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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 6942. Proofs should normally use mulcl 7008 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 6887 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 1393 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 1393 | . . 3 wff 𝐵 ∈ ℂ |
6 | 3, 5 | wa 97 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) |
7 | cmul 6894 | . . . 4 class · | |
8 | 1, 4, 7 | co 5512 | . . 3 class (𝐴 · 𝐵) |
9 | 8, 2 | wcel 1393 | . 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 7008 |
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