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Metamath Proof Explorer

Theorem addgt0d

Description: Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
ltnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
addgt0d.3 ⊢ ( 𝜑 → 0 < 𝐴 )
addgt0d.4 ⊢ ( 𝜑 → 0 < 𝐵 )
Assertion addgt0d ( 𝜑 → 0 < ( 𝐴 + 𝐵 ) )

Proof

Step Hyp Ref Expression
1 leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 ltnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 addgt0d.3 ⊢ ( 𝜑 → 0 < 𝐴 )
4 addgt0d.4 ⊢ ( 𝜑 → 0 < 𝐵 )
5 0red ⊢ ( 𝜑 → 0 ∈ ℝ )
6 5 1 3 ltled ⊢ ( 𝜑 → 0 ≤ 𝐴 )
7 1 2 6 4 addgegt0d ⊢ ( 𝜑 → 0 < ( 𝐴 + 𝐵 ) )