Metamath Proof Explorer


Theorem ltletrd

Description: Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006)

Ref Expression
Hypotheses ltd.1 ( 𝜑𝐴 ∈ ℝ )
ltd.2 ( 𝜑𝐵 ∈ ℝ )
letrd.3 ( 𝜑𝐶 ∈ ℝ )
ltletrd.4 ( 𝜑𝐴 < 𝐵 )
ltletrd.5 ( 𝜑𝐵𝐶 )
Assertion ltletrd ( 𝜑𝐴 < 𝐶 )

Proof

Step Hyp Ref Expression
1 ltd.1 ( 𝜑𝐴 ∈ ℝ )
2 ltd.2 ( 𝜑𝐵 ∈ ℝ )
3 letrd.3 ( 𝜑𝐶 ∈ ℝ )
4 ltletrd.4 ( 𝜑𝐴 < 𝐵 )
5 ltletrd.5 ( 𝜑𝐵𝐶 )
6 ltletr ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 < 𝐵𝐵𝐶 ) → 𝐴 < 𝐶 ) )
7 1 2 3 6 syl3anc ( 𝜑 → ( ( 𝐴 < 𝐵𝐵𝐶 ) → 𝐴 < 𝐶 ) )
8 4 5 7 mp2and ( 𝜑𝐴 < 𝐶 )