Metamath Proof Explorer
Description: ltmul1d for exponentiation of positive reals. (Contributed by Steven
Nguyen, 22-Aug-2023)
|
|
Ref |
Expression |
|
Hypotheses |
ltexp1d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ+ ) |
|
|
ltexp1d.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ+ ) |
|
|
ltexp1d.3 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
|
Assertion |
ltexp1d |
⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐴 ↑ 𝑁 ) < ( 𝐵 ↑ 𝑁 ) ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
ltexp1d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ+ ) |
2 |
|
ltexp1d.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ+ ) |
3 |
|
ltexp1d.3 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
4 |
|
rpexpmord |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 ↑ 𝑁 ) < ( 𝐵 ↑ 𝑁 ) ) ) |
5 |
3 1 2 4
|
syl3anc |
⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐴 ↑ 𝑁 ) < ( 𝐵 ↑ 𝑁 ) ) ) |