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

Axiom ax-addcl 9042
Description: Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 9018. Proofs should normally use addcl 9064 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  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )

Detailed syntax breakdown of Axiom ax-addcl
StepHypRef Expression
1 cA . . . 4  class  A
2 cc 8980 . . . 4  class  CC
31, 2wcel 1725 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1725 . . 3  wff  B  e.  CC
63, 5wa 359 . 2  wff  ( A  e.  CC  /\  B  e.  CC )
7 caddc 8985 . . . 4  class  +
81, 4, 7co 6073 . . 3  class  ( A  +  B )
98, 2wcel 1725 . 2  wff  ( A  +  B )  e.  CC
106, 9wi 4 1  wff  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
Colors of variables: wff set class
This axiom is referenced by:  addcl  9064
  Copyright terms: Public domain W3C validator