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Theorem dp15lemf 1157
 Description: Part of proof (1)=>(5) in Day/Pickering 1982.
Hypotheses
Ref Expression
dp15lema.1 d = (a2 ∪ (a0 ∩ (a1b1)))
dp15lema.2 p0 = ((a1b1) ∩ (a2b2))
dp15lema.3 e = (b0 ∩ (a0p0))
Assertion
Ref Expression
dp15lemf (((a0a2) ∩ ((b0 ∩ (a0p0)) ∪ b2)) ∪ (((a1a2) ∪ (b1 ∩ (a0a1))) ∩ (b1b2))) ≤ (((a1a2) ∩ (b1b2)) ∪ (((a0a2) ∩ (b0b2)) ∪ (b1 ∩ (a0a1))))

Proof of Theorem dp15lemf
StepHypRef Expression
1 lea 160 . . . . 5 (b0 ∩ (a0p0)) ≤ b0
21leror 152 . . . 4 ((b0 ∩ (a0p0)) ∪ b2) ≤ (b0b2)
32lelan 167 . . 3 ((a0a2) ∩ ((b0 ∩ (a0p0)) ∪ b2)) ≤ ((a0a2) ∩ (b0b2))
4 leao1 162 . . . . . 6 (b1 ∩ (a0a1)) ≤ (b1b2)
54mldual2i 1125 . . . . 5 ((b1b2) ∩ ((a1a2) ∪ (b1 ∩ (a0a1)))) = (((b1b2) ∩ (a1a2)) ∪ (b1 ∩ (a0a1)))
6 ancom 74 . . . . 5 ((b1b2) ∩ ((a1a2) ∪ (b1 ∩ (a0a1)))) = (((a1a2) ∪ (b1 ∩ (a0a1))) ∩ (b1b2))
7 ancom 74 . . . . . 6 ((b1b2) ∩ (a1a2)) = ((a1a2) ∩ (b1b2))
87ror 71 . . . . 5 (((b1b2) ∩ (a1a2)) ∪ (b1 ∩ (a0a1))) = (((a1a2) ∩ (b1b2)) ∪ (b1 ∩ (a0a1)))
95, 6, 83tr2 64 . . . 4 (((a1a2) ∪ (b1 ∩ (a0a1))) ∩ (b1b2)) = (((a1a2) ∩ (b1b2)) ∪ (b1 ∩ (a0a1)))
109bile 142 . . 3 (((a1a2) ∪ (b1 ∩ (a0a1))) ∩ (b1b2)) ≤ (((a1a2) ∩ (b1b2)) ∪ (b1 ∩ (a0a1)))
113, 10le2or 168 . 2 (((a0a2) ∩ ((b0 ∩ (a0p0)) ∪ b2)) ∪ (((a1a2) ∪ (b1 ∩ (a0a1))) ∩ (b1b2))) ≤ (((a0a2) ∩ (b0b2)) ∪ (((a1a2) ∩ (b1b2)) ∪ (b1 ∩ (a0a1))))
12 or12 80 . 2 (((a0a2) ∩ (b0b2)) ∪ (((a1a2) ∩ (b1b2)) ∪ (b1 ∩ (a0a1)))) = (((a1a2) ∩ (b1b2)) ∪ (((a0a2) ∩ (b0b2)) ∪ (b1 ∩ (a0a1))))
1311, 12lbtr 139 1 (((a0a2) ∩ ((b0 ∩ (a0p0)) ∪ b2)) ∪ (((a1a2) ∪ (b1 ∩ (a0a1))) ∩ (b1b2))) ≤ (((a1a2) ∩ (b1b2)) ∪ (((a0a2) ∩ (b0b2)) ∪ (b1 ∩ (a0a1))))
 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:  dp15lemh  1159
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