Axiom ax-addf 11213

Description: Addition is an operation on the complex numbers. This deprecated axiom is provided for historical compatibility but is not a bona fide axiom for complex numbers (independent of set theory) since it cannot be interpreted as a first-order or second-order statement (see https://us.metamath.org/downloads/schmidt-cnaxioms.pdf). It may be deleted in the future and should be avoided for new theorems. Instead, the less specific addcl 11216 should be used. Note that uses of ax-addf 11213 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) in place of +, from which this axiom (with the defined operation in place of +) follows as a theorem.

This axiom is justified by Theorem axaddf 11164. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

Assertion

Ref	Expression
ax-addf	⊢ + :(ℂ × ℂ)⟶ℂ

Detailed syntax breakdown of Axiom ax-addf

Step	Hyp	Ref	Expression
1		cc 11132	. . 3 class ℂ
2	1, 1	cxp 5657	. 2 class (ℂ × ℂ)
3		caddc 11137	. 2 class +
4	2, 1, 3	wf 6532	1 wff + :(ℂ × ℂ)⟶ℂ

This axiom is referenced by:  addex  13010  cnfldadd  21326  dfcnfldOLD  21336  cnfldplusf  21364  addcn  24810  itg1addlem4  25657  cnaddabloOLD  30567  cnidOLD  30568  cncvcOLD  30569  cnnv  30663  cnnvba  30665  cncph  30805  raddcn  33965  addcomgi  44455