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 ) )