. . . . . . 7
⊢ (𝐴 ∈ ℝ → -𝐴 ∈
ℝ) |
25 | 24 | reefcld 15806 |
. . . . . 6
⊢ (𝐴 ∈ ℝ →
(exp‘-𝐴) ∈
ℝ) |
26 | 23, 25 | resubcld 11412 |
. . . . 5
⊢ (𝐴 ∈ ℝ →
((exp‘𝐴) −
(exp‘-𝐴)) ∈
ℝ) |
27 | 26 | recnd 11012 |
. . . 4
⊢ (𝐴 ∈ ℝ →
((exp‘𝐴) −
(exp‘-𝐴)) ∈
ℂ) |
28 | 23, 25 | readdcld 11013 |
. . . . 5
⊢ (𝐴 ∈ ℝ →
((exp‘𝐴) +
(exp‘-𝐴)) ∈
ℝ) |
29 | 28 | recnd 11012 |
. . . 4
⊢ (𝐴 ∈ ℝ →
((exp‘𝐴) +
(exp‘-𝐴)) ∈
ℂ) |
30 | | 2cnd 12060 |
. . . 4
⊢ (𝐴 ∈ ℝ → 2 ∈
ℂ) |
31 | 20, 6 | eqnetrrd 3013 |
. . . . 5
⊢ (𝐴 ∈ ℝ →
(((exp‘𝐴) +
(exp‘-𝐴)) / 2) ≠
0) |
32 | | 2ne0 12086 |
. . . . . . 7
⊢ 2 ≠
0 |
33 | 32 | a1i 11 |
. . . . . 6
⊢ (𝐴 ∈ ℝ → 2 ≠
0) |
34 | 29, 30, 33 | divne0bd 11772 |
. . . . 5
⊢ (𝐴 ∈ ℝ →
(((exp‘𝐴) +
(exp‘-𝐴)) ≠ 0
↔ (((exp‘𝐴) +
(exp‘-𝐴)) / 2) ≠
0)) |
35 | 31, 34 | mpbird 256 |
. . . 4
⊢ (𝐴 ∈ ℝ →
((exp‘𝐴) +
(exp‘-𝐴)) ≠
0) |
36 | 27, 29, 30, 35, 33 | divcan7d 11788 |
. . 3
⊢ (𝐴 ∈ ℝ →
((((exp‘𝐴) −
(exp‘-𝐴)) / 2) /
(((exp‘𝐴) +
(exp‘-𝐴)) / 2)) =
(((exp‘𝐴) −
(exp‘-𝐴)) /
((exp‘𝐴) +
(exp‘-𝐴)))) |
37 | 22, 36 | eqtrd 2779 |
. 2
⊢ (𝐴 ∈ ℝ →
((tan‘(i · 𝐴))
/ i) = (((exp‘𝐴)
− (exp‘-𝐴)) /
((exp‘𝐴) +
(exp‘-𝐴)))) |
38 | 24 | rpefcld 15823 |
. . . . . 6
⊢ (𝐴 ∈ ℝ →
(exp‘-𝐴) ∈
ℝ+) |
39 | 23, 38 | ltsubrpd 12813 |
. . . . 5
⊢ (𝐴 ∈ ℝ →
((exp‘𝐴) −
(exp‘-𝐴)) <
(exp‘𝐴)) |
40 | 23, 38 | ltaddrpd 12814 |
. . . . 5
⊢ (𝐴 ∈ ℝ →
(exp‘𝐴) <
((exp‘𝐴) +
(exp‘-𝐴))) |
41 | 26, 23, 28, 39, 40 | lttrd 11145 |
. . . 4
⊢ (𝐴 ∈ ℝ →
((exp‘𝐴) −
(exp‘-𝐴)) <
((exp‘𝐴) +
(exp‘-𝐴))) |
42 | 29 | mulid1d 11001 |
. . . 4
⊢ (𝐴 ∈ ℝ →
(((exp‘𝐴) +
(exp‘-𝐴)) · 1)
= ((exp‘𝐴) +
(exp‘-𝐴))) |
43 | 41, 42 | breqtrrd 5103 |
. . 3
⊢ (𝐴 ∈ ℝ →
((exp‘𝐴) −
(exp‘-𝐴)) <
(((exp‘𝐴) +
(exp‘-𝐴)) ·
1)) |
44 | | 1red 10985 |
. . . 4
⊢ (𝐴 ∈ ℝ → 1 ∈
ℝ) |
45 | | efgt0 15821 |
. . . . 5
⊢ (𝐴 ∈ ℝ → 0 <
(exp‘𝐴)) |
46 | | efgt0 15821 |
. . . . . 6
⊢ (-𝐴 ∈ ℝ → 0 <
(exp‘-𝐴)) |
47 | 24, 46 | syl 17 |
. . . . 5
⊢ (𝐴 ∈ ℝ → 0 <
(exp‘-𝐴)) |
48 | 23, 25, 45, 47 | addgt0d 11559 |
. . . 4
⊢ (𝐴 ∈ ℝ → 0 <
((exp‘𝐴) +
(exp‘-𝐴))) |
49 | | ltdivmul 11859 |
. . . 4
⊢
((((exp‘𝐴)
− (exp‘-𝐴))
∈ ℝ ∧ 1 ∈ ℝ ∧ (((exp‘𝐴) + (exp‘-𝐴)) ∈ ℝ ∧ 0 <
((exp‘𝐴) +
(exp‘-𝐴)))) →
((((exp‘𝐴) −
(exp‘-𝐴)) /
((exp‘𝐴) +
(exp‘-𝐴))) < 1
↔ ((exp‘𝐴)
− (exp‘-𝐴))
< (((exp‘𝐴) +
(exp‘-𝐴)) ·
1))) |
50 | 26, 44, 28, 48, 49 | syl112anc 1373 |
. . 3
⊢ (𝐴 ∈ ℝ →
((((exp‘𝐴) −
(exp‘-𝐴)) /
((exp‘𝐴) +
(exp‘-𝐴))) < 1
↔ ((exp‘𝐴)
− (exp‘-𝐴))
< (((exp‘𝐴) +
(exp‘-𝐴)) ·
1))) |
51 | 43, 50 | mpbird 256 |
. 2
⊢ (𝐴 ∈ ℝ →
(((exp‘𝐴) −
(exp‘-𝐴)) /
((exp‘𝐴) +
(exp‘-𝐴))) <
1) |
52 | 37, 51 | eqbrtrd 5097 |
1
⊢ (𝐴 ∈ ℝ →
((tan‘(i · 𝐴))
/ i) < 1) |