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Metamath Proof Explorer

Theorem lesub1dd

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

Ref Expression
Hypotheses leidd.1 ⊢ φ → A ∈ ℝ
ltnegd.2 ⊢ φ → B ∈ ℝ
ltadd1d.3 ⊢ φ → C ∈ ℝ
leadd1dd.4 ⊢ φ → A ≤ B
Assertion lesub1dd ⊢ φ → A − C ≤ B − C

Proof

Step Hyp Ref Expression
1 leidd.1 ⊢ φ → A ∈ ℝ
2 ltnegd.2 ⊢ φ → B ∈ ℝ
3 ltadd1d.3 ⊢ φ → C ∈ ℝ
4 leadd1dd.4 ⊢ φ → A ≤ B
5 1 2 3 lesub1d ⊢ φ → A ≤ B ↔ A − C ≤ B − C
6 4 5 mpbid ⊢ φ → A − C ≤ B − C