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Metamath Proof Explorer

Theorem ltadd1i

Description: Addition to both sides of 'less than'. Theorem I.18 of Apostol p. 20. (Contributed by NM, 21-Jan-1997)

Ref Expression
Hypotheses lt2.1 ⊢ 𝐴 ∈ ℝ
lt2.2 ⊢ 𝐵 ∈ ℝ
lt2.3 ⊢ 𝐶 ∈ ℝ
Assertion ltadd1i ( 𝐴 < 𝐵 ↔ ( 𝐴 + 𝐶 ) < ( 𝐵 + 𝐶 ) )

Proof

Step Hyp Ref Expression
1 lt2.1 ⊢ 𝐴 ∈ ℝ
2 lt2.2 ⊢ 𝐵 ∈ ℝ
3 lt2.3 ⊢ 𝐶 ∈ ℝ
4 ltadd1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 + 𝐶 ) < ( 𝐵 + 𝐶 ) ) )
5 1 2 3 4 mp3an ⊢ ( 𝐴 < 𝐵 ↔ ( 𝐴 + 𝐶 ) < ( 𝐵 + 𝐶 ) )