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Metamath Proof Explorer

Theorem recgt1d

Description: The reciprocal of a positive number greater than 1 is less than 1. (Contributed by Mario Carneiro, 28-May-2016)

		Ref	Expression
	Hypothesis	rpred.1	$⊢ φ \to A \in ℝ^{+}$
	Assertion	recgt1d	$⊢ φ \to (1 < A \leftrightarrow \frac{1}{A} < 1)$

Step	Hyp	Ref	Expression
1		rpred.1	$⊢ φ \to A \in ℝ^{+}$
2	1	rpregt0d	$⊢ φ \to (A \in ℝ \land 0 < A)$
3		recgt1	$⊢ (A \in ℝ \land 0 < A) \to (1 < A \leftrightarrow \frac{1}{A} < 1)$
4	2 3	syl	$⊢ φ \to (1 < A \leftrightarrow \frac{1}{A} < 1)$