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Theorem anorabs2 224
 Description: Absorption law for ortholattices.
Assertion
Ref Expression
anorabs2 (a ∩ (b ∪ (a ∩ (bc)))) = (a ∩ (bc))

Proof of Theorem anorabs2
StepHypRef Expression
1 lea 160 . . 3 (a ∩ (b ∪ (a ∩ (bc)))) ≤ a
2 lear 161 . . . 4 (a ∩ (b ∪ (a ∩ (bc)))) ≤ (b ∪ (a ∩ (bc)))
3 leo 158 . . . . 5 b ≤ (bc)
4 lear 161 . . . . 5 (a ∩ (bc)) ≤ (bc)
53, 4lel2or 170 . . . 4 (b ∪ (a ∩ (bc))) ≤ (bc)
62, 5letr 137 . . 3 (a ∩ (b ∪ (a ∩ (bc)))) ≤ (bc)
71, 6ler2an 173 . 2 (a ∩ (b ∪ (a ∩ (bc)))) ≤ (a ∩ (bc))
8 lea 160 . . 3 (a ∩ (bc)) ≤ a
9 leor 159 . . 3 (a ∩ (bc)) ≤ (b ∪ (a ∩ (bc)))
108, 9ler2an 173 . 2 (a ∩ (bc)) ≤ (a ∩ (b ∪ (a ∩ (bc))))
117, 10lebi 145 1 (a ∩ (b ∪ (a ∩ (bc)))) = (a ∩ (bc))
 Colors of variables: term Syntax hints:   = wb 1   ∪ 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:  anorabs  225
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