Step |
Hyp |
Ref |
Expression |
1 |
|
efhalfpi |
⊢ ( exp ‘ ( i · ( π / 2 ) ) ) = i |
2 |
|
ax-icn |
⊢ i ∈ ℂ |
3 |
|
ine0 |
⊢ i ≠ 0 |
4 |
|
halfpire |
⊢ ( π / 2 ) ∈ ℝ |
5 |
4
|
recni |
⊢ ( π / 2 ) ∈ ℂ |
6 |
2 5
|
mulcli |
⊢ ( i · ( π / 2 ) ) ∈ ℂ |
7 |
|
pipos |
⊢ 0 < π |
8 |
|
pire |
⊢ π ∈ ℝ |
9 |
|
lt0neg2 |
⊢ ( π ∈ ℝ → ( 0 < π ↔ - π < 0 ) ) |
10 |
8 9
|
ax-mp |
⊢ ( 0 < π ↔ - π < 0 ) |
11 |
7 10
|
mpbi |
⊢ - π < 0 |
12 |
|
halfpos2 |
⊢ ( π ∈ ℝ → ( 0 < π ↔ 0 < ( π / 2 ) ) ) |
13 |
8 12
|
ax-mp |
⊢ ( 0 < π ↔ 0 < ( π / 2 ) ) |
14 |
7 13
|
mpbi |
⊢ 0 < ( π / 2 ) |
15 |
8
|
renegcli |
⊢ - π ∈ ℝ |
16 |
|
0re |
⊢ 0 ∈ ℝ |
17 |
15 16 4
|
lttri |
⊢ ( ( - π < 0 ∧ 0 < ( π / 2 ) ) → - π < ( π / 2 ) ) |
18 |
11 14 17
|
mp2an |
⊢ - π < ( π / 2 ) |
19 |
|
reim |
⊢ ( ( π / 2 ) ∈ ℂ → ( ℜ ‘ ( π / 2 ) ) = ( ℑ ‘ ( i · ( π / 2 ) ) ) ) |
20 |
5 19
|
ax-mp |
⊢ ( ℜ ‘ ( π / 2 ) ) = ( ℑ ‘ ( i · ( π / 2 ) ) ) |
21 |
|
rere |
⊢ ( ( π / 2 ) ∈ ℝ → ( ℜ ‘ ( π / 2 ) ) = ( π / 2 ) ) |
22 |
4 21
|
ax-mp |
⊢ ( ℜ ‘ ( π / 2 ) ) = ( π / 2 ) |
23 |
20 22
|
eqtr3i |
⊢ ( ℑ ‘ ( i · ( π / 2 ) ) ) = ( π / 2 ) |
24 |
18 23
|
breqtrri |
⊢ - π < ( ℑ ‘ ( i · ( π / 2 ) ) ) |
25 |
8
|
a1i |
⊢ ( ⊤ → π ∈ ℝ ) |
26 |
25 25
|
ltaddposd |
⊢ ( ⊤ → ( 0 < π ↔ π < ( π + π ) ) ) |
27 |
7 26
|
mpbii |
⊢ ( ⊤ → π < ( π + π ) ) |
28 |
|
picn |
⊢ π ∈ ℂ |
29 |
28
|
times2i |
⊢ ( π · 2 ) = ( π + π ) |
30 |
27 29
|
breqtrrdi |
⊢ ( ⊤ → π < ( π · 2 ) ) |
31 |
|
2rp |
⊢ 2 ∈ ℝ+ |
32 |
31
|
a1i |
⊢ ( ⊤ → 2 ∈ ℝ+ ) |
33 |
25 25 32
|
ltdivmul2d |
⊢ ( ⊤ → ( ( π / 2 ) < π ↔ π < ( π · 2 ) ) ) |
34 |
30 33
|
mpbird |
⊢ ( ⊤ → ( π / 2 ) < π ) |
35 |
34
|
mptru |
⊢ ( π / 2 ) < π |
36 |
4 8 35
|
ltleii |
⊢ ( π / 2 ) ≤ π |
37 |
23 36
|
eqbrtri |
⊢ ( ℑ ‘ ( i · ( π / 2 ) ) ) ≤ π |
38 |
|
ellogrn |
⊢ ( ( i · ( π / 2 ) ) ∈ ran log ↔ ( ( i · ( π / 2 ) ) ∈ ℂ ∧ - π < ( ℑ ‘ ( i · ( π / 2 ) ) ) ∧ ( ℑ ‘ ( i · ( π / 2 ) ) ) ≤ π ) ) |
39 |
6 24 37 38
|
mpbir3an |
⊢ ( i · ( π / 2 ) ) ∈ ran log |
40 |
|
logeftb |
⊢ ( ( i ∈ ℂ ∧ i ≠ 0 ∧ ( i · ( π / 2 ) ) ∈ ran log ) → ( ( log ‘ i ) = ( i · ( π / 2 ) ) ↔ ( exp ‘ ( i · ( π / 2 ) ) ) = i ) ) |
41 |
2 3 39 40
|
mp3an |
⊢ ( ( log ‘ i ) = ( i · ( π / 2 ) ) ↔ ( exp ‘ ( i · ( π / 2 ) ) ) = i ) |
42 |
1 41
|
mpbir |
⊢ ( log ‘ i ) = ( i · ( π / 2 ) ) |