Step |
Hyp |
Ref |
Expression |
1 |
|
exp11nnd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ+ ) |
2 |
|
exp11nnd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ+ ) |
3 |
|
exp11nnd.3 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
4 |
|
exp11nnd.4 |
⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) = ( 𝐵 ↑ 𝑁 ) ) |
5 |
1
|
rpred |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
6 |
3
|
nnnn0d |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
7 |
5 6
|
reexpcld |
⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℝ ) |
8 |
2
|
rpred |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
9 |
8 6
|
reexpcld |
⊢ ( 𝜑 → ( 𝐵 ↑ 𝑁 ) ∈ ℝ ) |
10 |
7 9
|
lttri3d |
⊢ ( 𝜑 → ( ( 𝐴 ↑ 𝑁 ) = ( 𝐵 ↑ 𝑁 ) ↔ ( ¬ ( 𝐴 ↑ 𝑁 ) < ( 𝐵 ↑ 𝑁 ) ∧ ¬ ( 𝐵 ↑ 𝑁 ) < ( 𝐴 ↑ 𝑁 ) ) ) ) |
11 |
4 10
|
mpbid |
⊢ ( 𝜑 → ( ¬ ( 𝐴 ↑ 𝑁 ) < ( 𝐵 ↑ 𝑁 ) ∧ ¬ ( 𝐵 ↑ 𝑁 ) < ( 𝐴 ↑ 𝑁 ) ) ) |
12 |
1 2 3
|
ltexp1d |
⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐴 ↑ 𝑁 ) < ( 𝐵 ↑ 𝑁 ) ) ) |
13 |
12
|
notbid |
⊢ ( 𝜑 → ( ¬ 𝐴 < 𝐵 ↔ ¬ ( 𝐴 ↑ 𝑁 ) < ( 𝐵 ↑ 𝑁 ) ) ) |
14 |
2 1 3
|
ltexp1d |
⊢ ( 𝜑 → ( 𝐵 < 𝐴 ↔ ( 𝐵 ↑ 𝑁 ) < ( 𝐴 ↑ 𝑁 ) ) ) |
15 |
14
|
notbid |
⊢ ( 𝜑 → ( ¬ 𝐵 < 𝐴 ↔ ¬ ( 𝐵 ↑ 𝑁 ) < ( 𝐴 ↑ 𝑁 ) ) ) |
16 |
13 15
|
anbi12d |
⊢ ( 𝜑 → ( ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ↔ ( ¬ ( 𝐴 ↑ 𝑁 ) < ( 𝐵 ↑ 𝑁 ) ∧ ¬ ( 𝐵 ↑ 𝑁 ) < ( 𝐴 ↑ 𝑁 ) ) ) ) |
17 |
11 16
|
mpbird |
⊢ ( 𝜑 → ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ) |
18 |
5 8
|
lttri3d |
⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ) ) |
19 |
17 18
|
mpbird |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |