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Metamath Proof Explorer

Theorem leadd2i

Description: Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999)

Ref Expression
Hypotheses lt2.1 ⊢ 𝐴 ∈ ℝ
lt2.2 ⊢ 𝐵 ∈ ℝ
lt2.3 ⊢ 𝐶 ∈ ℝ
Assertion leadd2i ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 + 𝐴 ) ≤ ( 𝐶 + 𝐵 ) )

Proof

Step Hyp Ref Expression
1 lt2.1 ⊢ 𝐴 ∈ ℝ
2 lt2.2 ⊢ 𝐵 ∈ ℝ
3 lt2.3 ⊢ 𝐶 ∈ ℝ
4 leadd2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 + 𝐴 ) ≤ ( 𝐶 + 𝐵 ) ) )
5 1 2 3 4 mp3an ⊢ ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 + 𝐴 ) ≤ ( 𝐶 + 𝐵 ) )