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Axiom ax-distr 9095
Description: Distributive law for complex numbers. Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 9071. Proofs should normally use adddi 9117 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 9026 . . . 4  class  CC
31, 2wcel 1728 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1728 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
76, 2wcel 1728 . . 3  wff  C  e.  CC
83, 5, 7w3a 937 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
9 caddc 9031 . . . . 5  class  +
104, 6, 9co 6117 . . . 4  class  ( B  +  C )
11 cmul 9033 . . . 4  class  x.
121, 10, 11co 6117 . . 3  class  ( A  x.  ( B  +  C ) )
131, 4, 11co 6117 . . . 4  class  ( A  x.  B )
141, 6, 11co 6117 . . . 4  class  ( A  x.  C )
1513, 14, 9co 6117 . . 3  class  ( ( A  x.  B )  +  ( A  x.  C ) )
1612, 15wceq 1654 . 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  9117
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