Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → 𝐴 ∈ ℂ ) |
2 |
|
simp3 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → 𝐵 ∈ ℂ ) |
3 |
|
cxpcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℂ ) |
4 |
1 2 3
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℂ ) |
5 |
2
|
negcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → - 𝐵 ∈ ℂ ) |
6 |
|
cxpcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ - 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 - 𝐵 ) ∈ ℂ ) |
7 |
1 5 6
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 - 𝐵 ) ∈ ℂ ) |
8 |
|
cxpne0 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) ≠ 0 ) |
9 |
2
|
negidd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐵 + - 𝐵 ) = 0 ) |
10 |
9
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 + - 𝐵 ) ) = ( 𝐴 ↑𝑐 0 ) ) |
11 |
|
simp2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → 𝐴 ≠ 0 ) |
12 |
|
cxpadd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ∧ - 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 + - 𝐵 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · ( 𝐴 ↑𝑐 - 𝐵 ) ) ) |
13 |
1 11 2 5 12
|
syl211anc |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 + - 𝐵 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · ( 𝐴 ↑𝑐 - 𝐵 ) ) ) |
14 |
|
cxp0 |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑𝑐 0 ) = 1 ) |
15 |
1 14
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 0 ) = 1 ) |
16 |
10 13 15
|
3eqtr3d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 ↑𝑐 𝐵 ) · ( 𝐴 ↑𝑐 - 𝐵 ) ) = 1 ) |
17 |
4 7 8 16
|
mvllmuld |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 - 𝐵 ) = ( 1 / ( 𝐴 ↑𝑐 𝐵 ) ) ) |