Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  lem3.3.7i2e1 GIF version

Theorem lem3.3.7i2e1 1063
 Description: Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to 2, and this is the first part of the equation. (Contributed by Roy F. Longton, 3-Jul-05.)
Assertion
Ref Expression
lem3.3.7i2e1 (a2 (ab)) = (a2 (ab))

Proof of Theorem lem3.3.7i2e1
StepHypRef Expression
1 or1r 105 . . . . . 6 (1 ∪ b ) = 1
21ax-r1 35 . . . . 5 1 = (1 ∪ b )
32ran 78 . . . 4 (1 ∩ ((ab) ∪ (a ∩ (ab) ))) = ((1 ∪ b ) ∩ ((ab) ∪ (a ∩ (ab) )))
4 an1r 107 . . . 4 (1 ∩ ((ab) ∪ (a ∩ (ab) ))) = ((ab) ∪ (a ∩ (ab) ))
5 df-t 41 . . . . . 6 1 = (aa )
65ax-r5 38 . . . . 5 (1 ∪ b ) = ((aa ) ∪ b )
76ran 78 . . . 4 ((1 ∪ b ) ∩ ((ab) ∪ (a ∩ (ab) ))) = (((aa ) ∪ b ) ∩ ((ab) ∪ (a ∩ (ab) )))
83, 4, 73tr2 64 . . 3 ((ab) ∪ (a ∩ (ab) )) = (((aa ) ∪ b ) ∩ ((ab) ∪ (a ∩ (ab) )))
9 ax-a3 32 . . . 4 ((aa ) ∪ b ) = (a ∪ (ab ))
109ran 78 . . 3 (((aa ) ∪ b ) ∩ ((ab) ∪ (a ∩ (ab) ))) = ((a ∪ (ab )) ∩ ((ab) ∪ (a ∩ (ab) )))
11 oran3 93 . . . . 5 (ab ) = (ab)
1211lor 70 . . . 4 (a ∪ (ab )) = (a ∪ (ab) )
1312ran 78 . . 3 ((a ∪ (ab )) ∩ ((ab) ∪ (a ∩ (ab) ))) = ((a ∪ (ab) ) ∩ ((ab) ∪ (a ∩ (ab) )))
148, 10, 133tr 65 . 2 ((ab) ∪ (a ∩ (ab) )) = ((a ∪ (ab) ) ∩ ((ab) ∪ (a ∩ (ab) )))
15 df-i2 45 . 2 (a2 (ab)) = ((ab) ∪ (a ∩ (ab) ))
16 df-id2 51 . 2 (a2 (ab)) = ((a ∪ (ab) ) ∩ ((ab) ∪ (a ∩ (ab) )))
1714, 15, 163tr1 63 1 (a2 (ab)) = (a2 (ab))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →2 wi2 13   ≡2 wid2 19 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 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i2 45  df-id2 51 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator