Theorem zgt1rpn0n1

Description: An integer greater than 1 is a positive real number not equal to 0 or 1. Useful for working with integer logarithm bases (which is a common case, e.g., base 2, base 3, or base 10). (Contributed by Thierry Arnoux, 26-Sep-2017) (Proof shortened by AV, 9-Jul-2022)

	Ref	Expression
Assertion	zgt1rpn0n1	⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) → ( 𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) )

Proof

Step	Hyp	Ref	Expression
1		eluz2nn	⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) → 𝐵 ∈ ℕ )
2	1	nnrpd	⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) → 𝐵 ∈ ℝ+ )
3		eluz2n0	⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) → 𝐵 ≠ 0 )
4		1nuz2	⊢ ¬ 1 ∈ ( ℤ≥ ‘ 2 )
5		nelne2	⊢ ( ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ ¬ 1 ∈ ( ℤ≥ ‘ 2 ) ) → 𝐵 ≠ 1 )
6	4 5	mpan2	⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) → 𝐵 ≠ 1 )
7	2 3 6	3jca	⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) → ( 𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) )