recgt1i

Description: The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005)

		Ref	Expression
	Assertion	recgt1i	$⊢ (A \in ℝ \land 1 < A) \to (0 < \frac{1}{A} \land \frac{1}{A} < 1)$

Step	Hyp	Ref	Expression
1		0lt1	$⊢ 0 < 1$
2		0re	$⊢ 0 \in ℝ$
3		1re	$⊢ 1 \in ℝ$
4		lttr	$⊢ (0 \in ℝ \land 1 \in ℝ \land A \in ℝ) \to ((0 < 1 \land 1 < A) \to 0 < A)$
5	2 3 4	mp3an12	$⊢ A \in ℝ \to ((0 < 1 \land 1 < A) \to 0 < A)$
6	1 5	mpani	$⊢ A \in ℝ \to (1 < A \to 0 < A)$
7	6	imdistani	$⊢ (A \in ℝ \land 1 < A) \to (A \in ℝ \land 0 < A)$
8		recgt0	$⊢ (A \in ℝ \land 0 < A) \to 0 < \frac{1}{A}$
9	7 8	syl	$⊢ (A \in ℝ \land 1 < A) \to 0 < \frac{1}{A}$
10		recgt1	$⊢ (A \in ℝ \land 0 < A) \to (1 < A \leftrightarrow \frac{1}{A} < 1)$
11	10	biimpa	$⊢ ((A \in ℝ \land 0 < A) \land 1 < A) \to \frac{1}{A} < 1$
12	7 11	sylancom	$⊢ (A \in ℝ \land 1 < A) \to \frac{1}{A} < 1$
13	9 12	jca	$⊢ (A \in ℝ \land 1 < A) \to (0 < \frac{1}{A} \land \frac{1}{A} < 1)$

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