Step |
Hyp |
Ref |
Expression |
1 |
|
efne0d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
3 |
|
oveq1 |
⊢ ( ( exp ‘ 𝐴 ) = 0 → ( ( exp ‘ 𝐴 ) · ( exp ‘ - 𝐴 ) ) = ( 0 · ( exp ‘ - 𝐴 ) ) ) |
4 |
|
efcan |
⊢ ( 𝐴 ∈ ℂ → ( ( exp ‘ 𝐴 ) · ( exp ‘ - 𝐴 ) ) = 1 ) |
5 |
1 4
|
syl |
⊢ ( 𝜑 → ( ( exp ‘ 𝐴 ) · ( exp ‘ - 𝐴 ) ) = 1 ) |
6 |
1
|
negcld |
⊢ ( 𝜑 → - 𝐴 ∈ ℂ ) |
7 |
6
|
efcld |
⊢ ( 𝜑 → ( exp ‘ - 𝐴 ) ∈ ℂ ) |
8 |
7
|
mul02d |
⊢ ( 𝜑 → ( 0 · ( exp ‘ - 𝐴 ) ) = 0 ) |
9 |
5 8
|
eqeq12d |
⊢ ( 𝜑 → ( ( ( exp ‘ 𝐴 ) · ( exp ‘ - 𝐴 ) ) = ( 0 · ( exp ‘ - 𝐴 ) ) ↔ 1 = 0 ) ) |
10 |
3 9
|
imbitrid |
⊢ ( 𝜑 → ( ( exp ‘ 𝐴 ) = 0 → 1 = 0 ) ) |
11 |
10
|
necon3d |
⊢ ( 𝜑 → ( 1 ≠ 0 → ( exp ‘ 𝐴 ) ≠ 0 ) ) |
12 |
2 11
|
mpi |
⊢ ( 𝜑 → ( exp ‘ 𝐴 ) ≠ 0 ) |