Description: Closure law for division of a real by a positive real. (Contributed by NM, 10-Nov-2008)
Ref | Expression | ||
---|---|---|---|
Assertion | rerpdivcl | ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 / 𝐵 ) ∈ ℝ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rprene0 | ⊢ ( 𝐵 ∈ ℝ+ → ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ) | |
2 | redivcl | ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 𝐴 / 𝐵 ) ∈ ℝ ) | |
3 | 2 | 3expb | ⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ) → ( 𝐴 / 𝐵 ) ∈ ℝ ) |
4 | 1 3 | sylan2 | ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 / 𝐵 ) ∈ ℝ ) |