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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 7829. Proofs should normally use remulcl 7902 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 7773 | . . . 4 class ℝ | |
3 | 1, 2 | wcel 2141 | . . 3 wff 𝐴 ∈ ℝ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 2141 | . . 3 wff 𝐵 ∈ ℝ |
6 | 3, 5 | wa 103 | . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) |
7 | cmul 7779 | . . . 4 class · | |
8 | 1, 4, 7 | co 5853 | . . 3 class (𝐴 · 𝐵) |
9 | 8, 2 | wcel 2141 | . 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 7902 |
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