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

Axiom ax-distr 8804
Description: Distributive law for complex numbers. Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 8780. Proofs should normally use adddi 8826 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-distr  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )

Detailed syntax breakdown of Axiom ax-distr
StepHypRef Expression
1 cA . . . 4  class  A
2 cc 8735 . . . 4  class  CC
31, 2wcel 1684 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1684 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
76, 2wcel 1684 . . 3  wff  C  e.  CC
83, 5, 7w3a 934 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
9 caddc 8740 . . . . 5  class  +
104, 6, 9co 5858 . . . 4  class  ( B  +  C )
11 cmul 8742 . . . 4  class  x.
121, 10, 11co 5858 . . 3  class  ( A  x.  ( B  +  C ) )
131, 4, 11co 5858 . . . 4  class  ( A  x.  B )
141, 6, 11co 5858 . . . 4  class  ( A  x.  C )
1513, 14, 9co 5858 . . 3  class  ( ( A  x.  B )  +  ( A  x.  C ) )
1612, 15wceq 1623 . 2  wff  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) )
178, 16wi 4 1  wff  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
Colors of variables: wff set class
This axiom is referenced by:  adddi  8826
  Copyright terms: Public domain W3C validator