Step |
Hyp |
Ref |
Expression |
1 |
|
cxp112d.c |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
2 |
|
cxp112d.a |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
3 |
|
cxp112d.b |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
4 |
|
cxp112d.0 |
⊢ ( 𝜑 → 𝐶 ≠ 0 ) |
5 |
|
cxp112d.1 |
⊢ ( 𝜑 → 𝐶 ≠ 1 ) |
6 |
1 4 2
|
cxpefd |
⊢ ( 𝜑 → ( 𝐶 ↑𝑐 𝐴 ) = ( exp ‘ ( 𝐴 · ( log ‘ 𝐶 ) ) ) ) |
7 |
1 4 3
|
cxpefd |
⊢ ( 𝜑 → ( 𝐶 ↑𝑐 𝐵 ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐶 ) ) ) ) |
8 |
6 7
|
eqeq12d |
⊢ ( 𝜑 → ( ( 𝐶 ↑𝑐 𝐴 ) = ( 𝐶 ↑𝑐 𝐵 ) ↔ ( exp ‘ ( 𝐴 · ( log ‘ 𝐶 ) ) ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐶 ) ) ) ) ) |
9 |
1 4
|
logcld |
⊢ ( 𝜑 → ( log ‘ 𝐶 ) ∈ ℂ ) |
10 |
2 9
|
mulcld |
⊢ ( 𝜑 → ( 𝐴 · ( log ‘ 𝐶 ) ) ∈ ℂ ) |
11 |
3 9
|
mulcld |
⊢ ( 𝜑 → ( 𝐵 · ( log ‘ 𝐶 ) ) ∈ ℂ ) |
12 |
10 11
|
ef11d |
⊢ ( 𝜑 → ( ( exp ‘ ( 𝐴 · ( log ‘ 𝐶 ) ) ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐶 ) ) ) ↔ ∃ 𝑛 ∈ ℤ ( 𝐴 · ( log ‘ 𝐶 ) ) = ( ( 𝐵 · ( log ‘ 𝐶 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) ) ) |
13 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → 𝐴 ∈ ℂ ) |
14 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( log ‘ 𝐶 ) ∈ ℂ ) |
15 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( 𝐵 · ( log ‘ 𝐶 ) ) ∈ ℂ ) |
16 |
|
ax-icn |
⊢ i ∈ ℂ |
17 |
|
2cn |
⊢ 2 ∈ ℂ |
18 |
|
picn |
⊢ π ∈ ℂ |
19 |
17 18
|
mulcli |
⊢ ( 2 · π ) ∈ ℂ |
20 |
16 19
|
mulcli |
⊢ ( i · ( 2 · π ) ) ∈ ℂ |
21 |
20
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( i · ( 2 · π ) ) ∈ ℂ ) |
22 |
|
zcn |
⊢ ( 𝑛 ∈ ℤ → 𝑛 ∈ ℂ ) |
23 |
22
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → 𝑛 ∈ ℂ ) |
24 |
21 23
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( ( i · ( 2 · π ) ) · 𝑛 ) ∈ ℂ ) |
25 |
15 24
|
addcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( ( 𝐵 · ( log ‘ 𝐶 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) ∈ ℂ ) |
26 |
1 4 5
|
logccne0d |
⊢ ( 𝜑 → ( log ‘ 𝐶 ) ≠ 0 ) |
27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( log ‘ 𝐶 ) ≠ 0 ) |
28 |
13 14 25 27
|
ldiv |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( ( 𝐴 · ( log ‘ 𝐶 ) ) = ( ( 𝐵 · ( log ‘ 𝐶 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) ↔ 𝐴 = ( ( ( 𝐵 · ( log ‘ 𝐶 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) / ( log ‘ 𝐶 ) ) ) ) |
29 |
15 24 14 27
|
divdird |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( ( ( 𝐵 · ( log ‘ 𝐶 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) / ( log ‘ 𝐶 ) ) = ( ( ( 𝐵 · ( log ‘ 𝐶 ) ) / ( log ‘ 𝐶 ) ) + ( ( ( i · ( 2 · π ) ) · 𝑛 ) / ( log ‘ 𝐶 ) ) ) ) |
30 |
3 9 26
|
divcan4d |
⊢ ( 𝜑 → ( ( 𝐵 · ( log ‘ 𝐶 ) ) / ( log ‘ 𝐶 ) ) = 𝐵 ) |
31 |
30
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( ( 𝐵 · ( log ‘ 𝐶 ) ) / ( log ‘ 𝐶 ) ) = 𝐵 ) |
32 |
31
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( ( ( 𝐵 · ( log ‘ 𝐶 ) ) / ( log ‘ 𝐶 ) ) + ( ( ( i · ( 2 · π ) ) · 𝑛 ) / ( log ‘ 𝐶 ) ) ) = ( 𝐵 + ( ( ( i · ( 2 · π ) ) · 𝑛 ) / ( log ‘ 𝐶 ) ) ) ) |
33 |
29 32
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( ( ( 𝐵 · ( log ‘ 𝐶 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) / ( log ‘ 𝐶 ) ) = ( 𝐵 + ( ( ( i · ( 2 · π ) ) · 𝑛 ) / ( log ‘ 𝐶 ) ) ) ) |
34 |
33
|
eqeq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( 𝐴 = ( ( ( 𝐵 · ( log ‘ 𝐶 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) / ( log ‘ 𝐶 ) ) ↔ 𝐴 = ( 𝐵 + ( ( ( i · ( 2 · π ) ) · 𝑛 ) / ( log ‘ 𝐶 ) ) ) ) ) |
35 |
28 34
|
bitrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℤ ) → ( ( 𝐴 · ( log ‘ 𝐶 ) ) = ( ( 𝐵 · ( log ‘ 𝐶 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) ↔ 𝐴 = ( 𝐵 + ( ( ( i · ( 2 · π ) ) · 𝑛 ) / ( log ‘ 𝐶 ) ) ) ) ) |
36 |
35
|
rexbidva |
⊢ ( 𝜑 → ( ∃ 𝑛 ∈ ℤ ( 𝐴 · ( log ‘ 𝐶 ) ) = ( ( 𝐵 · ( log ‘ 𝐶 ) ) + ( ( i · ( 2 · π ) ) · 𝑛 ) ) ↔ ∃ 𝑛 ∈ ℤ 𝐴 = ( 𝐵 + ( ( ( i · ( 2 · π ) ) · 𝑛 ) / ( log ‘ 𝐶 ) ) ) ) ) |
37 |
8 12 36
|
3bitrd |
⊢ ( 𝜑 → ( ( 𝐶 ↑𝑐 𝐴 ) = ( 𝐶 ↑𝑐 𝐵 ) ↔ ∃ 𝑛 ∈ ℤ 𝐴 = ( 𝐵 + ( ( ( i · ( 2 · π ) ) · 𝑛 ) / ( log ‘ 𝐶 ) ) ) ) ) |