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

 Description: Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by theorem axaddrcl 10011. Proofs should normally use readdcl 10057 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-addrcl ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)

Detailed syntax breakdown of Axiom ax-addrcl
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cr 9973 . . . 4 class
31, 2wcel 2030 . . 3 wff 𝐴 ∈ ℝ
4 cB . . . 4 class 𝐵
54, 2wcel 2030 . . 3 wff 𝐵 ∈ ℝ
63, 5wa 383 . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)
7 caddc 9977 . . . 4 class +
81, 4, 7co 6690 . . 3 class (𝐴 + 𝐵)
98, 2wcel 2030 . 2 wff (𝐴 + 𝐵) ∈ ℝ
106, 9wi 4 1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
 Colors of variables: wff setvar class This axiom is referenced by:  readdcl  10057
 Copyright terms: Public domain W3C validator