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Metamath Proof Explorer

Theorem subge0

Description: Nonnegative subtraction. (Contributed by NM, 14-Mar-2005) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion subge0 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 ≤ ( 𝐴 − 𝐵 ) ↔ 𝐵 ≤ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 0red ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → 0 ∈ ℝ )
2 simpr ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → 𝐵 ∈ ℝ )
3 simpl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → 𝐴 ∈ ℝ )
4 leaddsub ⊢ ( ( 0 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( ( 0 + 𝐵 ) ≤ 𝐴 ↔ 0 ≤ ( 𝐴 − 𝐵 ) ) )
5 1 2 3 4 syl3anc ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 0 + 𝐵 ) ≤ 𝐴 ↔ 0 ≤ ( 𝐴 − 𝐵 ) ) )
6 2 recnd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → 𝐵 ∈ ℂ )
7 6 addlidd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 + 𝐵 ) = 𝐵 )
8 7 breq1d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 0 + 𝐵 ) ≤ 𝐴 ↔ 𝐵 ≤ 𝐴 ) )
9 5 8 bitr3d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 ≤ ( 𝐴 − 𝐵 ) ↔ 𝐵 ≤ 𝐴 ) )