Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  dp41lemg GIF version

Theorem dp41lemg 1187
 Description: Part of proof (4)=>(1) in Day/Pickering 1982.
Hypotheses
Ref Expression
dp41lem.1 c0 = ((a1a2) ∩ (b1b2))
dp41lem.2 c1 = ((a0a2) ∩ (b0b2))
dp41lem.3 c2 = ((a0a1) ∩ (b0b1))
dp41lem.4 p = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
dp41lem.5 p2 = ((a0b0) ∩ (a1b1))
dp41lem.6 p2 ≤ (a2b2)
Assertion
Ref Expression
dp41lemg (((b1b2) ∩ ((a1a2) ∪ (b1 ∩ (a0a1)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (a0 ∩ (b0b1))))) = (((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a1b1)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a0b0)))))

Proof of Theorem dp41lemg
StepHypRef Expression
1 or32 82 . . . 4 ((a1a2) ∪ (b1 ∩ (a0a1))) = ((a1 ∪ (b1 ∩ (a0a1))) ∪ a2)
2 ml3 1128 . . . . . 6 (a1 ∪ (b1 ∩ (a0a1))) = (a1 ∪ (a0 ∩ (b1a1)))
3 orcom 73 . . . . . . . 8 (b1a1) = (a1b1)
43lan 77 . . . . . . 7 (a0 ∩ (b1a1)) = (a0 ∩ (a1b1))
54lor 70 . . . . . 6 (a1 ∪ (a0 ∩ (b1a1))) = (a1 ∪ (a0 ∩ (a1b1)))
62, 5tr 62 . . . . 5 (a1 ∪ (b1 ∩ (a0a1))) = (a1 ∪ (a0 ∩ (a1b1)))
76ror 71 . . . 4 ((a1 ∪ (b1 ∩ (a0a1))) ∪ a2) = ((a1 ∪ (a0 ∩ (a1b1))) ∪ a2)
8 or32 82 . . . 4 ((a1 ∪ (a0 ∩ (a1b1))) ∪ a2) = ((a1a2) ∪ (a0 ∩ (a1b1)))
91, 7, 83tr 65 . . 3 ((a1a2) ∪ (b1 ∩ (a0a1))) = ((a1a2) ∪ (a0 ∩ (a1b1)))
109lan 77 . 2 ((b1b2) ∩ ((a1a2) ∪ (b1 ∩ (a0a1)))) = ((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a1b1))))
11 or32 82 . . . 4 ((b0b2) ∪ (a0 ∩ (b0b1))) = ((b0 ∪ (a0 ∩ (b0b1))) ∪ b2)
12 orcom 73 . . . . . . . 8 (b0b1) = (b1b0)
1312lan 77 . . . . . . 7 (a0 ∩ (b0b1)) = (a0 ∩ (b1b0))
1413lor 70 . . . . . 6 (b0 ∪ (a0 ∩ (b0b1))) = (b0 ∪ (a0 ∩ (b1b0)))
15 ml3 1128 . . . . . 6 (b0 ∪ (a0 ∩ (b1b0))) = (b0 ∪ (b1 ∩ (a0b0)))
1614, 15tr 62 . . . . 5 (b0 ∪ (a0 ∩ (b0b1))) = (b0 ∪ (b1 ∩ (a0b0)))
1716ror 71 . . . 4 ((b0 ∪ (a0 ∩ (b0b1))) ∪ b2) = ((b0 ∪ (b1 ∩ (a0b0))) ∪ b2)
18 or32 82 . . . 4 ((b0 ∪ (b1 ∩ (a0b0))) ∪ b2) = ((b0b2) ∪ (b1 ∩ (a0b0)))
1911, 17, 183tr 65 . . 3 ((b0b2) ∪ (a0 ∩ (b0b1))) = ((b0b2) ∪ (b1 ∩ (a0b0)))
2019lan 77 . 2 ((a0a2) ∩ ((b0b2) ∪ (a0 ∩ (b0b1)))) = ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a0b0))))
2110, 202or 72 1 (((b1b2) ∩ ((a1a2) ∪ (b1 ∩ (a0a1)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (a0 ∩ (b0b1))))) = (((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a1b1)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a0b0)))))
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2   ∪ wo 6   ∩ wa 7 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-ml 1120 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131 This theorem is referenced by:  dp41lemm  1192
 Copyright terms: Public domain W3C validator