Step |
Hyp |
Ref |
Expression |
1 |
|
elznn0 |
⊢ ( 𝐶 ∈ ℤ ↔ ( 𝐶 ∈ ℝ ∧ ( 𝐶 ∈ ℕ0 ∨ - 𝐶 ∈ ℕ0 ) ) ) |
2 |
|
cxpmul2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℕ0 ) → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) ) |
3 |
2
|
3expia |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐶 ∈ ℕ0 → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) ) ) |
4 |
3
|
ad4ant13 |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ 𝐶 ∈ ℝ ) → ( 𝐶 ∈ ℕ0 → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) ) ) |
5 |
|
simplll |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → 𝐴 ∈ ℂ ) |
6 |
|
simplr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → 𝐵 ∈ ℂ ) |
7 |
|
simprr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → - 𝐶 ∈ ℕ0 ) |
8 |
|
cxpmul2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ - 𝐶 ∈ ℕ0 ) → ( 𝐴 ↑𝑐 ( 𝐵 · - 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ - 𝐶 ) ) |
9 |
5 6 7 8
|
syl3anc |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → ( 𝐴 ↑𝑐 ( 𝐵 · - 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ - 𝐶 ) ) |
10 |
9
|
oveq2d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → ( 1 / ( 𝐴 ↑𝑐 ( 𝐵 · - 𝐶 ) ) ) = ( 1 / ( ( 𝐴 ↑𝑐 𝐵 ) ↑ - 𝐶 ) ) ) |
11 |
|
simprl |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → 𝐶 ∈ ℝ ) |
12 |
11
|
recnd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → 𝐶 ∈ ℂ ) |
13 |
6 12
|
mulneg2d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → ( 𝐵 · - 𝐶 ) = - ( 𝐵 · 𝐶 ) ) |
14 |
13
|
negeqd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → - ( 𝐵 · - 𝐶 ) = - - ( 𝐵 · 𝐶 ) ) |
15 |
6 12
|
mulcld |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → ( 𝐵 · 𝐶 ) ∈ ℂ ) |
16 |
15
|
negnegd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → - - ( 𝐵 · 𝐶 ) = ( 𝐵 · 𝐶 ) ) |
17 |
14 16
|
eqtrd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → - ( 𝐵 · - 𝐶 ) = ( 𝐵 · 𝐶 ) ) |
18 |
17
|
oveq2d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → ( 𝐴 ↑𝑐 - ( 𝐵 · - 𝐶 ) ) = ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) ) |
19 |
|
simpllr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → 𝐴 ≠ 0 ) |
20 |
12
|
negcld |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → - 𝐶 ∈ ℂ ) |
21 |
6 20
|
mulcld |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → ( 𝐵 · - 𝐶 ) ∈ ℂ ) |
22 |
|
cxpneg |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ ( 𝐵 · - 𝐶 ) ∈ ℂ ) → ( 𝐴 ↑𝑐 - ( 𝐵 · - 𝐶 ) ) = ( 1 / ( 𝐴 ↑𝑐 ( 𝐵 · - 𝐶 ) ) ) ) |
23 |
5 19 21 22
|
syl3anc |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → ( 𝐴 ↑𝑐 - ( 𝐵 · - 𝐶 ) ) = ( 1 / ( 𝐴 ↑𝑐 ( 𝐵 · - 𝐶 ) ) ) ) |
24 |
18 23
|
eqtr3d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( 1 / ( 𝐴 ↑𝑐 ( 𝐵 · - 𝐶 ) ) ) ) |
25 |
|
cxpcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℂ ) |
26 |
25
|
ad4ant13 |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℂ ) |
27 |
|
expneg2 |
⊢ ( ( ( 𝐴 ↑𝑐 𝐵 ) ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ - 𝐶 ∈ ℕ0 ) → ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) = ( 1 / ( ( 𝐴 ↑𝑐 𝐵 ) ↑ - 𝐶 ) ) ) |
28 |
26 12 7 27
|
syl3anc |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) = ( 1 / ( ( 𝐴 ↑𝑐 𝐵 ) ↑ - 𝐶 ) ) ) |
29 |
10 24 28
|
3eqtr4d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℝ ∧ - 𝐶 ∈ ℕ0 ) ) → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) ) |
30 |
29
|
expr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ 𝐶 ∈ ℝ ) → ( - 𝐶 ∈ ℕ0 → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) ) ) |
31 |
4 30
|
jaod |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) ∧ 𝐶 ∈ ℝ ) → ( ( 𝐶 ∈ ℕ0 ∨ - 𝐶 ∈ ℕ0 ) → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) ) ) |
32 |
31
|
expimpd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) → ( ( 𝐶 ∈ ℝ ∧ ( 𝐶 ∈ ℕ0 ∨ - 𝐶 ∈ ℕ0 ) ) → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) ) ) |
33 |
1 32
|
syl5bi |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) → ( 𝐶 ∈ ℤ → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) ) ) |
34 |
33
|
impr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℤ ) ) → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) ) |