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Theorem dp35leme 1171
 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
dp35leme (b0 ∩ (a0p0)) ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))

Proof of Theorem dp35leme
StepHypRef Expression
1 leor 159 . . 3 b0 ≤ (a0b0)
2 dp35lem.4 . . . . 5 p0 = ((a1b1) ∩ (a2b2))
32lor 70 . . . 4 (a0p0) = (a0 ∪ ((a1b1) ∩ (a2b2)))
43bile 142 . . 3 (a0p0) ≤ (a0 ∪ ((a1b1) ∩ (a2b2)))
51, 4le2an 169 . 2 (b0 ∩ (a0p0)) ≤ ((a0b0) ∩ (a0 ∪ ((a1b1) ∩ (a2b2))))
6 ancom 74 . . . . . 6 (((a1b1) ∩ (a2b2)) ∩ (a0b0)) = ((a0b0) ∩ ((a1b1) ∩ (a2b2)))
7 anass 76 . . . . . . 7 (((a0b0) ∩ (a1b1)) ∩ (a2b2)) = ((a0b0) ∩ ((a1b1) ∩ (a2b2)))
87cm 61 . . . . . 6 ((a0b0) ∩ ((a1b1) ∩ (a2b2))) = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
96, 8tr 62 . . . . 5 (((a1b1) ∩ (a2b2)) ∩ (a0b0)) = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
109lor 70 . . . 4 (a0 ∪ (((a1b1) ∩ (a2b2)) ∩ (a0b0))) = (a0 ∪ (((a0b0) ∩ (a1b1)) ∩ (a2b2)))
11 ancom 74 . . . . 5 ((a0b0) ∩ (a0 ∪ ((a1b1) ∩ (a2b2)))) = ((a0 ∪ ((a1b1) ∩ (a2b2))) ∩ (a0b0))
12 leo 158 . . . . . 6 a0 ≤ (a0b0)
1312mlduali 1126 . . . . 5 ((a0 ∪ ((a1b1) ∩ (a2b2))) ∩ (a0b0)) = (a0 ∪ (((a1b1) ∩ (a2b2)) ∩ (a0b0)))
1411, 13tr 62 . . . 4 ((a0b0) ∩ (a0 ∪ ((a1b1) ∩ (a2b2)))) = (a0 ∪ (((a1b1) ∩ (a2b2)) ∩ (a0b0)))
15 dp35lem.5 . . . . 5 p = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
1615lor 70 . . . 4 (a0p) = (a0 ∪ (((a0b0) ∩ (a1b1)) ∩ (a2b2)))
1710, 14, 163tr1 63 . . 3 ((a0b0) ∩ (a0 ∪ ((a1b1) ∩ (a2b2)))) = (a0p)
18 dp35lem.1 . . . 4 c0 = ((a1a2) ∩ (b1b2))
19 dp35lem.2 . . . 4 c1 = ((a0a2) ∩ (b0b2))
20 dp35lem.3 . . . 4 c2 = ((a0a1) ∩ (b0b1))
2118, 19, 20, 2, 15dp35lemf 1170 . . 3 (a0p) ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))
2217, 21bltr 138 . 2 ((a0b0) ∩ (a0 ∪ ((a1b1) ∩ (a2b2)))) ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))
235, 22letr 137 1 (b0 ∩ (a0p0)) ≤ (a0 ∪ (b0 ∩ (b1 ∪ (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:  dp35lemd  1172
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