Theorem ltrec1d

Description: Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	rpaddcld.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
	ltrec1d.2	⊢ ( 𝜑 → ( 1 / 𝐴 ) < 𝐵 )
Assertion	ltrec1d	⊢ ( 𝜑 → ( 1 / 𝐵 ) < 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		rpred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		rpaddcld.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3		ltrec1d.2	⊢ ( 𝜑 → ( 1 / 𝐴 ) < 𝐵 )
4	1	rpregt0d	⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) )
5	2	rpregt0d	⊢ ( 𝜑 → ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) )
6		ltrec1	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( ( 1 / 𝐴 ) < 𝐵 ↔ ( 1 / 𝐵 ) < 𝐴 ) )
7	4 5 6	syl2anc	⊢ ( 𝜑 → ( ( 1 / 𝐴 ) < 𝐵 ↔ ( 1 / 𝐵 ) < 𝐴 ) )
8	3 7	mpbid	⊢ ( 𝜑 → ( 1 / 𝐵 ) < 𝐴 )