ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax-addcl GIF version

Axiom ax-addcl 7038
Description: Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by theorem axaddcl 6998. Proofs should normally use addcl 7064 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 6945 . . . 4 class
31, 2wcel 1409 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1409 . . 3 wff 𝐵 ∈ ℂ
63, 5wa 101 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7 caddc 6950 . . . 4 class +
81, 4, 7co 5540 . . 3 class (𝐴 + 𝐵)
98, 2wcel 1409 . 2 wff (𝐴 + 𝐵) ∈ ℂ
106, 9wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
Colors of variables: wff set class
This axiom is referenced by:  addcl  7064
  Copyright terms: Public domain W3C validator