Theorem 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 ∈ ℝ ∧ 1 < 𝐴 ) → ( 0 < ( 1 / 𝐴 ) ∧ ( 1 / 𝐴 ) < 1 ) )

Proof

Step	Hyp	Ref	Expression
1		0lt1	⊢ 0 < 1
2		0re	⊢ 0 ∈ ℝ
3		1re	⊢ 1 ∈ ℝ
4		lttr	⊢ ( ( 0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( ( 0 < 1 ∧ 1 < 𝐴 ) → 0 < 𝐴 ) )
5	2 3 4	mp3an12	⊢ ( 𝐴 ∈ ℝ → ( ( 0 < 1 ∧ 1 < 𝐴 ) → 0 < 𝐴 ) )
6	1 5	mpani	⊢ ( 𝐴 ∈ ℝ → ( 1 < 𝐴 → 0 < 𝐴 ) )
7	6	imdistani	⊢ ( ( 𝐴 ∈ ℝ ∧ 1 < 𝐴 ) → ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) )
8		recgt0	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → 0 < ( 1 / 𝐴 ) )
9	7 8	syl	⊢ ( ( 𝐴 ∈ ℝ ∧ 1 < 𝐴 ) → 0 < ( 1 / 𝐴 ) )
10		recgt1	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 1 < 𝐴 ↔ ( 1 / 𝐴 ) < 1 ) )
11	10	biimpa	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ 1 < 𝐴 ) → ( 1 / 𝐴 ) < 1 )
12	7 11	sylancom	⊢ ( ( 𝐴 ∈ ℝ ∧ 1 < 𝐴 ) → ( 1 / 𝐴 ) < 1 )
13	9 12	jca	⊢ ( ( 𝐴 ∈ ℝ ∧ 1 < 𝐴 ) → ( 0 < ( 1 / 𝐴 ) ∧ ( 1 / 𝐴 ) < 1 ) )