Axiom ax-mulf 11179

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 11458 or eff 16134. 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 11183. Note that uses of ax-mulf 11179 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) in place of ·, as seen in mpomulf 11194.

This axiom is justified by Theorem axmulf 11130. (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 11097	. . 3 class ℂ
2	1, 1	cxp 5659	. 2 class (ℂ × ℂ)
3		cmul 11104	. 2 class ·
4	2, 1, 3	wf 6532	1 wff · :(ℂ × ℂ)⟶ℂ

This axiom is referenced by:  mulnzcnf  11859  mulex  13014  cnfldmul  21509  mulcn  25004  dvdsmulf1o  27336  cncvcOLD  30901  xrge0pluscn  34296