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Theorem dp41lemj 1191
 Description: Part of proof (4)=>(1) in Day/Pickering 1982. (Contributed by NM, 3-Apr-2012.)
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
dp41lemj (((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a2b2)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a2b2))))) = (((b1b2) ∩ ((a1a2) ∪ (b2 ∩ (a0a2)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (a2 ∩ (b1b2)))))

Proof of Theorem dp41lemj
StepHypRef Expression
1 ax-a2 31 . . . . . . . 8 (a2b2) = (b2a2)
21lan 77 . . . . . . 7 (a0 ∩ (a2b2)) = (a0 ∩ (b2a2))
32lor 70 . . . . . 6 (a2 ∪ (a0 ∩ (a2b2))) = (a2 ∪ (a0 ∩ (b2a2)))
4 ml3 1130 . . . . . 6 (a2 ∪ (a0 ∩ (b2a2))) = (a2 ∪ (b2 ∩ (a0a2)))
53, 4tr 62 . . . . 5 (a2 ∪ (a0 ∩ (a2b2))) = (a2 ∪ (b2 ∩ (a0a2)))
65lor 70 . . . 4 (a1 ∪ (a2 ∪ (a0 ∩ (a2b2)))) = (a1 ∪ (a2 ∪ (b2 ∩ (a0a2))))
7 orass 75 . . . 4 ((a1a2) ∪ (a0 ∩ (a2b2))) = (a1 ∪ (a2 ∪ (a0 ∩ (a2b2))))
8 orass 75 . . . 4 ((a1a2) ∪ (b2 ∩ (a0a2))) = (a1 ∪ (a2 ∪ (b2 ∩ (a0a2))))
96, 7, 83tr1 63 . . 3 ((a1a2) ∪ (a0 ∩ (a2b2))) = ((a1a2) ∪ (b2 ∩ (a0a2)))
109lan 77 . 2 ((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a2b2)))) = ((b1b2) ∩ ((a1a2) ∪ (b2 ∩ (a0a2))))
11 ml3 1130 . . . . 5 (b2 ∪ (b1 ∩ (a2b2))) = (b2 ∪ (a2 ∩ (b1b2)))
1211lor 70 . . . 4 (b0 ∪ (b2 ∪ (b1 ∩ (a2b2)))) = (b0 ∪ (b2 ∪ (a2 ∩ (b1b2))))
13 orass 75 . . . 4 ((b0b2) ∪ (b1 ∩ (a2b2))) = (b0 ∪ (b2 ∪ (b1 ∩ (a2b2))))
14 orass 75 . . . 4 ((b0b2) ∪ (a2 ∩ (b1b2))) = (b0 ∪ (b2 ∪ (a2 ∩ (b1b2))))
1512, 13, 143tr1 63 . . 3 ((b0b2) ∪ (b1 ∩ (a2b2))) = ((b0b2) ∪ (a2 ∩ (b1b2)))
1615lan 77 . 2 ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a2b2)))) = ((a0a2) ∩ ((b0b2) ∪ (a2 ∩ (b1b2))))
1710, 162or 72 1 (((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a2b2)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a2b2))))) = (((b1b2) ∩ ((a1a2) ∪ (b2 ∩ (a0a2)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (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 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:  dp41lemm  1194
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