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Theorem nom13 310
 Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom13 (a3 (ab)) = (a1 b)

Proof of Theorem nom13
StepHypRef Expression
1 oran 87 . . . . . . . . 9 (a ∪ (ab)) = (a ∩ (ab) )
21ax-r1 35 . . . . . . . 8 (a ∩ (ab) ) = (a ∪ (ab))
3 orabs 120 . . . . . . . 8 (a ∪ (ab)) = a
42, 3ax-r2 36 . . . . . . 7 (a ∩ (ab) ) = a
54con3 68 . . . . . 6 (a ∩ (ab) ) = a
65lor 70 . . . . 5 ((a ∩ (ab)) ∪ (a ∩ (ab) )) = ((a ∩ (ab)) ∪ a )
7 lea 160 . . . . . 6 (a ∩ (ab)) ≤ a
87df-le2 131 . . . . 5 ((a ∩ (ab)) ∪ a ) = a
96, 8ax-r2 36 . . . 4 ((a ∩ (ab)) ∪ (a ∩ (ab) )) = a
109ax-r5 38 . . 3 (((a ∩ (ab)) ∪ (a ∩ (ab) )) ∪ (a ∩ (a ∪ (ab)))) = (a ∪ (a ∩ (a ∪ (ab))))
11 womaa 222 . . 3 (a ∪ (a ∩ (a ∪ (ab)))) = (a ∪ (ab))
1210, 11ax-r2 36 . 2 (((a ∩ (ab)) ∪ (a ∩ (ab) )) ∪ (a ∩ (a ∪ (ab)))) = (a ∪ (ab))
13 df-i3 46 . 2 (a3 (ab)) = (((a ∩ (ab)) ∪ (a ∩ (ab) )) ∪ (a ∩ (a ∪ (ab))))
14 df-i1 44 . 2 (a1 b) = (a ∪ (ab))
1512, 13, 143tr1 63 1 (a3 (ab)) = (a1 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12   →3 wi3 14 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-i3 46  df-le1 130  df-le2 131 This theorem is referenced by:  nom44  329  lem3.3.7i3e3  1068
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