MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-addcl Structured version   Visualization version   GIF version

Axiom ax-addcl 10789
Description: Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by Theorem axaddcl 10765. Proofs should normally use addcl 10811 instead, which asserts the same thing but follows our naming conventions for closures. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-addcl ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)

Detailed syntax breakdown of Axiom ax-addcl
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cc 10727 . . . 4 class
31, 2wcel 2110 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 2110 . . 3 wff 𝐵 ∈ ℂ
63, 5wa 399 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7 caddc 10732 . . . 4 class +
81, 4, 7co 7213 . . 3 class (𝐴 + 𝐵)
98, 2wcel 2110 . 2 wff (𝐴 + 𝐵) ∈ ℂ
106, 9wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
Colors of variables: wff setvar class
This axiom is referenced by:  addcl  10811
  Copyright terms: Public domain W3C validator