Axiom ax-mulf 11168

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 11447 or eff 16125. 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 11172. Note that uses of ax-mulf 11168 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) in place of ·, as seen in mpomulf 11183.

This axiom is justified by Theorem axmulf 11119. (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 11086	. . 3 class ℂ
2	1, 1	cxp 5650	. 2 class (ℂ × ℂ)
3		cmul 11093	. 2 class ·
4	2, 1, 3	wf 6521	1 wff · :(ℂ × ℂ)⟶ℂ

This axiom is referenced by:  mulnzcnf  11848  mulex  13006  cnfldmul  21490  mulcn  24986  dvdsmulf1o  27318  cncvcOLD  30844  xrge0pluscn  34247