Metamath Proof Explorer


Axiom ax-mulcl

Description: Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by Theorem axmulcl . Proofs should normally use mulcl instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994)

Ref Expression
Assertion ax-mulcl ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 · 𝐵 ) ∈ ℂ )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA 𝐴
1 cc
2 0 1 wcel 𝐴 ∈ ℂ
3 cB 𝐵
4 3 1 wcel 𝐵 ∈ ℂ
5 2 4 wa ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ )
6 cmul ·
7 0 3 6 co ( 𝐴 · 𝐵 )
8 7 1 wcel ( 𝐴 · 𝐵 ) ∈ ℂ
9 5 8 wi ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 · 𝐵 ) ∈ ℂ )