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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 7553. Proofs should normally use mulcl 7619 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 7498 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 1448 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 1448 | . . 3 wff 𝐵 ∈ ℂ |
6 | 3, 5 | wa 103 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) |
7 | cmul 7505 | . . . 4 class · | |
8 | 1, 4, 7 | co 5706 | . . 3 class (𝐴 · 𝐵) |
9 | 8, 2 | wcel 1448 | . 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 7619 |
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