QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  distlem Unicode version
Theorem distlem 188
Description: Distributive law inference (uses OL only).
Hypothesis
Ref Expression
distlem.1 (a ^ (b v c)) =< b
Assertion
Ref Expression
distlem (a ^ (b v c)) = ((a ^ b) v (a ^ c))
Proof of Theorem distlem
StepHypRef Expression
1 lea 160 . . . 4 (a ^ (b v c)) =< a
2 distlem.1 . . . 4 (a ^ (b v c)) =< b
31, 2ler2an 173 . . 3 (a ^ (b v c)) =< (a ^ b)
4 leo 158 . . 3 (a ^ b) =< ((a ^ b) v (a ^ c))
53, 4letr 137 . 2 (a ^ (b v c)) =< ((a ^ b) v (a ^ c))
6 ledi 174 . 2 ((a ^ b) v (a ^ c)) =< (a ^ (b v c))
75, 6lebi 145 1 (a ^ (b v c)) = ((a ^ b) v (a ^ c))
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2   v wo 6   ^ wa 7
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131
This theorem is referenced by:  oadist2a  1007
  Copyright terms: Public domain W3C validator