Metamath Proof Explorer

Theorem redivcl

Description: Closure law for division of reals. (Contributed by NM, 27-Sep-1999) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion redivcl ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 𝐴 / 𝐵 ) ∈ ℝ )

Proof

Step Hyp Ref Expression
1 simp1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → 𝐴 ∈ ℝ )
2 1 recnd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → 𝐴 ∈ ℂ )
3 simp2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → 𝐵 ∈ ℝ )
4 3 recnd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → 𝐵 ∈ ℂ )
5 simp3 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → 𝐵 ≠ 0 )
6 divrec ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 / 𝐵 ) = ( 𝐴 · ( 1 / 𝐵 ) ) )
7 2 4 5 6 syl3anc ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 𝐴 / 𝐵 ) = ( 𝐴 · ( 1 / 𝐵 ) ) )
8 rereccl ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 1 / 𝐵 ) ∈ ℝ )
9 8 3adant1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 1 / 𝐵 ) ∈ ℝ )
10 1 9 remulcld ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 𝐴 · ( 1 / 𝐵 ) ) ∈ ℝ )
11 7 10 eqeltrd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 𝐴 / 𝐵 ) ∈ ℝ )