ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  adddii Unicode version

Theorem adddii 7788
Description: Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.)
Hypotheses
Ref Expression
axi.1  |-  A  e.  CC
axi.2  |-  B  e.  CC
axi.3  |-  C  e.  CC
Assertion
Ref Expression
adddii  |-  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) )

Proof of Theorem adddii
StepHypRef Expression
1 axi.1 . 2  |-  A  e.  CC
2 axi.2 . 2  |-  B  e.  CC
3 axi.3 . 2  |-  C  e.  CC
4 adddi 7764 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
51, 2, 3, 4mp3an 1315 1  |-  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1331    e. wcel 1480  (class class class)co 5774   CCcc 7630    + caddc 7635    x. cmul 7637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-distr 7736
This theorem depends on definitions:  df-bi 116  df-3an 964
This theorem is referenced by:  3t3e9  8889  numltc  9219  numsucc  9233  numma  9237  decmul10add  9262  4t3lem  9290  9t11e99  9323  decbin2  9334  binom2i  10413  3dec  10473  3dvds2dec  11574
  Copyright terms: Public domain W3C validator