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Axiom ax-distr 8800
Description: Distributive law for complex numbers. Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 8776. Proofs should normally use adddi 8822 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 8731 . . . 4  class  CC
31, 2wcel 1685 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1685 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
76, 2wcel 1685 . . 3  wff  C  e.  CC
83, 5, 7w3a 936 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
9 caddc 8736 . . . . 5  class  +
104, 6, 9co 5820 . . . 4  class  ( B  +  C )
11 cmul 8738 . . . 4  class  x.
121, 10, 11co 5820 . . 3  class  ( A  x.  ( B  +  C ) )
131, 4, 11co 5820 . . . 4  class  ( A  x.  B )
141, 6, 11co 5820 . . . 4  class  ( A  x.  C )
1513, 14, 9co 5820 . . 3  class  ( ( A  x.  B )  +  ( A  x.  C ) )
1612, 15wceq 1624 . 2  wff  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) )
178, 16wi 6 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  8822
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