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| Description: Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.) | 
| Ref | Expression | 
|---|---|
| resubcl | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | recn 11246 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
| 2 | recn 11246 | . . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
| 3 | negsub 11558 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | |
| 4 | 1, 2, 3 | syl2an 596 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | 
| 5 | renegcl 11573 | . . 3 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ) | |
| 6 | readdcl 11239 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ -𝐵 ∈ ℝ) → (𝐴 + -𝐵) ∈ ℝ) | |
| 7 | 5, 6 | sylan2 593 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + -𝐵) ∈ ℝ) | 
| 8 | 4, 7 | eqeltrrd 2841 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ) | 
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