Step |
Hyp |
Ref |
Expression |
1 |
|
cxple2 |
⊢ ( ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐵 ≤ 𝐴 ↔ ( 𝐵 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐶 ) ) ) |
2 |
1
|
3com12 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐵 ≤ 𝐴 ↔ ( 𝐵 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐶 ) ) ) |
3 |
2
|
notbid |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( ¬ 𝐵 ≤ 𝐴 ↔ ¬ ( 𝐵 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐶 ) ) ) |
4 |
|
simp1l |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → 𝐴 ∈ ℝ ) |
5 |
|
simp2l |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → 𝐵 ∈ ℝ ) |
6 |
4 5
|
ltnled |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴 ) ) |
7 |
|
simp1r |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → 0 ≤ 𝐴 ) |
8 |
|
rpre |
⊢ ( 𝐶 ∈ ℝ+ → 𝐶 ∈ ℝ ) |
9 |
8
|
3ad2ant3 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → 𝐶 ∈ ℝ ) |
10 |
|
recxpcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ↑𝑐 𝐶 ) ∈ ℝ ) |
11 |
4 7 9 10
|
syl3anc |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 ↑𝑐 𝐶 ) ∈ ℝ ) |
12 |
|
simp2r |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → 0 ≤ 𝐵 ) |
13 |
|
recxpcl |
⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐶 ∈ ℝ ) → ( 𝐵 ↑𝑐 𝐶 ) ∈ ℝ ) |
14 |
5 12 9 13
|
syl3anc |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐵 ↑𝑐 𝐶 ) ∈ ℝ ) |
15 |
11 14
|
ltnled |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( ( 𝐴 ↑𝑐 𝐶 ) < ( 𝐵 ↑𝑐 𝐶 ) ↔ ¬ ( 𝐵 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐶 ) ) ) |
16 |
3 6 15
|
3bitr4d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 ↑𝑐 𝐶 ) < ( 𝐵 ↑𝑐 𝐶 ) ) ) |