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Metamath Proof Explorer

Theorem expge0

Description: A nonnegative real raised to a nonnegative integer is nonnegative. (Contributed by NM, 16-Dec-2005) (Revised by Mario Carneiro, 4-Jun-2014)

Ref Expression
Assertion expge0 ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴 ) → 0 ≤ ( 𝐴 ↑ 𝑁 ) )

Proof

Step Hyp Ref Expression
1 breq2 ⊢ ( 𝑧 = 𝐴 → ( 0 ≤ 𝑧 ↔ 0 ≤ 𝐴 ) )
2 1 elrab ⊢ ( 𝐴 ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ↔ ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) )
3 ssrab2 ⊢ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ⊆ ℝ
4 ax-resscn ⊢ ℝ ⊆ ℂ
5 3 4 sstri ⊢ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ⊆ ℂ
6 breq2 ⊢ ( 𝑧 = 𝑥 → ( 0 ≤ 𝑧 ↔ 0 ≤ 𝑥 ) )
7 6 elrab ⊢ ( 𝑥 ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ↔ ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ) )
8 breq2 ⊢ ( 𝑧 = 𝑦 → ( 0 ≤ 𝑧 ↔ 0 ≤ 𝑦 ) )
9 8 elrab ⊢ ( 𝑦 ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ↔ ( 𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ) )
10 breq2 ⊢ ( 𝑧 = ( 𝑥 · 𝑦 ) → ( 0 ≤ 𝑧 ↔ 0 ≤ ( 𝑥 · 𝑦 ) ) )
11 remulcl ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 · 𝑦 ) ∈ ℝ )
12 11 ad2ant2r ⊢ ( ( ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ) ∧ ( 𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ) ) → ( 𝑥 · 𝑦 ) ∈ ℝ )
13 mulge0 ⊢ ( ( ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ) ∧ ( 𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ) ) → 0 ≤ ( 𝑥 · 𝑦 ) )
14 10 12 13 elrabd ⊢ ( ( ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ) ∧ ( 𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ) ) → ( 𝑥 · 𝑦 ) ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } )
15 7 9 14 syl2anb ⊢ ( ( 𝑥 ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ∧ 𝑦 ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ) → ( 𝑥 · 𝑦 ) ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } )
16 1re ⊢ 1 ∈ ℝ
17 0le1 ⊢ 0 ≤ 1
18 breq2 ⊢ ( 𝑧 = 1 → ( 0 ≤ 𝑧 ↔ 0 ≤ 1 ) )
19 18 elrab ⊢ ( 1 ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ↔ ( 1 ∈ ℝ ∧ 0 ≤ 1 ) )
20 16 17 19 mpbir2an ⊢ 1 ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 }
21 5 15 20 expcllem ⊢ ( ( 𝐴 ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑁 ) ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } )
22 breq2 ⊢ ( 𝑧 = ( 𝐴 ↑ 𝑁 ) → ( 0 ≤ 𝑧 ↔ 0 ≤ ( 𝐴 ↑ 𝑁 ) ) )
23 22 elrab ⊢ ( ( 𝐴 ↑ 𝑁 ) ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ↔ ( ( 𝐴 ↑ 𝑁 ) ∈ ℝ ∧ 0 ≤ ( 𝐴 ↑ 𝑁 ) ) )
24 23 simprbi ⊢ ( ( 𝐴 ↑ 𝑁 ) ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } → 0 ≤ ( 𝐴 ↑ 𝑁 ) )
25 21 24 syl ⊢ ( ( 𝐴 ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ∧ 𝑁 ∈ ℕ0 ) → 0 ≤ ( 𝐴 ↑ 𝑁 ) )
26 2 25 sylanbr ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝑁 ∈ ℕ0 ) → 0 ≤ ( 𝐴 ↑ 𝑁 ) )
27 26 3impa ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝑁 ∈ ℕ0 ) → 0 ≤ ( 𝐴 ↑ 𝑁 ) )
28 27 3com23 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴 ) → 0 ≤ ( 𝐴 ↑ 𝑁 ) )