Theorem 2logb3irr

Description: Example for logbprmirr . The logarithm of three to base two is irrational. (Contributed by AV, 31-Dec-2022)

	Ref	Expression
Assertion	2logb3irr	⊢ ( 2 logb 3 ) ∈ ( ℝ ∖ ℚ )

Proof

Step	Hyp	Ref	Expression
1		3prm	⊢ 3 ∈ ℙ
2		2prm	⊢ 2 ∈ ℙ
3		2re	⊢ 2 ∈ ℝ
4		2lt3	⊢ 2 < 3
5	3 4	gtneii	⊢ 3 ≠ 2
6		logbprmirr	⊢ ( ( 3 ∈ ℙ ∧ 2 ∈ ℙ ∧ 3 ≠ 2 ) → ( 2 logb 3 ) ∈ ( ℝ ∖ ℚ ) )
7	1 2 5 6	mp3an	⊢ ( 2 logb 3 ) ∈ ( ℝ ∖ ℚ )