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Metamath Proof Explorer

Theorem ltsubposd

Description: Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses leidd.1 ⊢ φ → A ∈ ℝ
ltnegd.2 ⊢ φ → B ∈ ℝ
Assertion ltsubposd ⊢ φ → 0 < A ↔ B − A < B

Proof

Step Hyp Ref Expression
1 leidd.1 ⊢ φ → A ∈ ℝ
2 ltnegd.2 ⊢ φ → B ∈ ℝ
3 ltsubpos ⊢ A ∈ ℝ ∧ B ∈ ℝ → 0 < A ↔ B − A < B
4 1 2 3 syl2anc ⊢ φ → 0 < A ↔ B − A < B