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Theorem dp35lem0 1177
 Description: Part of proof (3)=>(5) in Day/Pickering 1982.
Hypotheses
Ref Expression
dp35lem.1 c0 = ((a1a2) ∩ (b1b2))
dp35lem.2 c1 = ((a0a2) ∩ (b0b2))
dp35lem.3 c2 = ((a0a1) ∩ (b0b1))
dp35lem.4 p0 = ((a1b1) ∩ (a2b2))
dp35lem.5 p = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
Assertion
Ref Expression
dp35lem0 ((a0a1) ∩ ((b0 ∩ (a0p0)) ∪ b1)) ≤ ((c0c1) ∪ (b1 ∩ (a0a1)))

Proof of Theorem dp35lem0
StepHypRef Expression
1 orcom 73 . . . . . 6 ((b0 ∩ (a0p0)) ∪ b1) = (b1 ∪ (b0 ∩ (a0p0)))
2 leid 148 . . . . . 6 (b1 ∪ (b0 ∩ (a0p0))) ≤ (b1 ∪ (b0 ∩ (a0p0)))
31, 2bltr 138 . . . . 5 ((b0 ∩ (a0p0)) ∪ b1) ≤ (b1 ∪ (b0 ∩ (a0p0)))
4 dp35lem.1 . . . . . 6 c0 = ((a1a2) ∩ (b1b2))
5 dp35lem.2 . . . . . 6 c1 = ((a0a2) ∩ (b0b2))
6 dp35lem.3 . . . . . 6 c2 = ((a0a1) ∩ (b0b1))
7 dp35lem.4 . . . . . 6 p0 = ((a1b1) ∩ (a2b2))
8 dp35lem.5 . . . . . 6 p = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
94, 5, 6, 7, 8dp35lema 1176 . . . . 5 (b1 ∪ (b0 ∩ (a0p0))) ≤ (b1 ∪ ((a0a1) ∩ (c0c1)))
103, 9letr 137 . . . 4 ((b0 ∩ (a0p0)) ∪ b1) ≤ (b1 ∪ ((a0a1) ∩ (c0c1)))
1110lelan 167 . . 3 ((a0a1) ∩ ((b0 ∩ (a0p0)) ∪ b1)) ≤ ((a0a1) ∩ (b1 ∪ ((a0a1) ∩ (c0c1))))
12 id 59 . . . . 5 ((a0a1) ∩ (b1 ∪ ((a0a1) ∩ (c0c1)))) = ((a0a1) ∩ (b1 ∪ ((a0a1) ∩ (c0c1))))
13 lea 160 . . . . . 6 ((a0a1) ∩ (c0c1)) ≤ (a0a1)
1413mldual2i 1125 . . . . 5 ((a0a1) ∩ (b1 ∪ ((a0a1) ∩ (c0c1)))) = (((a0a1) ∩ b1) ∪ ((a0a1) ∩ (c0c1)))
1512, 14tr 62 . . . 4 ((a0a1) ∩ (b1 ∪ ((a0a1) ∩ (c0c1)))) = (((a0a1) ∩ b1) ∪ ((a0a1) ∩ (c0c1)))
16 ancom 74 . . . . 5 ((a0a1) ∩ b1) = (b1 ∩ (a0a1))
1716ror 71 . . . 4 (((a0a1) ∩ b1) ∪ ((a0a1) ∩ (c0c1))) = ((b1 ∩ (a0a1)) ∪ ((a0a1) ∩ (c0c1)))
1815, 17tr 62 . . 3 ((a0a1) ∩ (b1 ∪ ((a0a1) ∩ (c0c1)))) = ((b1 ∩ (a0a1)) ∪ ((a0a1) ∩ (c0c1)))
1911, 18lbtr 139 . 2 ((a0a1) ∩ ((b0 ∩ (a0p0)) ∪ b1)) ≤ ((b1 ∩ (a0a1)) ∪ ((a0a1) ∩ (c0c1)))
20 lear 161 . . . 4 ((a0a1) ∩ (c0c1)) ≤ (c0c1)
2120lelor 166 . . 3 ((b1 ∩ (a0a1)) ∪ ((a0a1) ∩ (c0c1))) ≤ ((b1 ∩ (a0a1)) ∪ (c0c1))
22 orcom 73 . . 3 ((b1 ∩ (a0a1)) ∪ (c0c1)) = ((c0c1) ∪ (b1 ∩ (a0a1)))
2321, 22lbtr 139 . 2 ((b1 ∩ (a0a1)) ∪ ((a0a1) ∩ (c0c1))) ≤ ((c0c1) ∪ (b1 ∩ (a0a1)))
2419, 23letr 137 1 ((a0a1) ∩ ((b0 ∩ (a0p0)) ∪ b1)) ≤ ((c0c1) ∪ (b1 ∩ (a0a1)))
 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:  dp35  1178  oadp35  1210
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