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Theorem womaan 223
 Description: Weak OM-like absorption law for ortholattices. (Contributed by NM, 8-Nov-1998.)
Assertion
Ref Expression
womaan (a ∪ (a ∩ (a ∪ (ab)))) = (a ∪ (ab))

Proof of Theorem womaan
StepHypRef Expression
1 leo 158 . . 3 a ≤ (a ∪ (ab))
2 lear 161 . . 3 (a ∩ (a ∪ (ab))) ≤ (a ∪ (ab))
31, 2lel2or 170 . 2 (a ∪ (a ∩ (a ∪ (ab)))) ≤ (a ∪ (ab))
4 leo 158 . . 3 a ≤ (a ∪ (a ∩ (a ∪ (ab))))
5 lea 160 . . . . 5 (ab) ≤ a
6 leor 159 . . . . 5 (ab) ≤ (a ∪ (ab))
75, 6ler2an 173 . . . 4 (ab) ≤ (a ∩ (a ∪ (ab)))
8 leor 159 . . . 4 (a ∩ (a ∪ (ab))) ≤ (a ∪ (a ∩ (a ∪ (ab))))
97, 8letr 137 . . 3 (ab) ≤ (a ∪ (a ∩ (a ∪ (ab))))
104, 9lel2or 170 . 2 (a ∪ (ab)) ≤ (a ∪ (a ∩ (a ∪ (ab))))
113, 10lebi 145 1 (a ∪ (a ∩ (a ∪ (ab)))) = (a ∪ (ab))
 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 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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