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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 7034. Proofs should normally use mulcl 7100 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 6979 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 1433 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 1433 | . . 3 wff 𝐵 ∈ ℂ |
6 | 3, 5 | wa 102 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) |
7 | cmul 6986 | . . . 4 class · | |
8 | 1, 4, 7 | co 5532 | . . 3 class (𝐴 · 𝐵) |
9 | 8, 2 | wcel 1433 | . 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 7100 |
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