Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  oml5 GIF version

Theorem oml5 449
 Description: Orthomodular law.
Assertion
Ref Expression
oml5 ((ab) ∪ ((ab) ∩ (bc))) = (bc)

Proof of Theorem oml5
StepHypRef Expression
1 oml 445 . . 3 ((ab) ∪ ((ab) ∩ ((ab) ∪ (bc)))) = ((ab) ∪ (bc))
2 ax-a3 32 . . . . . 6 ((b ∪ (ab)) ∪ c) = (b ∪ ((ab) ∪ c))
3 ancom 74 . . . . . . . . 9 (ab) = (ba)
43lor 70 . . . . . . . 8 (b ∪ (ab)) = (b ∪ (ba))
5 orabs 120 . . . . . . . 8 (b ∪ (ba)) = b
64, 5ax-r2 36 . . . . . . 7 (b ∪ (ab)) = b
76ax-r5 38 . . . . . 6 ((b ∪ (ab)) ∪ c) = (bc)
8 or12 80 . . . . . 6 (b ∪ ((ab) ∪ c)) = ((ab) ∪ (bc))
92, 7, 83tr2 64 . . . . 5 (bc) = ((ab) ∪ (bc))
109lan 77 . . . 4 ((ab) ∩ (bc)) = ((ab) ∩ ((ab) ∪ (bc)))
1110lor 70 . . 3 ((ab) ∪ ((ab) ∩ (bc))) = ((ab) ∪ ((ab) ∩ ((ab) ∪ (bc))))
122, 8ax-r2 36 . . 3 ((b ∪ (ab)) ∪ c) = ((ab) ∪ (bc))
131, 11, 123tr1 63 . 2 ((ab) ∪ ((ab) ∩ (bc))) = ((b ∪ (ab)) ∪ c)
1413, 7ax-r2 36 1 ((ab) ∪ ((ab) ∩ (bc))) = (bc)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ 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  ax-r3 439 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42 This theorem is referenced by:  i3th1  543
 Copyright terms: Public domain W3C validator