Description: Closure law for extended division of positive reals. (Contributed by Thierry Arnoux, 18-Dec-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | rpxdivcld.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℝ+ ) | |
rpxdivcld.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℝ+ ) | ||
Assertion | rpxdivcld | ⊢ ( 𝜑 → ( 𝐴 /𝑒 𝐵 ) ∈ ℝ+ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpxdivcld.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℝ+ ) | |
2 | rpxdivcld.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℝ+ ) | |
3 | 1 | rpred | ⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
4 | 2 | rpred | ⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
5 | 2 | rpne0d | ⊢ ( 𝜑 → 𝐵 ≠ 0 ) |
6 | rexdiv | ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 𝐴 /𝑒 𝐵 ) = ( 𝐴 / 𝐵 ) ) | |
7 | 3 4 5 6 | syl3anc | ⊢ ( 𝜑 → ( 𝐴 /𝑒 𝐵 ) = ( 𝐴 / 𝐵 ) ) |
8 | 1 2 | rpdivcld | ⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) ∈ ℝ+ ) |
9 | 7 8 | eqeltrd | ⊢ ( 𝜑 → ( 𝐴 /𝑒 𝐵 ) ∈ ℝ+ ) |