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Theorem dp41lemf 1188
 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
dp41lemf ((c0c1) ∪ ((a0 ∩ (b0b1)) ∪ (b1 ∩ (a0a1)))) = (((b1b2) ∩ ((a1a2) ∪ (b1 ∩ (a0a1)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (a0 ∩ (b0b1)))))

Proof of Theorem dp41lemf
StepHypRef Expression
1 orcom 73 . . 3 ((a0 ∩ (b0b1)) ∪ (b1 ∩ (a0a1))) = ((b1 ∩ (a0a1)) ∪ (a0 ∩ (b0b1)))
21lor 70 . 2 ((c0c1) ∪ ((a0 ∩ (b0b1)) ∪ (b1 ∩ (a0a1)))) = ((c0c1) ∪ ((b1 ∩ (a0a1)) ∪ (a0 ∩ (b0b1))))
3 or4 84 . . 3 ((c0c1) ∪ ((b1 ∩ (a0a1)) ∪ (a0 ∩ (b0b1)))) = ((c0 ∪ (b1 ∩ (a0a1))) ∪ (c1 ∪ (a0 ∩ (b0b1))))
4 dp41lem.1 . . . . . 6 c0 = ((a1a2) ∩ (b1b2))
5 ancom 74 . . . . . 6 ((a1a2) ∩ (b1b2)) = ((b1b2) ∩ (a1a2))
64, 5tr 62 . . . . 5 c0 = ((b1b2) ∩ (a1a2))
76ror 71 . . . 4 (c0 ∪ (b1 ∩ (a0a1))) = (((b1b2) ∩ (a1a2)) ∪ (b1 ∩ (a0a1)))
8 dp41lem.2 . . . . 5 c1 = ((a0a2) ∩ (b0b2))
98ror 71 . . . 4 (c1 ∪ (a0 ∩ (b0b1))) = (((a0a2) ∩ (b0b2)) ∪ (a0 ∩ (b0b1)))
107, 92or 72 . . 3 ((c0 ∪ (b1 ∩ (a0a1))) ∪ (c1 ∪ (a0 ∩ (b0b1)))) = ((((b1b2) ∩ (a1a2)) ∪ (b1 ∩ (a0a1))) ∪ (((a0a2) ∩ (b0b2)) ∪ (a0 ∩ (b0b1))))
113, 10tr 62 . 2 ((c0c1) ∪ ((b1 ∩ (a0a1)) ∪ (a0 ∩ (b0b1)))) = ((((b1b2) ∩ (a1a2)) ∪ (b1 ∩ (a0a1))) ∪ (((a0a2) ∩ (b0b2)) ∪ (a0 ∩ (b0b1))))
12 leao1 162 . . . 4 (b1 ∩ (a0a1)) ≤ (b1b2)
1312mli 1126 . . 3 (((b1b2) ∩ (a1a2)) ∪ (b1 ∩ (a0a1))) = ((b1b2) ∩ ((a1a2) ∪ (b1 ∩ (a0a1))))
14 leao1 162 . . . 4 (a0 ∩ (b0b1)) ≤ (a0a2)
1514mli 1126 . . 3 (((a0a2) ∩ (b0b2)) ∪ (a0 ∩ (b0b1))) = ((a0a2) ∩ ((b0b2) ∪ (a0 ∩ (b0b1))))
1613, 152or 72 . 2 ((((b1b2) ∩ (a1a2)) ∪ (b1 ∩ (a0a1))) ∪ (((a0a2) ∩ (b0b2)) ∪ (a0 ∩ (b0b1)))) = (((b1b2) ∩ ((a1a2) ∪ (b1 ∩ (a0a1)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (a0 ∩ (b0b1)))))
172, 11, 163tr 65 1 ((c0c1) ∪ ((a0 ∩ (b0b1)) ∪ (b1 ∩ (a0a1)))) = (((b1b2) ∩ ((a1a2) ∪ (b1 ∩ (a0a1)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (a0 ∩ (b0b1)))))
 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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