Axiom ax-mulcl 8030

Description: Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulcl 7986. Proofs should normally use mulcl 8059 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

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

Detailed syntax breakdown of Axiom ax-mulcl

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 7930	. . . 4 class ℂ
3	1, 2	wcel 2177	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2177	. . 3 wff 𝐵 ∈ ℂ
6	3, 5	wa 104	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7		cmul 7937	. . . 4 class ·
8	1, 4, 7	co 5951	. . 3 class (𝐴 · 𝐵)
9	8, 2	wcel 2177	. 2 wff (𝐴 · 𝐵) ∈ ℂ
10	6, 9	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)

This axiom is referenced by:  mulcl  8059