This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem exple1

Description: A real between 0 and 1 inclusive raised to a nonnegative integer power 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 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑁 ) ≤ 1 )

Proof

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 )