Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  dp41lemh GIF version

Theorem dp41lemh 1188
 Description: Part of proof (4)=>(1) in Day/Pickering 1982. "By CP(a,b)".
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
dp41lemh (((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a1b1)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a0b0))))) ≤ (((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a2b2)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a2b2)))))

Proof of Theorem dp41lemh
StepHypRef Expression
1 lea 160 . . . . 5 (a0 ∩ (a1b1)) ≤ a0
2 leo 158 . . . . . . 7 a0 ≤ (a0b0)
32leran 153 . . . . . 6 (a0 ∩ (a1b1)) ≤ ((a0b0) ∩ (a1b1))
4 dp41lem.5 . . . . . . . 8 p2 = ((a0b0) ∩ (a1b1))
54cm 61 . . . . . . 7 ((a0b0) ∩ (a1b1)) = p2
6 dp41lem.6 . . . . . . 7 p2 ≤ (a2b2)
75, 6bltr 138 . . . . . 6 ((a0b0) ∩ (a1b1)) ≤ (a2b2)
83, 7letr 137 . . . . 5 (a0 ∩ (a1b1)) ≤ (a2b2)
91, 8ler2an 173 . . . 4 (a0 ∩ (a1b1)) ≤ (a0 ∩ (a2b2))
109lelor 166 . . 3 ((a1a2) ∪ (a0 ∩ (a1b1))) ≤ ((a1a2) ∪ (a0 ∩ (a2b2)))
1110lelan 167 . 2 ((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a1b1)))) ≤ ((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a2b2))))
12 lea 160 . . . . 5 (b1 ∩ (a0b0)) ≤ b1
13 lear 161 . . . . . . 7 (b1 ∩ (a0b0)) ≤ (a0b0)
14 leao3 164 . . . . . . 7 (b1 ∩ (a0b0)) ≤ (a1b1)
1513, 14ler2an 173 . . . . . 6 (b1 ∩ (a0b0)) ≤ ((a0b0) ∩ (a1b1))
1615, 7letr 137 . . . . 5 (b1 ∩ (a0b0)) ≤ (a2b2)
1712, 16ler2an 173 . . . 4 (b1 ∩ (a0b0)) ≤ (b1 ∩ (a2b2))
1817lelor 166 . . 3 ((b0b2) ∪ (b1 ∩ (a0b0))) ≤ ((b0b2) ∪ (b1 ∩ (a2b2)))
1918lelan 167 . 2 ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a0b0)))) ≤ ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a2b2))))
2011, 19le2or 168 1 (((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a1b1)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a0b0))))) ≤ (((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a2b2)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a2b2)))))
 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 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