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Metamath Proof Explorer

Theorem ltsubaddi

Description: 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997) (Proof shortened by Andrew Salmon, 19-Nov-2011)

Ref Expression
Hypotheses lt2.1 ⊢ A ∈ ℝ
lt2.2 ⊢ B ∈ ℝ
lt2.3 ⊢ C ∈ ℝ
Assertion ltsubaddi ⊢ 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 ltsubadd ⊢ A ∈ ℝ ∧ B ∈ ℝ ∧ C ∈ ℝ → A − B < C ↔ A < C + B
5 1 2 3 4 mp3an ⊢ A − B < C ↔ A < C + B