Step |
Hyp |
Ref |
Expression |
1 |
|
elznn0nn |
⊢ ( 𝑁 ∈ ℤ ↔ ( 𝑁 ∈ ℕ0 ∨ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) ) |
2 |
|
absexp |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) → ( abs ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( abs ‘ 𝐴 ) ↑ 𝑁 ) ) |
3 |
2
|
ex |
⊢ ( 𝐴 ∈ ℂ → ( 𝑁 ∈ ℕ0 → ( abs ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( abs ‘ 𝐴 ) ↑ 𝑁 ) ) ) |
4 |
3
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 𝑁 ∈ ℕ0 → ( abs ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( abs ‘ 𝐴 ) ↑ 𝑁 ) ) ) |
5 |
|
1cnd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → 1 ∈ ℂ ) |
6 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → 𝐴 ∈ ℂ ) |
7 |
|
nnnn0 |
⊢ ( - 𝑁 ∈ ℕ → - 𝑁 ∈ ℕ0 ) |
8 |
7
|
ad2antll |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → - 𝑁 ∈ ℕ0 ) |
9 |
6 8
|
expcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → ( 𝐴 ↑ - 𝑁 ) ∈ ℂ ) |
10 |
|
simplr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → 𝐴 ≠ 0 ) |
11 |
|
nnz |
⊢ ( - 𝑁 ∈ ℕ → - 𝑁 ∈ ℤ ) |
12 |
11
|
ad2antll |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → - 𝑁 ∈ ℤ ) |
13 |
6 10 12
|
expne0d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → ( 𝐴 ↑ - 𝑁 ) ≠ 0 ) |
14 |
|
absdiv |
⊢ ( ( 1 ∈ ℂ ∧ ( 𝐴 ↑ - 𝑁 ) ∈ ℂ ∧ ( 𝐴 ↑ - 𝑁 ) ≠ 0 ) → ( abs ‘ ( 1 / ( 𝐴 ↑ - 𝑁 ) ) ) = ( ( abs ‘ 1 ) / ( abs ‘ ( 𝐴 ↑ - 𝑁 ) ) ) ) |
15 |
5 9 13 14
|
syl3anc |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → ( abs ‘ ( 1 / ( 𝐴 ↑ - 𝑁 ) ) ) = ( ( abs ‘ 1 ) / ( abs ‘ ( 𝐴 ↑ - 𝑁 ) ) ) ) |
16 |
|
abs1 |
⊢ ( abs ‘ 1 ) = 1 |
17 |
16
|
oveq1i |
⊢ ( ( abs ‘ 1 ) / ( abs ‘ ( 𝐴 ↑ - 𝑁 ) ) ) = ( 1 / ( abs ‘ ( 𝐴 ↑ - 𝑁 ) ) ) |
18 |
|
absexp |
⊢ ( ( 𝐴 ∈ ℂ ∧ - 𝑁 ∈ ℕ0 ) → ( abs ‘ ( 𝐴 ↑ - 𝑁 ) ) = ( ( abs ‘ 𝐴 ) ↑ - 𝑁 ) ) |
19 |
6 8 18
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → ( abs ‘ ( 𝐴 ↑ - 𝑁 ) ) = ( ( abs ‘ 𝐴 ) ↑ - 𝑁 ) ) |
20 |
19
|
oveq2d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → ( 1 / ( abs ‘ ( 𝐴 ↑ - 𝑁 ) ) ) = ( 1 / ( ( abs ‘ 𝐴 ) ↑ - 𝑁 ) ) ) |
21 |
17 20
|
eqtrid |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → ( ( abs ‘ 1 ) / ( abs ‘ ( 𝐴 ↑ - 𝑁 ) ) ) = ( 1 / ( ( abs ‘ 𝐴 ) ↑ - 𝑁 ) ) ) |
22 |
15 21
|
eqtrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → ( abs ‘ ( 1 / ( 𝐴 ↑ - 𝑁 ) ) ) = ( 1 / ( ( abs ‘ 𝐴 ) ↑ - 𝑁 ) ) ) |
23 |
|
simprl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → 𝑁 ∈ ℝ ) |
24 |
23
|
recnd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → 𝑁 ∈ ℂ ) |
25 |
|
expneg2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ - 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑁 ) = ( 1 / ( 𝐴 ↑ - 𝑁 ) ) ) |
26 |
6 24 8 25
|
syl3anc |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → ( 𝐴 ↑ 𝑁 ) = ( 1 / ( 𝐴 ↑ - 𝑁 ) ) ) |
27 |
26
|
fveq2d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → ( abs ‘ ( 𝐴 ↑ 𝑁 ) ) = ( abs ‘ ( 1 / ( 𝐴 ↑ - 𝑁 ) ) ) ) |
28 |
|
abscl |
⊢ ( 𝐴 ∈ ℂ → ( abs ‘ 𝐴 ) ∈ ℝ ) |
29 |
28
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → ( abs ‘ 𝐴 ) ∈ ℝ ) |
30 |
29
|
recnd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → ( abs ‘ 𝐴 ) ∈ ℂ ) |
31 |
|
expneg2 |
⊢ ( ( ( abs ‘ 𝐴 ) ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ - 𝑁 ∈ ℕ0 ) → ( ( abs ‘ 𝐴 ) ↑ 𝑁 ) = ( 1 / ( ( abs ‘ 𝐴 ) ↑ - 𝑁 ) ) ) |
32 |
30 24 8 31
|
syl3anc |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → ( ( abs ‘ 𝐴 ) ↑ 𝑁 ) = ( 1 / ( ( abs ‘ 𝐴 ) ↑ - 𝑁 ) ) ) |
33 |
22 27 32
|
3eqtr4d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → ( abs ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( abs ‘ 𝐴 ) ↑ 𝑁 ) ) |
34 |
33
|
ex |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) → ( abs ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( abs ‘ 𝐴 ) ↑ 𝑁 ) ) ) |
35 |
4 34
|
jaod |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( 𝑁 ∈ ℕ0 ∨ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) → ( abs ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( abs ‘ 𝐴 ) ↑ 𝑁 ) ) ) |
36 |
35
|
3impia |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ ( 𝑁 ∈ ℕ0 ∨ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ ) ) ) → ( abs ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( abs ‘ 𝐴 ) ↑ 𝑁 ) ) |
37 |
1 36
|
syl3an3b |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( abs ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( abs ‘ 𝐴 ) ↑ 𝑁 ) ) |