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Metamath Proof Explorer

Theorem subge0d

Description: Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
ltnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion subge0d ( 𝜑 → ( 0 ≤ ( 𝐴 − 𝐵 ) ↔ 𝐵 ≤ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 ltnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 subge0 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 ≤ ( 𝐴 − 𝐵 ) ↔ 𝐵 ≤ 𝐴 ) )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 0 ≤ ( 𝐴 − 𝐵 ) ↔ 𝐵 ≤ 𝐴 ) )