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Axiom ax-distr 9047
Description: Distributive law for complex numbers. Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 9023. Proofs should normally use adddi 9069 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 8978 . . . 4  class  CC
31, 2wcel 1725 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1725 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
76, 2wcel 1725 . . 3  wff  C  e.  CC
83, 5, 7w3a 936 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
9 caddc 8983 . . . . 5  class  +
104, 6, 9co 6073 . . . 4  class  ( B  +  C )
11 cmul 8985 . . . 4  class  x.
121, 10, 11co 6073 . . 3  class  ( A  x.  ( B  +  C ) )
131, 4, 11co 6073 . . . 4  class  ( A  x.  B )
141, 6, 11co 6073 . . . 4  class  ( A  x.  C )
1513, 14, 9co 6073 . . 3  class  ( ( A  x.  B )  +  ( A  x.  C ) )
1612, 15wceq 1652 . 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  9069
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