Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  2oath1 GIF version

Theorem 2oath1 826
 Description: OA-like theorem with →2 instead of →0 . (Contributed by NM, 15-Nov-1998.)
Assertion
Ref Expression
2oath1 ((a2 b) ∩ ((bc) →2 ((a2 b) ∩ (a2 c)))) = ((a2 b) ∩ (a2 c))

Proof of Theorem 2oath1
StepHypRef Expression
1 df-i2 45 . . 3 ((bc) →2 ((a2 b) ∩ (a2 c))) = (((a2 b) ∩ (a2 c)) ∪ ((bc) ∩ ((a2 b) ∩ (a2 c)) ))
21lan 77 . 2 ((a2 b) ∩ ((bc) →2 ((a2 b) ∩ (a2 c)))) = ((a2 b) ∩ (((a2 b) ∩ (a2 c)) ∪ ((bc) ∩ ((a2 b) ∩ (a2 c)) )))
3 coman1 185 . . 3 ((a2 b) ∩ (a2 c)) C (a2 b)
4 comorr2 463 . . . . 5 ((a2 b) ∩ (a2 c)) C ((bc) ∪ ((a2 b) ∩ (a2 c)))
54comcom2 183 . . . 4 ((a2 b) ∩ (a2 c)) C ((bc) ∪ ((a2 b) ∩ (a2 c)))
6 anor3 90 . . . . 5 ((bc) ∩ ((a2 b) ∩ (a2 c)) ) = ((bc) ∪ ((a2 b) ∩ (a2 c)))
76ax-r1 35 . . . 4 ((bc) ∪ ((a2 b) ∩ (a2 c))) = ((bc) ∩ ((a2 b) ∩ (a2 c)) )
85, 7cbtr 182 . . 3 ((a2 b) ∩ (a2 c)) C ((bc) ∩ ((a2 b) ∩ (a2 c)) )
93, 8fh2 470 . 2 ((a2 b) ∩ (((a2 b) ∩ (a2 c)) ∪ ((bc) ∩ ((a2 b) ∩ (a2 c)) ))) = (((a2 b) ∩ ((a2 b) ∩ (a2 c))) ∪ ((a2 b) ∩ ((bc) ∩ ((a2 b) ∩ (a2 c)) )))
10 anass 76 . . . . . 6 (((a2 b) ∩ (a2 b)) ∩ (a2 c)) = ((a2 b) ∩ ((a2 b) ∩ (a2 c)))
1110ax-r1 35 . . . . 5 ((a2 b) ∩ ((a2 b) ∩ (a2 c))) = (((a2 b) ∩ (a2 b)) ∩ (a2 c))
12 anidm 111 . . . . . 6 ((a2 b) ∩ (a2 b)) = (a2 b)
1312ran 78 . . . . 5 (((a2 b) ∩ (a2 b)) ∩ (a2 c)) = ((a2 b) ∩ (a2 c))
1411, 13ax-r2 36 . . . 4 ((a2 b) ∩ ((a2 b) ∩ (a2 c))) = ((a2 b) ∩ (a2 c))
15 oran 87 . . . . . . . . 9 ((bc) ∪ ((a2 b) ∩ (a2 c))) = ((bc) ∩ ((a2 b) ∩ (a2 c)) )
1615lor 70 . . . . . . . 8 ((a2 b) ∪ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = ((a2 b) ∪ ((bc) ∩ ((a2 b) ∩ (a2 c)) ) )
1716ax-r1 35 . . . . . . 7 ((a2 b) ∪ ((bc) ∩ ((a2 b) ∩ (a2 c)) ) ) = ((a2 b) ∪ ((bc) ∪ ((a2 b) ∩ (a2 c))))
18 2oalem1 825 . . . . . . 7 ((a2 b) ∪ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = 1
1917, 18ax-r2 36 . . . . . 6 ((a2 b) ∪ ((bc) ∩ ((a2 b) ∩ (a2 c)) ) ) = 1
2019ax-r4 37 . . . . 5 ((a2 b) ∪ ((bc) ∩ ((a2 b) ∩ (a2 c)) ) ) = 1
21 df-a 40 . . . . 5 ((a2 b) ∩ ((bc) ∩ ((a2 b) ∩ (a2 c)) )) = ((a2 b) ∪ ((bc) ∩ ((a2 b) ∩ (a2 c)) ) )
22 df-f 42 . . . . 5 0 = 1
2320, 21, 223tr1 63 . . . 4 ((a2 b) ∩ ((bc) ∩ ((a2 b) ∩ (a2 c)) )) = 0
2414, 232or 72 . . 3 (((a2 b) ∩ ((a2 b) ∩ (a2 c))) ∪ ((a2 b) ∩ ((bc) ∩ ((a2 b) ∩ (a2 c)) ))) = (((a2 b) ∩ (a2 c)) ∪ 0)
25 or0 102 . . 3 (((a2 b) ∩ (a2 c)) ∪ 0) = ((a2 b) ∩ (a2 c))
2624, 25ax-r2 36 . 2 (((a2 b) ∩ ((a2 b) ∩ (a2 c))) ∪ ((a2 b) ∩ ((bc) ∩ ((a2 b) ∩ (a2 c)) ))) = ((a2 b) ∩ (a2 c))
272, 9, 263tr 65 1 ((a2 b) ∩ ((bc) →2 ((a2 b) ∩ (a2 c)))) = ((a2 b) ∩ (a2 c))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →2 wi2 13 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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  2oath1i1  827  oale  829  oaeqv  830  distoa  944
 Copyright terms: Public domain W3C validator