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Theorem dp41lemk 1190
 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
dp41lemk (((b1b2) ∩ ((a1a2) ∪ (b2 ∩ (a0a2)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (a2 ∩ (b1b2))))) = ((c0 ∪ (b2 ∩ (a0a2))) ∪ (c1 ∪ (a2 ∩ (b1b2))))

Proof of Theorem dp41lemk
StepHypRef Expression
1 leao3 164 . . . 4 (b2 ∩ (a0a2)) ≤ (b1b2)
21mldual2i 1125 . . 3 ((b1b2) ∩ ((a1a2) ∪ (b2 ∩ (a0a2)))) = (((b1b2) ∩ (a1a2)) ∪ (b2 ∩ (a0a2)))
3 dp41lem.1 . . . . . 6 c0 = ((a1a2) ∩ (b1b2))
4 ancom 74 . . . . . 6 ((a1a2) ∩ (b1b2)) = ((b1b2) ∩ (a1a2))
53, 4tr 62 . . . . 5 c0 = ((b1b2) ∩ (a1a2))
65ror 71 . . . 4 (c0 ∪ (b2 ∩ (a0a2))) = (((b1b2) ∩ (a1a2)) ∪ (b2 ∩ (a0a2)))
76cm 61 . . 3 (((b1b2) ∩ (a1a2)) ∪ (b2 ∩ (a0a2))) = (c0 ∪ (b2 ∩ (a0a2)))
82, 7tr 62 . 2 ((b1b2) ∩ ((a1a2) ∪ (b2 ∩ (a0a2)))) = (c0 ∪ (b2 ∩ (a0a2)))
9 leao3 164 . . . 4 (a2 ∩ (b1b2)) ≤ (a0a2)
109mldual2i 1125 . . 3 ((a0a2) ∩ ((b0b2) ∪ (a2 ∩ (b1b2)))) = (((a0a2) ∩ (b0b2)) ∪ (a2 ∩ (b1b2)))
11 dp41lem.2 . . . . 5 c1 = ((a0a2) ∩ (b0b2))
1211ror 71 . . . 4 (c1 ∪ (a2 ∩ (b1b2))) = (((a0a2) ∩ (b0b2)) ∪ (a2 ∩ (b1b2)))
1312cm 61 . . 3 (((a0a2) ∩ (b0b2)) ∪ (a2 ∩ (b1b2))) = (c1 ∪ (a2 ∩ (b1b2)))
1410, 13tr 62 . 2 ((a0a2) ∩ ((b0b2) ∪ (a2 ∩ (b1b2)))) = (c1 ∪ (a2 ∩ (b1b2)))
158, 142or 72 1 (((b1b2) ∩ ((a1a2) ∪ (b2 ∩ (a0a2)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (a2 ∩ (b1b2))))) = ((c0 ∪ (b2 ∩ (a0a2))) ∪ (c1 ∪ (a2 ∩ (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 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
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