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Theorem oml6 488
 Description: Orthomodular law.
Assertion
Ref Expression
oml6 (a ∪ (b ∩ (ab ))) = (ab)

Proof of Theorem oml6
StepHypRef Expression
1 comor1 461 . . . 4 (ab ) C a
21comcom7 460 . . 3 (ab ) C a
3 comor2 462 . . . 4 (ab ) C b
43comcom7 460 . . 3 (ab ) C b
52, 4fh4c 478 . 2 (a ∪ (b ∩ (ab ))) = ((ab) ∩ (a ∪ (ab )))
6 df-t 41 . . . . . 6 1 = (aa )
76ax-r5 38 . . . . 5 (1 ∪ b ) = ((aa ) ∪ b )
8 ax-a2 31 . . . . . 6 (1 ∪ b ) = (b ∪ 1)
9 or1 104 . . . . . 6 (b ∪ 1) = 1
108, 9ax-r2 36 . . . . 5 (1 ∪ b ) = 1
11 ax-a3 32 . . . . 5 ((aa ) ∪ b ) = (a ∪ (ab ))
127, 10, 113tr2 64 . . . 4 1 = (a ∪ (ab ))
1312ax-r1 35 . . 3 (a ∪ (ab )) = 1
1413lan 77 . 2 ((ab) ∩ (a ∪ (ab ))) = ((ab) ∩ 1)
15 an1 106 . 2 ((ab) ∩ 1) = (ab)
165, 14, 153tr 65 1 (a ∪ (b ∩ (ab ))) = (ab)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  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  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  sa5  836
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