Axiom ax-mulf 11116

Description: Multiplication is an operation on the complex numbers. This axiom tells us that · is defined only on complex numbers which is analogous to the way that other operations are defined, for example see subf 11393 or eff 16044. However, while Metamath can handle this axiom, if we wish to work with weaker complex number axioms, we can avoid it by using the less specific mulcl 11120. Note that uses of ax-mulf 11116 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) in place of ·, as seen in mpomulf 11131.

This axiom is justified by Theorem axmulf 11067. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

Assertion

Ref	Expression
ax-mulf	⊢ · :(ℂ × ℂ)⟶ℂ

Detailed syntax breakdown of Axiom ax-mulf

Step	Hyp	Ref	Expression
1		cc 11034	. . 3 class ℂ
2	1, 1	cxp 5623	. 2 class (ℂ × ℂ)
3		cmul 11041	. 2 class ·
4	2, 1, 3	wf 6488	1 wff · :(ℂ × ℂ)⟶ℂ

This axiom is referenced by:  mulnzcnf  11794  mulex  12939  cnfldmul  21362  mulcn  24858  dvdsmulf1o  27184  cncvcOLD  30679  xrge0pluscn  34131