Theorem loglt1b

Description: The logarithm of a number is less than 1 iff the number is less than Euler's constant. (Contributed by AV, 30-May-2020)

	Ref	Expression
Assertion	loglt1b	⊢ ( 𝐴 ∈ ℝ+ → ( ( log ‘ 𝐴 ) < 1 ↔ 𝐴 < e ) )

Proof

Step	Hyp	Ref	Expression
1		epr	⊢ e ∈ ℝ+
2		logltb	⊢ ( ( 𝐴 ∈ ℝ+ ∧ e ∈ ℝ+ ) → ( 𝐴 < e ↔ ( log ‘ 𝐴 ) < ( log ‘ e ) ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ ℝ+ → ( 𝐴 < e ↔ ( log ‘ 𝐴 ) < ( log ‘ e ) ) )
4		loge	⊢ ( log ‘ e ) = 1
5	4	a1i	⊢ ( 𝐴 ∈ ℝ+ → ( log ‘ e ) = 1 )
6	5	breq2d	⊢ ( 𝐴 ∈ ℝ+ → ( ( log ‘ 𝐴 ) < ( log ‘ e ) ↔ ( log ‘ 𝐴 ) < 1 ) )
7	3 6	bitr2d	⊢ ( 𝐴 ∈ ℝ+ → ( ( log ‘ 𝐴 ) < 1 ↔ 𝐴 < e ) )