This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem loge

Description: The natural logarithm of _e . One case of Property 1b of Cohen p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007)

		Ref	Expression
	Assertion	loge	$⊢ \log e = 1$

Step	Hyp	Ref	Expression
1		df-e	$⊢ e = e^{1}$
2	1	eqcomi	$⊢ e^{1} = e$
3		epr	$⊢ e \in ℝ^{+}$
4		1re	$⊢ 1 \in ℝ$
5		relogeftb	$⊢ (e \in ℝ^{+} \land 1 \in ℝ) \to (\log e = 1 \leftrightarrow e^{1} = e)$
6	3 4 5	mp2an	$⊢ \log e = 1 \leftrightarrow e^{1} = e$
7	2 6	mpbir	$⊢ \log e = 1$