Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  i3th1 GIF version

Theorem i3th1 543
 Description: Theorem for Kalmbach implication.
Assertion
Ref Expression
i3th1 (a3 (a3 (b3 a))) = 1

Proof of Theorem i3th1
StepHypRef Expression
1 df2i3 498 . . 3 (b3 a) = ((ba ) ∪ ((ba) ∩ (b ∪ (ba))))
21lor 70 . 2 (a ∪ (b3 a)) = (a ∪ ((ba ) ∪ ((ba) ∩ (b ∪ (ba)))))
3 lem4 511 . 2 (a3 (a3 (b3 a))) = (a ∪ (b3 a))
4 ax-a3 32 . . . . 5 ((ab) ∪ (ab )) = (a ∪ (b ∪ (ab )))
5 anor1 88 . . . . . 6 (ab ) = (ab)
65lor 70 . . . . 5 ((ab) ∪ (ab )) = ((ab) ∪ (ab) )
7 ax-a3 32 . . . . . . . 8 ((a ∪ (ab)) ∪ ((ba) ∩ (b ∪ (ba)))) = (a ∪ ((ab) ∪ ((ba) ∩ (b ∪ (ba)))))
8 ax-a2 31 . . . . . . . . . . . . 13 (ba) = (ab )
9 anor2 89 . . . . . . . . . . . . . . 15 (ab) = (ab )
109con2 67 . . . . . . . . . . . . . 14 (ab) = (ab )
1110ax-r1 35 . . . . . . . . . . . . 13 (ab ) = (ab)
128, 11ax-r2 36 . . . . . . . . . . . 12 (ba) = (ab)
13 ancom 74 . . . . . . . . . . . . 13 (ba) = (ab )
1413lor 70 . . . . . . . . . . . 12 (b ∪ (ba)) = (b ∪ (ab ))
1512, 142an 79 . . . . . . . . . . 11 ((ba) ∩ (b ∪ (ba))) = ((ab) ∩ (b ∪ (ab )))
1615lor 70 . . . . . . . . . 10 ((ab) ∪ ((ba) ∩ (b ∪ (ba)))) = ((ab) ∪ ((ab) ∩ (b ∪ (ab ))))
17 oml5 449 . . . . . . . . . 10 ((ab) ∪ ((ab) ∩ (b ∪ (ab )))) = (b ∪ (ab ))
1816, 17ax-r2 36 . . . . . . . . 9 ((ab) ∪ ((ba) ∩ (b ∪ (ba)))) = (b ∪ (ab ))
1918lor 70 . . . . . . . 8 (a ∪ ((ab) ∪ ((ba) ∩ (b ∪ (ba))))) = (a ∪ (b ∪ (ab )))
207, 19ax-r2 36 . . . . . . 7 ((a ∪ (ab)) ∪ ((ba) ∩ (b ∪ (ba)))) = (a ∪ (b ∪ (ab )))
2120ax-r1 35 . . . . . 6 (a ∪ (b ∪ (ab ))) = ((a ∪ (ab)) ∪ ((ba) ∩ (b ∪ (ba))))
22 orabs 120 . . . . . . 7 (a ∪ (ab)) = a
2322ax-r5 38 . . . . . 6 ((a ∪ (ab)) ∪ ((ba) ∩ (b ∪ (ba)))) = (a ∪ ((ba) ∩ (b ∪ (ba))))
2421, 23ax-r2 36 . . . . 5 (a ∪ (b ∪ (ab ))) = (a ∪ ((ba) ∩ (b ∪ (ba))))
254, 6, 243tr2 64 . . . 4 ((ab) ∪ (ab) ) = (a ∪ ((ba) ∩ (b ∪ (ba))))
26 df-t 41 . . . 4 1 = ((ab) ∪ (ab) )
27 ancom 74 . . . . . . 7 (ba ) = (ab )
2827lor 70 . . . . . 6 (a ∪ (ba )) = (a ∪ (ab ))
29 orabs 120 . . . . . 6 (a ∪ (ab )) = a
3028, 29ax-r2 36 . . . . 5 (a ∪ (ba )) = a
3130ax-r5 38 . . . 4 ((a ∪ (ba )) ∪ ((ba) ∩ (b ∪ (ba)))) = (a ∪ ((ba) ∩ (b ∪ (ba))))
3225, 26, 313tr1 63 . . 3 1 = ((a ∪ (ba )) ∪ ((ba) ∩ (b ∪ (ba))))
33 ax-a3 32 . . 3 ((a ∪ (ba )) ∪ ((ba) ∩ (b ∪ (ba)))) = (a ∪ ((ba ) ∪ ((ba) ∩ (b ∪ (ba)))))
3432, 33ax-r2 36 . 2 1 = (a ∪ ((ba ) ∪ ((ba) ∩ (b ∪ (ba)))))
352, 3, 343tr1 63 1 (a3 (a3 (b3 a))) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →3 wi3 14 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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  u3lem14aa  792
 Copyright terms: Public domain W3C validator