# Metamath Proof Explorer

## Theorem rmspecpos

Description: The discriminant used to define the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014)

Ref Expression
Assertion rmspecpos ( 𝐴 ∈ ( ℤ ‘ 2 ) → ( ( 𝐴 ↑ 2 ) − 1 ) ∈ ℝ+ )

### Proof

Step Hyp Ref Expression
1 eluzelre ( 𝐴 ∈ ( ℤ ‘ 2 ) → 𝐴 ∈ ℝ )
2 1 resqcld ( 𝐴 ∈ ( ℤ ‘ 2 ) → ( 𝐴 ↑ 2 ) ∈ ℝ )
3 1red ( 𝐴 ∈ ( ℤ ‘ 2 ) → 1 ∈ ℝ )
4 2 3 resubcld ( 𝐴 ∈ ( ℤ ‘ 2 ) → ( ( 𝐴 ↑ 2 ) − 1 ) ∈ ℝ )
5 sq1 ( 1 ↑ 2 ) = 1
6 eluz2b1 ( 𝐴 ∈ ( ℤ ‘ 2 ) ↔ ( 𝐴 ∈ ℤ ∧ 1 < 𝐴 ) )
7 6 simprbi ( 𝐴 ∈ ( ℤ ‘ 2 ) → 1 < 𝐴 )
8 0le1 0 ≤ 1
9 8 a1i ( 𝐴 ∈ ( ℤ ‘ 2 ) → 0 ≤ 1 )
10 eluzge2nn0 ( 𝐴 ∈ ( ℤ ‘ 2 ) → 𝐴 ∈ ℕ0 )
11 10 nn0ge0d ( 𝐴 ∈ ( ℤ ‘ 2 ) → 0 ≤ 𝐴 )
12 3 1 9 11 lt2sqd ( 𝐴 ∈ ( ℤ ‘ 2 ) → ( 1 < 𝐴 ↔ ( 1 ↑ 2 ) < ( 𝐴 ↑ 2 ) ) )
13 7 12 mpbid ( 𝐴 ∈ ( ℤ ‘ 2 ) → ( 1 ↑ 2 ) < ( 𝐴 ↑ 2 ) )
14 5 13 eqbrtrrid ( 𝐴 ∈ ( ℤ ‘ 2 ) → 1 < ( 𝐴 ↑ 2 ) )
15 3 2 posdifd ( 𝐴 ∈ ( ℤ ‘ 2 ) → ( 1 < ( 𝐴 ↑ 2 ) ↔ 0 < ( ( 𝐴 ↑ 2 ) − 1 ) ) )
16 14 15 mpbid ( 𝐴 ∈ ( ℤ ‘ 2 ) → 0 < ( ( 𝐴 ↑ 2 ) − 1 ) )
17 4 16 elrpd ( 𝐴 ∈ ( ℤ ‘ 2 ) → ( ( 𝐴 ↑ 2 ) − 1 ) ∈ ℝ+ )