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

Axiom ax-addass 7942
Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 7900. Proofs should normally use addass 7970 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-addass  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  +  C )  =  ( A  +  ( B  +  C
) ) )

Detailed syntax breakdown of Axiom ax-addass
StepHypRef Expression
1 cA . . . 4  class  A
2 cc 7838 . . . 4  class  CC
31, 2wcel 2160 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 2160 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
76, 2wcel 2160 . . 3  wff  C  e.  CC
83, 5, 7w3a 980 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
9 caddc 7843 . . . . 5  class  +
101, 4, 9co 5895 . . . 4  class  ( A  +  B )
1110, 6, 9co 5895 . . 3  class  ( ( A  +  B )  +  C )
124, 6, 9co 5895 . . . 4  class  ( B  +  C )
131, 12, 9co 5895 . . 3  class  ( A  +  ( B  +  C ) )
1411, 13wceq 1364 . 2  wff  ( ( A  +  B )  +  C )  =  ( A  +  ( B  +  C ) )
158, 14wi 4 1  wff  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  +  C )  =  ( A  +  ( B  +  C
) ) )
Colors of variables: wff set class
This axiom is referenced by:  addass  7970
  Copyright terms: Public domain W3C validator