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Metamath Proof Explorer

Theorem ltnegd

Description: Negative of both sides of 'less than'. Theorem I.23 of Apostol p. 20. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
ltnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion ltnegd ( 𝜑 → ( 𝐴 < 𝐵 ↔ - 𝐵 < - 𝐴 ) )

Proof

Step Hyp Ref Expression
1 leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 ltnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 ltneg ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ - 𝐵 < - 𝐴 ) )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ - 𝐵 < - 𝐴 ) )