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Axiom ax-distr 8849
Description: Distributive law for complex numbers. Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 8825. Proofs should normally use adddi 8871 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 8780 . . . 4  class  CC
31, 2wcel 1701 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1701 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
76, 2wcel 1701 . . 3  wff  C  e.  CC
83, 5, 7w3a 934 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
9 caddc 8785 . . . . 5  class  +
104, 6, 9co 5900 . . . 4  class  ( B  +  C )
11 cmul 8787 . . . 4  class  x.
121, 10, 11co 5900 . . 3  class  ( A  x.  ( B  +  C ) )
131, 4, 11co 5900 . . . 4  class  ( A  x.  B )
141, 6, 11co 5900 . . . 4  class  ( A  x.  C )
1513, 14, 9co 5900 . . 3  class  ( ( A  x.  B )  +  ( A  x.  C ) )
1612, 15wceq 1633 . 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  8871
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