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

Axiom ax-addass 7043
Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axaddass 7003. Proofs should normally use addass 7068 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 6944 . . . 4  class  CC
31, 2wcel 1409 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1409 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
76, 2wcel 1409 . . 3  wff  C  e.  CC
83, 5, 7w3a 896 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
9 caddc 6949 . . . . 5  class  +
101, 4, 9co 5539 . . . 4  class  ( A  +  B )
1110, 6, 9co 5539 . . 3  class  ( ( A  +  B )  +  C )
124, 6, 9co 5539 . . . 4  class  ( B  +  C )
131, 12, 9co 5539 . . 3  class  ( A  +  ( B  +  C ) )
1411, 13wceq 1259 . 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  7068
  Copyright terms: Public domain W3C validator