Step |
Hyp |
Ref |
Expression |
1 |
|
cxp111d.a |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
cxp111d.b |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
3 |
|
cxp111d.c |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
4 |
|
cxp111d.1 |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
5 |
|
cxp111d.2 |
⊢ ( 𝜑 → 𝐵 ≠ 0 ) |
6 |
|
cxp111d.3 |
⊢ ( 𝜑 → 𝐶 ≠ 0 ) |
7 |
1 4 3
|
cxpefd |
⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐶 ) = ( exp ‘ ( 𝐶 · ( log ‘ 𝐴 ) ) ) ) |
8 |
2 5 3
|
cxpefd |
⊢ ( 𝜑 → ( 𝐵 ↑𝑐 𝐶 ) = ( exp ‘ ( 𝐶 · ( log ‘ 𝐵 ) ) ) ) |
9 |
7 8
|
eqeq12d |
⊢ ( 𝜑 → ( ( 𝐴 ↑𝑐 𝐶 ) = ( 𝐵 ↑𝑐 𝐶 ) ↔ ( exp ‘ ( 𝐶 · ( log ‘ 𝐴 ) ) ) = ( exp ‘ ( 𝐶 · ( log ‘ 𝐵 ) ) ) ) ) |
10 |
1 4
|
logcld |
⊢ ( 𝜑 → ( log ‘ 𝐴 ) ∈ ℂ ) |
11 |
3 10
|
mulcld |
⊢ ( 𝜑 → ( 𝐶 · ( log ‘ 𝐴 ) ) ∈ ℂ ) |
12 |
2 5
|
logcld |
⊢ ( 𝜑 → ( log ‘ 𝐵 ) ∈ ℂ ) |
13 |
3 12
|
mulcld |
⊢ ( 𝜑 → ( 𝐶 · ( log ‘ 𝐵 ) ) ∈ ℂ ) |
14 |
11 13
|
ef11d |
⊢ ( 𝜑 → ( ( exp ‘ ( 𝐶 · ( log ‘ 𝐴 ) ) ) = ( exp ‘ ( 𝐶 · ( log ‘ 𝐵 ) ) ) ↔ ∃ 𝑛 ∈ ℤ ( 𝐶 · ( log ‘ 𝐴 ) ) = ( ( 𝐶 · ( log ‘ 𝐵 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) ) ) |
15 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( 𝐶 · ( log ‘ 𝐴 ) ) ∈ ℂ ) |
16 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( 𝐶 · ( log ‘ 𝐵 ) ) ∈ ℂ ) |
17 |
|
ax-icn |
⊢ i ∈ ℂ |
18 |
|
2cn |
⊢ 2 ∈ ℂ |
19 |
|
picn |
⊢ π ∈ ℂ |
20 |
18 19
|
mulcli |
⊢ ( 2 · π ) ∈ ℂ |
21 |
17 20
|
mulcli |
⊢ ( i · ( 2 · π ) ) ∈ ℂ |
22 |
21
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( i · ( 2 · π ) ) ∈ ℂ ) |
23 |
|
zcn |
⊢ ( 𝑛 ∈ ℤ → 𝑛 ∈ ℂ ) |
24 |
23
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → 𝑛 ∈ ℂ ) |
25 |
22 24
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( ( i · ( 2 · π ) ) · 𝑛 ) ∈ ℂ ) |
26 |
16 25
|
addcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( ( 𝐶 · ( log ‘ 𝐵 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) ∈ ℂ ) |
27 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → 𝐶 ∈ ℂ ) |
28 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → 𝐶 ≠ 0 ) |
29 |
|
div11 |
⊢ ( ( ( 𝐶 · ( log ‘ 𝐴 ) ) ∈ ℂ ∧ ( ( 𝐶 · ( log ‘ 𝐵 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( ( 𝐶 · ( log ‘ 𝐴 ) ) / 𝐶 ) = ( ( ( 𝐶 · ( log ‘ 𝐵 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) / 𝐶 ) ↔ ( 𝐶 · ( log ‘ 𝐴 ) ) = ( ( 𝐶 · ( log ‘ 𝐵 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) ) ) |
