Metamath Proof Explorer


Theorem le2subd

Description: Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses leidd.1 ( 𝜑𝐴 ∈ ℝ )
ltnegd.2 ( 𝜑𝐵 ∈ ℝ )
ltadd1d.3 ( 𝜑𝐶 ∈ ℝ )
lt2addd.4 ( 𝜑𝐷 ∈ ℝ )
le2addd.5 ( 𝜑𝐴𝐶 )
le2addd.6 ( 𝜑𝐵𝐷 )
Assertion le2subd ( 𝜑 → ( 𝐴𝐷 ) ≤ ( 𝐶𝐵 ) )

Proof

Step Hyp Ref Expression
1 leidd.1 ( 𝜑𝐴 ∈ ℝ )
2 ltnegd.2 ( 𝜑𝐵 ∈ ℝ )
3 ltadd1d.3 ( 𝜑𝐶 ∈ ℝ )
4 lt2addd.4 ( 𝜑𝐷 ∈ ℝ )
5 le2addd.5 ( 𝜑𝐴𝐶 )
6 le2addd.6 ( 𝜑𝐵𝐷 )
7 le2sub ( ( ( 𝐴 ∈ ℝ ∧ 𝐷 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) → ( ( 𝐴𝐶𝐵𝐷 ) → ( 𝐴𝐷 ) ≤ ( 𝐶𝐵 ) ) )
8 1 4 3 2 7 syl22anc ( 𝜑 → ( ( 𝐴𝐶𝐵𝐷 ) → ( 𝐴𝐷 ) ≤ ( 𝐶𝐵 ) ) )
9 5 6 8 mp2and ( 𝜑 → ( 𝐴𝐷 ) ≤ ( 𝐶𝐵 ) )