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Mirrors > Home > ILE Home > Th. List > ax-mulrcl | GIF version |
Description: Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulrcl 7844. Proofs should normally use remulcl 7917 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Ref | Expression |
---|---|
ax-mulrcl | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class 𝐴 | |
2 | cr 7788 | . . . 4 class ℝ | |
3 | 1, 2 | wcel 2148 | . . 3 wff 𝐴 ∈ ℝ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 2148 | . . 3 wff 𝐵 ∈ ℝ |
6 | 3, 5 | wa 104 | . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) |
7 | cmul 7794 | . . . 4 class · | |
8 | 1, 4, 7 | co 5868 | . . 3 class (𝐴 · 𝐵) |
9 | 8, 2 | wcel 2148 | . 2 wff (𝐴 · 𝐵) ∈ ℝ |
10 | 6, 9 | wi 4 | 1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) |
Colors of variables: wff set class |
This axiom is referenced by: remulcl 7917 |
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