Theorem xdivcl

Description: Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017)

	Ref	Expression
Assertion	xdivcl	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 𝐴 /𝑒 𝐵 ) ∈ ℝ* )

Proof

Step	Hyp	Ref	Expression
1		simp1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → 𝐴 ∈ ℝ* )
2		simp2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → 𝐵 ∈ ℝ )
3		simp3	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → 𝐵 ≠ 0 )
4	1 2 3	xdivcld	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 𝐴 /𝑒 𝐵 ) ∈ ℝ* )