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Theorem nom24 317
 Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom24 (a4 (ab)) = (a1 b)

Proof of Theorem nom24
StepHypRef Expression
1 leo 158 . . . . 5 a ≤ (ab )
21leror 152 . . . 4 (a ∪ (ab)) ≤ ((ab ) ∪ (ab))
3 oran3 93 . . . . 5 (ab ) = (ab)
4 anidm 111 . . . . . . . 8 (aa) = a
54ran 78 . . . . . . 7 ((aa) ∩ b) = (ab)
65ax-r1 35 . . . . . 6 (ab) = ((aa) ∩ b)
7 anass 76 . . . . . 6 ((aa) ∩ b) = (a ∩ (ab))
86, 7ax-r2 36 . . . . 5 (ab) = (a ∩ (ab))
93, 82or 72 . . . 4 ((ab ) ∪ (ab)) = ((ab) ∪ (a ∩ (ab)))
102, 9lbtr 139 . . 3 (a ∪ (ab)) ≤ ((ab) ∪ (a ∩ (ab)))
1110df2le2 136 . 2 ((a ∪ (ab)) ∩ ((ab) ∪ (a ∩ (ab)))) = (a ∪ (ab))
12 df-id4 53 . 2 (a4 (ab)) = ((a ∪ (ab)) ∩ ((ab) ∪ (a ∩ (ab))))
13 df-i1 44 . 2 (a1 b) = (a ∪ (ab))
1411, 12, 133tr1 63 1 (a4 (ab)) = (a1 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ 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
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