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Mirrors > Home > MPE Home > Th. List > rpdivcld | Structured version Visualization version GIF version |
Description: Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpred.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
rpaddcld.1 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
Ref | Expression |
---|---|
rpdivcld | ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
2 | rpaddcld.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
3 | rpdivcl 12510 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ+) | |
4 | 1, 2, 3 | syl2anc 587 | 1 ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ+) |
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