Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  gomaex3lem4 GIF version

Theorem gomaex3lem4 917
 Description: Lemma for Godowski 6-var -> Mayet Example 3.
Hypothesis
Ref Expression
gomaex3lem4.9 p = ((ab) →1 (de) )
Assertion
Ref Expression
gomaex3lem4 ((ab) ∩ (de) ) ≤ p

Proof of Theorem gomaex3lem4
StepHypRef Expression
1 leor 159 . 2 ((ab) ∩ (de) ) ≤ ((ab) ∪ ((ab) ∩ (de) ))
2 ax-a1 30 . . 3 ((ab) →1 (de) ) = ((ab) →1 (de) )
3 df-i1 44 . . . 4 ((ab) →1 (de) ) = ((ab) ∪ ((ab) ∩ (de) ))
43ax-r1 35 . . 3 ((ab) ∪ ((ab) ∩ (de) )) = ((ab) →1 (de) )
5 gomaex3lem4.9 . . . 4 p = ((ab) →1 (de) )
65ax-r4 37 . . 3 p = ((ab) →1 (de) )
72, 4, 63tr1 63 . 2 ((ab) ∪ ((ab) ∩ (de) )) = p
81, 7lbtr 139 1 ((ab) ∩ (de) ) ≤ p
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-a 40  df-i1 44  df-le1 130  df-le2 131 This theorem is referenced by:  gomaex3lem9  922
 Copyright terms: Public domain W3C validator