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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 7799. Proofs should normally use remulcl 7872 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 7743 | . . . 4 class ℝ | |
3 | 1, 2 | wcel 2135 | . . 3 wff 𝐴 ∈ ℝ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 2135 | . . 3 wff 𝐵 ∈ ℝ |
6 | 3, 5 | wa 103 | . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) |
7 | cmul 7749 | . . . 4 class · | |
8 | 1, 4, 7 | co 5836 | . . 3 class (𝐴 · 𝐵) |
9 | 8, 2 | wcel 2135 | . 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 7872 |
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