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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 7642. Proofs should normally use mulcl 7715 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 7586 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 1465 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 1465 | . . 3 wff 𝐵 ∈ ℂ |
6 | 3, 5 | wa 103 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) |
7 | cmul 7593 | . . . 4 class · | |
8 | 1, 4, 7 | co 5742 | . . 3 class (𝐴 · 𝐵) |
9 | 8, 2 | wcel 1465 | . 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 7715 |
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