Step |
Hyp |
Ref |
Expression |
1 |
|
simp3 |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → 𝐶 ∈ ℂ ) |
2 |
|
simp2 |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → 𝐵 ∈ ℝ ) |
3 |
2
|
recnd |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → 𝐵 ∈ ℂ ) |
4 |
|
relogcl |
⊢ ( 𝐴 ∈ ℝ+ → ( log ‘ 𝐴 ) ∈ ℝ ) |
5 |
4
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( log ‘ 𝐴 ) ∈ ℝ ) |
6 |
5
|
recnd |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( log ‘ 𝐴 ) ∈ ℂ ) |
7 |
1 3 6
|
mulassd |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐶 · 𝐵 ) · ( log ‘ 𝐴 ) ) = ( 𝐶 · ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) |
8 |
3 1
|
mulcomd |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) ) |
9 |
8
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐵 · 𝐶 ) · ( log ‘ 𝐴 ) ) = ( ( 𝐶 · 𝐵 ) · ( log ‘ 𝐴 ) ) ) |
10 |
|
rpcn |
⊢ ( 𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ ) |
11 |
10
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → 𝐴 ∈ ℂ ) |
12 |
|
rpne0 |
⊢ ( 𝐴 ∈ ℝ+ → 𝐴 ≠ 0 ) |
13 |
12
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → 𝐴 ≠ 0 ) |
14 |
|
cxpef |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) |
15 |
11 13 3 14
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) |
16 |
15
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( log ‘ ( 𝐴 ↑𝑐 𝐵 ) ) = ( log ‘ ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) ) |
17 |
2 5
|
remulcld |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · ( log ‘ 𝐴 ) ) ∈ ℝ ) |
18 |
17
|
relogefd |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( log ‘ ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) = ( 𝐵 · ( log ‘ 𝐴 ) ) ) |
19 |
16 18
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( log ‘ ( 𝐴 ↑𝑐 𝐵 ) ) = ( 𝐵 · ( log ‘ 𝐴 ) ) ) |
20 |
19
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( 𝐶 · ( log ‘ ( 𝐴 ↑𝑐 𝐵 ) ) ) = ( 𝐶 · ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) |
21 |
7 9 20
|
3eqtr4d |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐵 · 𝐶 ) · ( log ‘ 𝐴 ) ) = ( 𝐶 · ( log ‘ ( 𝐴 ↑𝑐 𝐵 ) ) ) ) |
22 |
21
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( exp ‘ ( ( 𝐵 · 𝐶 ) · ( log ‘ 𝐴 ) ) ) = ( exp ‘ ( 𝐶 · ( log ‘ ( 𝐴 ↑𝑐 𝐵 ) ) ) ) ) |
23 |
3 1
|
mulcld |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · 𝐶 ) ∈ ℂ ) |
24 |
|
cxpef |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ ( 𝐵 · 𝐶 ) ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( exp ‘ ( ( 𝐵 · 𝐶 ) · ( log ‘ 𝐴 ) ) ) ) |
25 |
11 13 23 24
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( exp ‘ ( ( 𝐵 · 𝐶 ) · ( log ‘ 𝐴 ) ) ) ) |
26 |
|
cxpcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℂ ) |
27 |
11 3 26
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℂ ) |
28 |
|
cxpne0 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) ≠ 0 ) |
29 |
11 13 3 28
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) ≠ 0 ) |
30 |
|
cxpef |
⊢ ( ( ( 𝐴 ↑𝑐 𝐵 ) ∈ ℂ ∧ ( 𝐴 ↑𝑐 𝐵 ) ≠ 0 ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 ↑𝑐 𝐵 ) ↑𝑐 𝐶 ) = ( exp ‘ ( 𝐶 · ( log ‘ ( 𝐴 ↑𝑐 𝐵 ) ) ) ) ) |
31 |
27 29 1 30
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 ↑𝑐 𝐵 ) ↑𝑐 𝐶 ) = ( exp ‘ ( 𝐶 · ( log ‘ ( 𝐴 ↑𝑐 𝐵 ) ) ) ) ) |
32 |
22 25 31
|
3eqtr4d |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑𝑐 𝐶 ) ) |