Theorem loggt0b

Description: The logarithm of a number is positive iff the number is greater than 1. (Contributed by AV, 30-May-2020)

	Ref	Expression
Assertion	loggt0b	⊢ ( 𝐴 ∈ ℝ+ → ( 0 < ( log ‘ 𝐴 ) ↔ 1 < 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		1rp	⊢ 1 ∈ ℝ+
2		logltb	⊢ ( ( 1 ∈ ℝ+ ∧ 𝐴 ∈ ℝ+ ) → ( 1 < 𝐴 ↔ ( log ‘ 1 ) < ( log ‘ 𝐴 ) ) )
3	1 2	mpan	⊢ ( 𝐴 ∈ ℝ+ → ( 1 < 𝐴 ↔ ( log ‘ 1 ) < ( log ‘ 𝐴 ) ) )
4		log1	⊢ ( log ‘ 1 ) = 0
5	4	a1i	⊢ ( 𝐴 ∈ ℝ+ → ( log ‘ 1 ) = 0 )
6	5	breq1d	⊢ ( 𝐴 ∈ ℝ+ → ( ( log ‘ 1 ) < ( log ‘ 𝐴 ) ↔ 0 < ( log ‘ 𝐴 ) ) )
7	3 6	bitr2d	⊢ ( 𝐴 ∈ ℝ+ → ( 0 < ( log ‘ 𝐴 ) ↔ 1 < 𝐴 ) )