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Metamath Proof Explorer

Theorem ltadd2dd

Description: Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016)

Ref Expression
Hypotheses ltd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
ltd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
letrd.3 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
ltletrd.4 ⊢ ( 𝜑 → 𝐴 < 𝐵 )
Assertion ltadd2dd ( 𝜑 → ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) )

Proof

Step Hyp Ref Expression
1 ltd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 ltd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 letrd.3 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4 ltletrd.4 ⊢ ( 𝜑 → 𝐴 < 𝐵 )
5 1 2 3 ltadd2d ⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) ) )
6 4 5 mpbid ⊢ ( 𝜑 → ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) )