30 |
15 26 27 28 29
|
syl112anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( ( ( 𝐶 · ( log ‘ 𝐴 ) ) / 𝐶 ) = ( ( ( 𝐶 · ( log ‘ 𝐵 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) / 𝐶 ) ↔ ( 𝐶 · ( log ‘ 𝐴 ) ) = ( ( 𝐶 · ( log ‘ 𝐵 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) ) ) |
31 |
10 3 6
|
divcan3d |
⊢ ( 𝜑 → ( ( 𝐶 · ( log ‘ 𝐴 ) ) / 𝐶 ) = ( log ‘ 𝐴 ) ) |
32 |
31
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( ( 𝐶 · ( log ‘ 𝐴 ) ) / 𝐶 ) = ( log ‘ 𝐴 ) ) |
33 |
16 25 27 28
|
divdird |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( ( ( 𝐶 · ( log ‘ 𝐵 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) / 𝐶 ) = ( ( ( 𝐶 · ( log ‘ 𝐵 ) ) / 𝐶 ) + ( ( ( i · ( 2 · π ) ) · 𝑛 ) / 𝐶 ) ) ) |
34 |
12 3 6
|
divcan3d |
⊢ ( 𝜑 → ( ( 𝐶 · ( log ‘ 𝐵 ) ) / 𝐶 ) = ( log ‘ 𝐵 ) ) |
35 |
34
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( ( 𝐶 · ( log ‘ 𝐵 ) ) / 𝐶 ) = ( log ‘ 𝐵 ) ) |
36 |
35
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( ( ( 𝐶 · ( log ‘ 𝐵 ) ) / 𝐶 ) + ( ( ( i · ( 2 · π ) ) · 𝑛 ) / 𝐶 ) ) = ( ( log ‘ 𝐵 ) + ( ( ( i · ( 2 · π ) ) · 𝑛 ) / 𝐶 ) ) ) |
37 |
33 36
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( ( ( 𝐶 · ( log ‘ 𝐵 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) / 𝐶 ) = ( ( log ‘ 𝐵 ) + ( ( ( i · ( 2 · π ) ) · 𝑛 ) / 𝐶 ) ) ) |
38 |
32 37
|
eqeq12d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( ( ( 𝐶 · ( log ‘ 𝐴 ) ) / 𝐶 ) = ( ( ( 𝐶 · ( log ‘ 𝐵 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) / 𝐶 ) ↔ ( log ‘ 𝐴 ) = ( ( log ‘ 𝐵 ) + ( ( ( i · ( 2 · π ) ) · 𝑛 ) / 𝐶 ) ) ) ) |
39 |
30 38
|
bitr3d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( ( 𝐶 · ( log ‘ 𝐴 ) ) = ( ( 𝐶 · ( log ‘ 𝐵 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) ↔ ( log ‘ 𝐴 ) = ( ( log ‘ 𝐵 ) + ( ( ( i · ( 2 · π ) ) · 𝑛 ) / 𝐶 ) ) ) ) |
40 |
39
|
rexbidva |
⊢ ( 𝜑 → ( ∃ 𝑛 ∈ ℤ ( 𝐶 · ( log ‘ 𝐴 ) ) = ( ( 𝐶 · ( log ‘ 𝐵 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) ↔ ∃ 𝑛 ∈ ℤ ( log ‘ 𝐴 ) = ( ( log ‘ 𝐵 ) + ( ( ( i · ( 2 · π ) ) · 𝑛 ) / 𝐶 ) ) ) ) |
41 |
9 14 40
|
3bitrd |
⊢ ( 𝜑 → ( ( 𝐴 ↑𝑐 𝐶 ) = ( 𝐵 ↑𝑐 𝐶 ) ↔ ∃ 𝑛 ∈ ℤ ( log ‘ 𝐴 ) = ( ( log ‘ 𝐵 ) + ( ( ( i · ( 2 · π ) ) · 𝑛 ) / 𝐶 ) ) ) ) |