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Metamath Proof Explorer

Theorem lt2addd

Description: Adding both side of two inequalities. Theorem I.25 of Apostol p. 20. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
ltnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
ltadd1d.3 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
lt2addd.4 ⊢ ( 𝜑 → 𝐷 ∈ ℝ )
lt2addd.5 ⊢ ( 𝜑 → 𝐴 < 𝐶 )
lt2addd.6 ⊢ ( 𝜑 → 𝐵 < 𝐷 )
Assertion lt2addd ( 𝜑 → ( 𝐴 + 𝐵 ) < ( 𝐶 + 𝐷 ) )

Proof

Step Hyp Ref Expression
1 leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 ltnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 ltadd1d.3 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4 lt2addd.4 ⊢ ( 𝜑 → 𝐷 ∈ ℝ )
5 lt2addd.5 ⊢ ( 𝜑 → 𝐴 < 𝐶 )
6 lt2addd.6 ⊢ ( 𝜑 → 𝐵 < 𝐷 )
7 2 4 6 ltled ⊢ ( 𝜑 → 𝐵 ≤ 𝐷 )
8 1 2 3 4 5 7 ltleaddd ⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) < ( 𝐶 + 𝐷 ) )