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Theorem dp34 1179
 Description: Part of theorem from Alan Day and Doug Pickering, "A note on the Arguesian lattice identity," Studia Sci. Math. Hungar. 19:303-305 (1982). (3)=>(4)
Hypotheses
Ref Expression
dp34.1 c0 = ((a1a2) ∩ (b1b2))
dp34.2 c1 = ((a0a2) ∩ (b0b2))
dp34.3 c2 = ((a0a1) ∩ (b0b1))
dp34.4 p = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
Assertion
Ref Expression
dp34 p ≤ ((a0b1) ∪ (c2 ∩ (c0c1)))

Proof of Theorem dp34
StepHypRef Expression
1 dp34.1 . . . 4 c0 = ((a1a2) ∩ (b1b2))
2 dp34.2 . . . 4 c1 = ((a0a2) ∩ (b0b2))
3 dp34.3 . . . 4 c2 = ((a0a1) ∩ (b0b1))
4 dp34.4 . . . 4 p = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
51, 2, 3, 4dp53 1168 . . 3 p ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))
6 lear 161 . . . 4 (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))) ≤ (b1 ∪ (c2 ∩ (c0c1)))
76lelor 166 . . 3 (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))) ≤ (a0 ∪ (b1 ∪ (c2 ∩ (c0c1))))
85, 7letr 137 . 2 p ≤ (a0 ∪ (b1 ∪ (c2 ∩ (c0c1))))
9 orass 75 . . 3 ((a0b1) ∪ (c2 ∩ (c0c1))) = (a0 ∪ (b1 ∪ (c2 ∩ (c0c1))))
109cm 61 . 2 (a0 ∪ (b1 ∪ (c2 ∩ (c0c1)))) = ((a0b1) ∪ (c2 ∩ (c0c1)))
118, 10lbtr 139 1 p ≤ ((a0b1) ∪ (c2 ∩ (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-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-ml 1120  ax-arg 1151 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:  dp41lema  1180  xdp41  1196  xxdp41  1199  xdp45lem  1202  xdp43lem  1203  xdp45  1204  xdp43  1205  3dp43  1206
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