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Mirrors > Home > ILE Home > Th. List > ax-mulcl | GIF version |
Description: Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by theorem axmulcl 7698. Proofs should normally use mulcl 7771 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Ref | Expression |
---|---|
ax-mulcl | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class 𝐴 | |
2 | cc 7642 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 1481 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 1481 | . . 3 wff 𝐵 ∈ ℂ |
6 | 3, 5 | wa 103 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) |
7 | cmul 7649 | . . . 4 class · | |
8 | 1, 4, 7 | co 5782 | . . 3 class (𝐴 · 𝐵) |
9 | 8, 2 | wcel 1481 | . 2 wff (𝐴 · 𝐵) ∈ ℂ |
10 | 6, 9 | wi 4 | 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ) |
Colors of variables: wff set class |
This axiom is referenced by: mulcl 7771 |
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