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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 7596. Proofs should normally use remulcl 7666 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 7540 | . . . 4 class ℝ | |
3 | 1, 2 | wcel 1461 | . . 3 wff 𝐴 ∈ ℝ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 1461 | . . 3 wff 𝐵 ∈ ℝ |
6 | 3, 5 | wa 103 | . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) |
7 | cmul 7546 | . . . 4 class · | |
8 | 1, 4, 7 | co 5726 | . . 3 class (𝐴 · 𝐵) |
9 | 8, 2 | wcel 1461 | . 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 7666 |
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