QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  nom24 Unicode version
Theorem nom24 317
Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom24 (a ==4 (a ^ b)) = (a ->1 b)
Proof of Theorem nom24
StepHypRef Expression
1 leo 158 . . . . 5 a' =< (a' v b')
21leror 152 . . . 4 (a' v (a ^ b)) =< ((a' v b') v (a ^ b))
3 oran3 93 . . . . 5 (a' v b') = (a ^ b)'
4 anidm 111 . . . . . . . 8 (a ^ a) = a
54ran 78 . . . . . . 7 ((a ^ a) ^ b) = (a ^ b)
65ax-r1 35 . . . . . 6 (a ^ b) = ((a ^ a) ^ b)
7 anass 76 . . . . . 6 ((a ^ a) ^ b) = (a ^ (a ^ b))
86, 7ax-r2 36 . . . . 5 (a ^ b) = (a ^ (a ^ b))
93, 82or 72 . . . 4 ((a' v b') v (a ^ b)) = ((a ^ b)' v (a ^ (a ^ b)))
102, 9lbtr 139 . . 3 (a' v (a ^ b)) =< ((a ^ b)' v (a ^ (a ^ b)))
1110df2le2 136 . 2 ((a' v (a ^ b)) ^ ((a ^ b)' v (a ^ (a ^ b)))) = (a' v (a ^ b))
12 df-id4 53 . 2 (a ==4 (a ^ b)) = ((a' v (a ^ b)) ^ ((a ^ b)' v (a ^ (a ^ b))))
13 df-i1 44 . 2 (a ->1 b) = (a' v (a ^ b))
1411, 12, 133tr1 63 1 (a ==4 (a ^ b)) = (a ->1 b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12   ==4 wid4 21
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-i1 44  df-id4 53  df-le1 130  df-le2 131
This theorem is referenced by:  nom31  320  nom53  334
  Copyright terms: Public domain W3C validator