Metamath Proof Explorer
Description: Difference of exponents law for exponential function, deduction form.
(Contributed by SN, 25-Apr-2025)
|
|
Ref |
Expression |
|
Hypotheses |
efsubd.a |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
efsubd.b |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
|
Assertion |
efsubd |
⊢ ( 𝜑 → ( exp ‘ ( 𝐴 − 𝐵 ) ) = ( ( exp ‘ 𝐴 ) / ( exp ‘ 𝐵 ) ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
efsubd.a |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
efsubd.b |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
3 |
|
efsub |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( exp ‘ ( 𝐴 − 𝐵 ) ) = ( ( exp ‘ 𝐴 ) / ( exp ‘ 𝐵 ) ) ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( exp ‘ ( 𝐴 − 𝐵 ) ) = ( ( exp ‘ 𝐴 ) / ( exp ‘ 𝐵 ) ) ) |