Metamath Proof Explorer


Theorem exple1

Description: A real between 0 and 1 inclusive raised to a nonnegative integer is less than or equal to 1. (Contributed by Paul Chapman, 29-Dec-2007) (Revised by Mario Carneiro, 5-Jun-2014)

Ref Expression
Assertion exple1 ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴𝑁 ) ≤ 1 )

Proof

Step Hyp Ref Expression
1 simpl1 ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → 𝐴 ∈ ℝ )
2 0nn0 0 ∈ ℕ0
3 2 a1i ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → 0 ∈ ℕ0 )
4 simpr ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → 𝑁 ∈ ℕ0 )
5 nn0uz 0 = ( ℤ ‘ 0 )
6 4 5 eleqtrdi ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → 𝑁 ∈ ( ℤ ‘ 0 ) )
7 simpl2 ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → 0 ≤ 𝐴 )
8 simpl3 ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → 𝐴 ≤ 1 )
9 leexp2r ( ( ( 𝐴 ∈ ℝ ∧ 0 ∈ ℕ0𝑁 ∈ ( ℤ ‘ 0 ) ) ∧ ( 0 ≤ 𝐴𝐴 ≤ 1 ) ) → ( 𝐴𝑁 ) ≤ ( 𝐴 ↑ 0 ) )
10 1 3 6 7 8 9 syl32anc ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴𝑁 ) ≤ ( 𝐴 ↑ 0 ) )
11 1 recnd ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → 𝐴 ∈ ℂ )
12 exp0 ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 0 ) = 1 )
13 11 12 syl ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ 0 ) = 1 )
14 10 13 breqtrd ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴𝑁 ) ≤ 1 )