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| Mirrors > Home > MPE Home > Th. List > resubcld | Structured version Visualization version GIF version | ||
| Description: Closure law for subtraction of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| renegcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| resubcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
| Ref | Expression |
|---|---|
| resubcld | ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℝ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | renegcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 2 | resubcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 3 | resubcl 11573 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℝ) |
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