Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  dp15lema GIF version

Theorem dp15lema 1154
 Description: Part of proof (1)=>(5) in Day/Pickering 1982. (Contributed by NM, 1-Apr-2012.)
Hypotheses
Ref Expression
dp15lema.1 d = (a2 ∪ (a0 ∩ (a1b1)))
dp15lema.2 p0 = ((a1b1) ∩ (a2b2))
dp15lema.3 e = (b0 ∩ (a0p0))
Assertion
Ref Expression
dp15lema ((a0e) ∩ (a1b1)) ≤ (db2)

Proof of Theorem dp15lema
StepHypRef Expression
1 dp15lema.3 . . . . 5 e = (b0 ∩ (a0p0))
2 dp15lema.2 . . . . . . 7 p0 = ((a1b1) ∩ (a2b2))
32lor 70 . . . . . 6 (a0p0) = (a0 ∪ ((a1b1) ∩ (a2b2)))
43lan 77 . . . . 5 (b0 ∩ (a0p0)) = (b0 ∩ (a0 ∪ ((a1b1) ∩ (a2b2))))
51, 4tr 62 . . . 4 e = (b0 ∩ (a0 ∪ ((a1b1) ∩ (a2b2))))
65lor 70 . . 3 (a0e) = (a0 ∪ (b0 ∩ (a0 ∪ ((a1b1) ∩ (a2b2)))))
76ran 78 . 2 ((a0e) ∩ (a1b1)) = ((a0 ∪ (b0 ∩ (a0 ∪ ((a1b1) ∩ (a2b2))))) ∩ (a1b1))
8 le1 146 . . . . . 6 b0 ≤ 1
98leran 153 . . . . 5 (b0 ∩ (a0 ∪ ((a1b1) ∩ (a2b2)))) ≤ (1 ∩ (a0 ∪ ((a1b1) ∩ (a2b2))))
109lelor 166 . . . 4 (a0 ∪ (b0 ∩ (a0 ∪ ((a1b1) ∩ (a2b2))))) ≤ (a0 ∪ (1 ∩ (a0 ∪ ((a1b1) ∩ (a2b2)))))
1110leran 153 . . 3 ((a0 ∪ (b0 ∩ (a0 ∪ ((a1b1) ∩ (a2b2))))) ∩ (a1b1)) ≤ ((a0 ∪ (1 ∩ (a0 ∪ ((a1b1) ∩ (a2b2))))) ∩ (a1b1))
12 an1r 107 . . . . . . . . 9 (1 ∩ (a0 ∪ ((a1b1) ∩ (a2b2)))) = (a0 ∪ ((a1b1) ∩ (a2b2)))
1312lor 70 . . . . . . . 8 (a0 ∪ (1 ∩ (a0 ∪ ((a1b1) ∩ (a2b2))))) = (a0 ∪ (a0 ∪ ((a1b1) ∩ (a2b2))))
14 orass 75 . . . . . . . . 9 ((a0a0) ∪ ((a1b1) ∩ (a2b2))) = (a0 ∪ (a0 ∪ ((a1b1) ∩ (a2b2))))
1514cm 61 . . . . . . . 8 (a0 ∪ (a0 ∪ ((a1b1) ∩ (a2b2)))) = ((a0a0) ∪ ((a1b1) ∩ (a2b2)))
16 oridm 110 . . . . . . . . . 10 (a0a0) = a0
1716ror 71 . . . . . . . . 9 ((a0a0) ∪ ((a1b1) ∩ (a2b2))) = (a0 ∪ ((a1b1) ∩ (a2b2)))
18 orcom 73 . . . . . . . . 9 (a0 ∪ ((a1b1) ∩ (a2b2))) = (((a1b1) ∩ (a2b2)) ∪ a0)
1917, 18tr 62 . . . . . . . 8 ((a0a0) ∪ ((a1b1) ∩ (a2b2))) = (((a1b1) ∩ (a2b2)) ∪ a0)
2013, 15, 193tr 65 . . . . . . 7 (a0 ∪ (1 ∩ (a0 ∪ ((a1b1) ∩ (a2b2))))) = (((a1b1) ∩ (a2b2)) ∪ a0)
2120ran 78 . . . . . 6 ((a0 ∪ (1 ∩ (a0 ∪ ((a1b1) ∩ (a2b2))))) ∩ (a1b1)) = ((((a1b1) ∩ (a2b2)) ∪ a0) ∩ (a1b1))
22 lea 160 . . . . . . 7 ((a1b1) ∩ (a2b2)) ≤ (a1b1)
2322mlduali 1128 . . . . . 6 ((((a1b1) ∩ (a2b2)) ∪ a0) ∩ (a1b1)) = (((a1b1) ∩ (a2b2)) ∪ (a0 ∩ (a1b1)))
2421, 23tr 62 . . . . 5 ((a0 ∪ (1 ∩ (a0 ∪ ((a1b1) ∩ (a2b2))))) ∩ (a1b1)) = (((a1b1) ∩ (a2b2)) ∪ (a0 ∩ (a1b1)))
25 lear 161 . . . . . 6 ((a1b1) ∩ (a2b2)) ≤ (a2b2)
2625leror 152 . . . . 5 (((a1b1) ∩ (a2b2)) ∪ (a0 ∩ (a1b1))) ≤ ((a2b2) ∪ (a0 ∩ (a1b1)))
2724, 26bltr 138 . . . 4 ((a0 ∪ (1 ∩ (a0 ∪ ((a1b1) ∩ (a2b2))))) ∩ (a1b1)) ≤ ((a2b2) ∪ (a0 ∩ (a1b1)))
28 or32 82 . . . . 5 ((a2b2) ∪ (a0 ∩ (a1b1))) = ((a2 ∪ (a0 ∩ (a1b1))) ∪ b2)
29 dp15lema.1 . . . . . . 7 d = (a2 ∪ (a0 ∩ (a1b1)))
3029ror 71 . . . . . 6 (db2) = ((a2 ∪ (a0 ∩ (a1b1))) ∪ b2)
3130cm 61 . . . . 5 ((a2 ∪ (a0 ∩ (a1b1))) ∪ b2) = (db2)
3228, 31tr 62 . . . 4 ((a2b2) ∪ (a0 ∩ (a1b1))) = (db2)
3327, 32lbtr 139 . . 3 ((a0 ∪ (1 ∩ (a0 ∪ ((a1b1) ∩ (a2b2))))) ∩ (a1b1)) ≤ (db2)
3411, 33letr 137 . 2 ((a0 ∪ (b0 ∩ (a0 ∪ ((a1b1) ∩ (a2b2))))) ∩ (a1b1)) ≤ (db2)
357, 34bltr 138 1 ((a0e) ∩ (a1b1)) ≤ (db2)
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2   ∪ wo 6   ∩ wa 7  1wt 8 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-ml 1122 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:  dp15lemb  1155
 Copyright terms: Public domain W3C validator