Axiom ax-mulf 11143

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 11422 or eff 16087. 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 11147. Note that uses of ax-mulf 11143 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) in place of ·, as seen in mpomulf 11158.

This axiom is justified by Theorem axmulf 11094. (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 11061	. . 3 class ℂ
2	1, 1	cxp 5638	. 2 class (ℂ × ℂ)
3		cmul 11068	. 2 class ·
4	2, 1, 3	wf 6506	1 wff · :(ℂ × ℂ)⟶ℂ

This axiom is referenced by:  mulnzcnf  11823  mulex  12982  cnfldmul  21405  mulcn  24901  dvdsmulf1o  27230  cncvcOLD  30725  xrge0pluscn  34191