QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  oi3ai3 GIF version

Theorem oi3ai3 503
Description: Theorem for Kalmbach implication. (Contributed by NM, 9-Nov-1997.)
Assertion
Ref Expression
oi3ai3 ((ab) ∪ (a3 b) ) = ((ab) ∩ (a3 b ))

Proof of Theorem oi3ai3
StepHypRef Expression
1 lea 160 . . . . . 6 (ab) ≤ a
2 leo 158 . . . . . 6 a ≤ (ab)
31, 2letr 137 . . . . 5 (ab) ≤ (ab)
43lecom 180 . . . 4 (ab) C (ab)
5 coman1 185 . . . . . 6 (ab) C a
6 ancom 74 . . . . . . . 8 (ab) = (ba)
7 coman1 185 . . . . . . . 8 (ba) C b
86, 7bctr 181 . . . . . . 7 (ab) C b
98comcom2 183 . . . . . 6 (ab) C b
105, 9com2an 484 . . . . 5 (ab) C (ab )
115comcom2 183 . . . . . 6 (ab) C a
125, 9com2or 483 . . . . . 6 (ab) C (ab )
1311, 12com2an 484 . . . . 5 (ab) C (a ∩ (ab ))
1410, 13com2or 483 . . . 4 (ab) C ((ab ) ∪ (a ∩ (ab )))
154, 14fh3 471 . . 3 ((ab) ∪ ((ab) ∩ ((ab ) ∪ (a ∩ (ab ))))) = (((ab) ∪ (ab)) ∩ ((ab) ∪ ((ab ) ∪ (a ∩ (ab )))))
163df-le2 131 . . . 4 ((ab) ∪ (ab)) = (ab)
17 ax-a3 32 . . . . . 6 (((ab) ∪ (ab )) ∪ (a ∩ (ab ))) = ((ab) ∪ ((ab ) ∪ (a ∩ (ab ))))
1817ax-r1 35 . . . . 5 ((ab) ∪ ((ab ) ∪ (a ∩ (ab )))) = (((ab) ∪ (ab )) ∪ (a ∩ (ab )))
19 ax-a2 31 . . . . . 6 ((ab) ∪ (ab )) = ((ab ) ∪ (ab))
2019ax-r5 38 . . . . 5 (((ab) ∪ (ab )) ∪ (a ∩ (ab ))) = (((ab ) ∪ (ab)) ∪ (a ∩ (ab )))
2118, 20ax-r2 36 . . . 4 ((ab) ∪ ((ab ) ∪ (a ∩ (ab )))) = (((ab ) ∪ (ab)) ∪ (a ∩ (ab )))
2216, 212an 79 . . 3 (((ab) ∪ (ab)) ∩ ((ab) ∪ ((ab ) ∪ (a ∩ (ab ))))) = ((ab) ∩ (((ab ) ∪ (ab)) ∪ (a ∩ (ab ))))
2315, 22ax-r2 36 . 2 ((ab) ∪ ((ab) ∩ ((ab ) ∪ (a ∩ (ab ))))) = ((ab) ∩ (((ab ) ∪ (ab)) ∪ (a ∩ (ab ))))
24 ni32 502 . . 3 (a3 b) = ((ab) ∩ ((ab ) ∪ (a ∩ (ab ))))
2524lor 70 . 2 ((ab) ∪ (a3 b) ) = ((ab) ∪ ((ab) ∩ ((ab ) ∪ (a ∩ (ab )))))
26 i3n1 249 . . 3 (a3 b ) = (((ab ) ∪ (ab)) ∪ (a ∩ (ab )))
2726lan 77 . 2 ((ab) ∩ (a3 b )) = ((ab) ∩ (((ab ) ∪ (ab)) ∪ (a ∩ (ab ))))
2823, 25, 273tr1 63 1 ((ab) ∪ (a3 b) ) = ((ab) ∩ (a3 b ))
Colors of variables: term
Syntax hints:   = wb 1   wn 4  wo 6  wa 7  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:  i3orlem6  557
  Copyright terms: Public domain W3C validator