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Theorem dp41leml 1191
 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
dp41leml ((c0 ∪ (b2 ∩ (a0a2))) ∪ (c1 ∪ (a2 ∩ (b1b2)))) = (c0c1)

Proof of Theorem dp41leml
StepHypRef Expression
1 or4 84 . 2 ((c0 ∪ (b2 ∩ (a0a2))) ∪ (c1 ∪ (a2 ∩ (b1b2)))) = ((c0c1) ∪ ((b2 ∩ (a0a2)) ∪ (a2 ∩ (b1b2))))
2 orcom 73 . 2 ((c0c1) ∪ ((b2 ∩ (a0a2)) ∪ (a2 ∩ (b1b2)))) = (((b2 ∩ (a0a2)) ∪ (a2 ∩ (b1b2))) ∪ (c0c1))
3 ancom 74 . . . . . 6 (b2 ∩ (a0a2)) = ((a0a2) ∩ b2)
4 leor 159 . . . . . . 7 b2 ≤ (b0b2)
54lelan 167 . . . . . 6 ((a0a2) ∩ b2) ≤ ((a0a2) ∩ (b0b2))
63, 5bltr 138 . . . . 5 (b2 ∩ (a0a2)) ≤ ((a0a2) ∩ (b0b2))
7 leor 159 . . . . . 6 a2 ≤ (a1a2)
87leran 153 . . . . 5 (a2 ∩ (b1b2)) ≤ ((a1a2) ∩ (b1b2))
96, 8le2or 168 . . . 4 ((b2 ∩ (a0a2)) ∪ (a2 ∩ (b1b2))) ≤ (((a0a2) ∩ (b0b2)) ∪ ((a1a2) ∩ (b1b2)))
10 dp41lem.2 . . . . . . 7 c1 = ((a0a2) ∩ (b0b2))
11 dp41lem.1 . . . . . . 7 c0 = ((a1a2) ∩ (b1b2))
1210, 112or 72 . . . . . 6 (c1c0) = (((a0a2) ∩ (b0b2)) ∪ ((a1a2) ∩ (b1b2)))
1312cm 61 . . . . 5 (((a0a2) ∩ (b0b2)) ∪ ((a1a2) ∩ (b1b2))) = (c1c0)
14 orcom 73 . . . . 5 (c1c0) = (c0c1)
1513, 14tr 62 . . . 4 (((a0a2) ∩ (b0b2)) ∪ ((a1a2) ∩ (b1b2))) = (c0c1)
169, 15lbtr 139 . . 3 ((b2 ∩ (a0a2)) ∪ (a2 ∩ (b1b2))) ≤ (c0c1)
1716df-le2 131 . 2 (((b2 ∩ (a0a2)) ∪ (a2 ∩ (b1b2))) ∪ (c0c1)) = (c0c1)
181, 2, 173tr 65 1 ((c0 ∪ (b2 ∩ (a0a2))) ∪ (c1 ∪ (a2 ∩ (b1b2)))) = (c0c1)
 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 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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