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Theorem dp15leme 1158
 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
dp15leme (((a0a2) ∩ ((b0 ∩ (a0p0)) ∪ b2)) ∪ (((a1a2) ∪ (a0 ∩ (a1b1))) ∩ (b1b2))) ≤ (((a0a2) ∩ ((b0 ∩ (a0p0)) ∪ b2)) ∪ (((a1a2) ∪ (b1 ∩ (a0a1))) ∩ (b1b2)))

Proof of Theorem dp15leme
StepHypRef Expression
1 ax-a2 31 . . . . . . 7 (a1a2) = (a2a1)
2 ax-a2 31 . . . . . . . 8 (a1b1) = (b1a1)
32lan 77 . . . . . . 7 (a0 ∩ (a1b1)) = (a0 ∩ (b1a1))
41, 32or 72 . . . . . 6 ((a1a2) ∪ (a0 ∩ (a1b1))) = ((a2a1) ∪ (a0 ∩ (b1a1)))
5 orass 75 . . . . . 6 ((a2a1) ∪ (a0 ∩ (b1a1))) = (a2 ∪ (a1 ∪ (a0 ∩ (b1a1))))
64, 5tr 62 . . . . 5 ((a1a2) ∪ (a0 ∩ (a1b1))) = (a2 ∪ (a1 ∪ (a0 ∩ (b1a1))))
7 ml3le 1129 . . . . . 6 (a1 ∪ (a0 ∩ (b1a1))) ≤ (a1 ∪ (b1 ∩ (a0a1)))
87lelor 166 . . . . 5 (a2 ∪ (a1 ∪ (a0 ∩ (b1a1)))) ≤ (a2 ∪ (a1 ∪ (b1 ∩ (a0a1))))
96, 8bltr 138 . . . 4 ((a1a2) ∪ (a0 ∩ (a1b1))) ≤ (a2 ∪ (a1 ∪ (b1 ∩ (a0a1))))
10 orass 75 . . . . . 6 ((a2a1) ∪ (b1 ∩ (a0a1))) = (a2 ∪ (a1 ∪ (b1 ∩ (a0a1))))
1110cm 61 . . . . 5 (a2 ∪ (a1 ∪ (b1 ∩ (a0a1)))) = ((a2a1) ∪ (b1 ∩ (a0a1)))
12 ax-a2 31 . . . . . 6 (a2a1) = (a1a2)
1312ror 71 . . . . 5 ((a2a1) ∪ (b1 ∩ (a0a1))) = ((a1a2) ∪ (b1 ∩ (a0a1)))
1411, 13tr 62 . . . 4 (a2 ∪ (a1 ∪ (b1 ∩ (a0a1)))) = ((a1a2) ∪ (b1 ∩ (a0a1)))
159, 14lbtr 139 . . 3 ((a1a2) ∪ (a0 ∩ (a1b1))) ≤ ((a1a2) ∪ (b1 ∩ (a0a1)))
1615leran 153 . 2 (((a1a2) ∪ (a0 ∩ (a1b1))) ∩ (b1b2)) ≤ (((a1a2) ∪ (b1 ∩ (a0a1))) ∩ (b1b2))
1716lelor 166 1 (((a0a2) ∩ ((b0 ∩ (a0p0)) ∪ b2)) ∪ (((a1a2) ∪ (a0 ∩ (a1b1))) ∩ (b1b2))) ≤ (((a0a2) ∩ ((b0 ∩ (a0p0)) ∪ b2)) ∪ (((a1a2) ∪ (b1 ∩ (a0a1))) ∩ (b1b2)))
 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 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:  dp15lemh  1161
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