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Axiom ax-distr 9062
Description: Distributive law for complex numbers. Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 9038. Proofs should normally use adddi 9084 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 8993 . . . 4  class  CC
31, 2wcel 1726 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1726 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
76, 2wcel 1726 . . 3  wff  C  e.  CC
83, 5, 7w3a 937 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
9 caddc 8998 . . . . 5  class  +
104, 6, 9co 6084 . . . 4  class  ( B  +  C )
11 cmul 9000 . . . 4  class  x.
121, 10, 11co 6084 . . 3  class  ( A  x.  ( B  +  C ) )
131, 4, 11co 6084 . . . 4  class  ( A  x.  B )
141, 6, 11co 6084 . . . 4  class  ( A  x.  C )
1513, 14, 9co 6084 . . 3  class  ( ( A  x.  B )  +  ( A  x.  C ) )
1612, 15wceq 1653 . 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  9084
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