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+ )