Axiom ax-mulf 11107

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 11384 or eff 16035. 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 11111. Note that uses of ax-mulf 11107 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) in place of ·, as seen in mpomulf 11122.

This axiom is justified by Theorem axmulf 11058. (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 11025	. . 3 class ℂ
2	1, 1	cxp 5620	. 2 class (ℂ × ℂ)
3		cmul 11032	. 2 class ·
4	2, 1, 3	wf 6486	1 wff · :(ℂ × ℂ)⟶ℂ

This axiom is referenced by:  mulnzcnf  11785  mulex  12930  cnfldmul  21350  dfcnfldOLD  21358  mulcn  24842  dvdsmulf1o  27177  fsumdvdsmulOLD  27178  cncvcOLD  30674  xrge0pluscn  34105