Step |
Hyp |
Ref |
Expression |
1 |
|
simpl1 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → 𝐴 ∈ ℝ ) |
2 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
3 |
2
|
a1i |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → 0 ∈ ℕ0 ) |
4 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → 𝑁 ∈ ℕ0 ) |
5 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
6 |
4 5
|
eleqtrdi |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → 𝑁 ∈ ( ℤ≥ ‘ 0 ) ) |
7 |
|
simpl2 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → 0 ≤ 𝐴 ) |
8 |
|
simpl3 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → 𝐴 ≤ 1 ) |
9 |
|
leexp2r |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ∈ ℕ0 ∧ 𝑁 ∈ ( ℤ≥ ‘ 0 ) ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) → ( 𝐴 ↑ 𝑁 ) ≤ ( 𝐴 ↑ 0 ) ) |
10 |
1 3 6 7 8 9
|
syl32anc |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑁 ) ≤ ( 𝐴 ↑ 0 ) ) |
11 |
1
|
recnd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → 𝐴 ∈ ℂ ) |
12 |
|
exp0 |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 0 ) = 1 ) |
13 |
11 12
|
syl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ 0 ) = 1 ) |
14 |
10 13
|
breqtrd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑁 ) ≤ 1 ) |