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Axiom ax-distr 9013
Description: Distributive law for complex numbers. Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 8989. Proofs should normally use adddi 9035 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 8944 . . . 4  class  CC
31, 2wcel 1721 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1721 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
76, 2wcel 1721 . . 3  wff  C  e.  CC
83, 5, 7w3a 936 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
9 caddc 8949 . . . . 5  class  +
104, 6, 9co 6040 . . . 4  class  ( B  +  C )
11 cmul 8951 . . . 4  class  x.
121, 10, 11co 6040 . . 3  class  ( A  x.  ( B  +  C ) )
131, 4, 11co 6040 . . . 4  class  ( A  x.  B )
141, 6, 11co 6040 . . . 4  class  ( A  x.  C )
1513, 14, 9co 6040 . . 3  class  ( ( A  x.  B )  +  ( A  x.  C ) )
1612, 15wceq 1649 . 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  9035
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