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Theorem oml5a 450
 Description: Orthomodular law. (Contributed by NM, 16-Nov-1997.)
Assertion
Ref Expression
oml5a ((ab) ∩ ((ab) ∪ (bc))) = (bc)

Proof of Theorem oml5a
StepHypRef Expression
1 omla 447 . . 3 ((ab) ∩ ((ab) ∪ ((ab) ∩ (bc)))) = ((ab) ∩ (bc))
2 anass 76 . . . . . 6 ((b ∩ (ab)) ∩ c) = (b ∩ ((ab) ∩ c))
3 ax-a2 31 . . . . . . . . 9 (ab) = (ba)
43lan 77 . . . . . . . 8 (b ∩ (ab)) = (b ∩ (ba))
5 anabs 121 . . . . . . . 8 (b ∩ (ba)) = b
64, 5ax-r2 36 . . . . . . 7 (b ∩ (ab)) = b
76ran 78 . . . . . 6 ((b ∩ (ab)) ∩ c) = (bc)
8 an12 81 . . . . . 6 (b ∩ ((ab) ∩ c)) = ((ab) ∩ (bc))
92, 7, 83tr2 64 . . . . 5 (bc) = ((ab) ∩ (bc))
109lor 70 . . . 4 ((ab) ∪ (bc)) = ((ab) ∪ ((ab) ∩ (bc)))
1110lan 77 . . 3 ((ab) ∩ ((ab) ∪ (bc))) = ((ab) ∩ ((ab) ∪ ((ab) ∩ (bc))))
122, 8ax-r2 36 . . 3 ((b ∩ (ab)) ∩ c) = ((ab) ∩ (bc))
131, 11, 123tr1 63 . 2 ((ab) ∩ ((ab) ∪ (bc))) = ((b ∩ (ab)) ∩ c)
1413, 7ax-r2 36 1 ((ab) ∩ ((ab) ∪ (bc))) = (bc)
 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  ax-r3 439 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42 This theorem is referenced by: (None)
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