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Metamath Proof Explorer

Theorem expgt1

Description: A real greater than 1 raised to a positive integer is greater than 1. (Contributed by NM, 13-Feb-2005) (Revised by Mario Carneiro, 4-Jun-2014)

Ref Expression
Assertion expgt1 ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → 1 < ( 𝐴 ↑ 𝑁 ) )

Proof

Step Hyp Ref Expression
1 1re ⊢ 1 ∈ ℝ
2 1 a1i ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → 1 ∈ ℝ )
3 simp1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → 𝐴 ∈ ℝ )
4 simp2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → 𝑁 ∈ ℕ )
5 4 nnnn0d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → 𝑁 ∈ ℕ0 )
6 reexpcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑁 ) ∈ ℝ )
7 3 5 6 syl2anc ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → ( 𝐴 ↑ 𝑁 ) ∈ ℝ )
8 simp3 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → 1 < 𝐴 )
9 nnm1nn0 ⊢ ( 𝑁 ∈ ℕ → ( 𝑁 − 1 ) ∈ ℕ0 )
10 4 9 syl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → ( 𝑁 − 1 ) ∈ ℕ0 )
11 ltle ⊢ ( ( 1 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 1 < 𝐴 → 1 ≤ 𝐴 ) )
12 1 3 11 sylancr ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → ( 1 < 𝐴 → 1 ≤ 𝐴 ) )
13 8 12 mpd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → 1 ≤ 𝐴 )
14 expge1 ⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝑁 − 1 ) ∈ ℕ0 ∧ 1 ≤ 𝐴 ) → 1 ≤ ( 𝐴 ↑ ( 𝑁 − 1 ) ) )
15 3 10 13 14 syl3anc ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → 1 ≤ ( 𝐴 ↑ ( 𝑁 − 1 ) ) )
16 reexpcl ⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝑁 − 1 ) ∈ ℕ0 ) → ( 𝐴 ↑ ( 𝑁 − 1 ) ) ∈ ℝ )
17 3 10 16 syl2anc ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → ( 𝐴 ↑ ( 𝑁 − 1 ) ) ∈ ℝ )
18 0red ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → 0 ∈ ℝ )
19 0lt1 ⊢ 0 < 1
20 19 a1i ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → 0 < 1 )
21 18 2 3 20 8 lttrd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → 0 < 𝐴 )
22 lemul1 ⊢ ( ( 1 ∈ ℝ ∧ ( 𝐴 ↑ ( 𝑁 − 1 ) ) ∈ ℝ ∧ ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ) → ( 1 ≤ ( 𝐴 ↑ ( 𝑁 − 1 ) ) ↔ ( 1 · 𝐴 ) ≤ ( ( 𝐴 ↑ ( 𝑁 − 1 ) ) · 𝐴 ) ) )
23 2 17 3 21 22 syl112anc ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → ( 1 ≤ ( 𝐴 ↑ ( 𝑁 − 1 ) ) ↔ ( 1 · 𝐴 ) ≤ ( ( 𝐴 ↑ ( 𝑁 − 1 ) ) · 𝐴 ) ) )
24 15 23 mpbid ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → ( 1 · 𝐴 ) ≤ ( ( 𝐴 ↑ ( 𝑁 − 1 ) ) · 𝐴 ) )
25 recn ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
26 25 3ad2ant1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → 𝐴 ∈ ℂ )
27 26 mullidd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → ( 1 · 𝐴 ) = 𝐴 )
28 27 eqcomd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → 𝐴 = ( 1 · 𝐴 ) )
29 expm1t ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ) → ( 𝐴 ↑ 𝑁 ) = ( ( 𝐴 ↑ ( 𝑁 − 1 ) ) · 𝐴 ) )
30 26 4 29 syl2anc ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → ( 𝐴 ↑ 𝑁 ) = ( ( 𝐴 ↑ ( 𝑁 − 1 ) ) · 𝐴 ) )
31 24 28 30 3brtr4d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → 𝐴 ≤ ( 𝐴 ↑ 𝑁 ) )
32 2 3 7 8 31 ltletrd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴 ) → 1 < ( 𝐴 ↑ 𝑁 ) )