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Theorem dp41leme 1187
 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
dp41leme (c2 ∩ ((c0c1) ∪ (c2 ∩ (a0b1)))) ≤ ((c0c1) ∪ ((a0 ∩ (b0b1)) ∪ (b1 ∩ (a0a1))))

Proof of Theorem dp41leme
StepHypRef Expression
1 mldual 1124 . . 3 (c2 ∩ ((c0c1) ∪ (c2 ∩ (a0b1)))) = ((c2 ∩ (c0c1)) ∪ (c2 ∩ (a0b1)))
2 dp41lem.3 . . . . . 6 c2 = ((a0a1) ∩ (b0b1))
32ran 78 . . . . 5 (c2 ∩ (a0b1)) = (((a0a1) ∩ (b0b1)) ∩ (a0b1))
4 anass 76 . . . . 5 (((a0a1) ∩ (b0b1)) ∩ (a0b1)) = ((a0a1) ∩ ((b0b1) ∩ (a0b1)))
5 leor 159 . . . . . . . . 9 b1 ≤ (b0b1)
65mldual2i 1127 . . . . . . . 8 ((b0b1) ∩ (a0b1)) = (((b0b1) ∩ a0) ∪ b1)
7 orcom 73 . . . . . . . 8 (((b0b1) ∩ a0) ∪ b1) = (b1 ∪ ((b0b1) ∩ a0))
8 ancom 74 . . . . . . . . 9 ((b0b1) ∩ a0) = (a0 ∩ (b0b1))
98lor 70 . . . . . . . 8 (b1 ∪ ((b0b1) ∩ a0)) = (b1 ∪ (a0 ∩ (b0b1)))
106, 7, 93tr 65 . . . . . . 7 ((b0b1) ∩ (a0b1)) = (b1 ∪ (a0 ∩ (b0b1)))
1110lan 77 . . . . . 6 ((a0a1) ∩ ((b0b1) ∩ (a0b1))) = ((a0a1) ∩ (b1 ∪ (a0 ∩ (b0b1))))
12 leao1 162 . . . . . . 7 (a0 ∩ (b0b1)) ≤ (a0a1)
1312mldual2i 1127 . . . . . 6 ((a0a1) ∩ (b1 ∪ (a0 ∩ (b0b1)))) = (((a0a1) ∩ b1) ∪ (a0 ∩ (b0b1)))
14 orcom 73 . . . . . . 7 (((a0a1) ∩ b1) ∪ (a0 ∩ (b0b1))) = ((a0 ∩ (b0b1)) ∪ ((a0a1) ∩ b1))
15 ancom 74 . . . . . . . 8 ((a0a1) ∩ b1) = (b1 ∩ (a0a1))
1615lor 70 . . . . . . 7 ((a0 ∩ (b0b1)) ∪ ((a0a1) ∩ b1)) = ((a0 ∩ (b0b1)) ∪ (b1 ∩ (a0a1)))
1714, 16tr 62 . . . . . 6 (((a0a1) ∩ b1) ∪ (a0 ∩ (b0b1))) = ((a0 ∩ (b0b1)) ∪ (b1 ∩ (a0a1)))
1811, 13, 173tr 65 . . . . 5 ((a0a1) ∩ ((b0b1) ∩ (a0b1))) = ((a0 ∩ (b0b1)) ∪ (b1 ∩ (a0a1)))
193, 4, 183tr 65 . . . 4 (c2 ∩ (a0b1)) = ((a0 ∩ (b0b1)) ∪ (b1 ∩ (a0a1)))
2019lor 70 . . 3 ((c2 ∩ (c0c1)) ∪ (c2 ∩ (a0b1))) = ((c2 ∩ (c0c1)) ∪ ((a0 ∩ (b0b1)) ∪ (b1 ∩ (a0a1))))
211, 20tr 62 . 2 (c2 ∩ ((c0c1) ∪ (c2 ∩ (a0b1)))) = ((c2 ∩ (c0c1)) ∪ ((a0 ∩ (b0b1)) ∪ (b1 ∩ (a0a1))))
22 lear 161 . . 3 (c2 ∩ (c0c1)) ≤ (c0c1)
2322leror 152 . 2 ((c2 ∩ (c0c1)) ∪ ((a0 ∩ (b0b1)) ∪ (b1 ∩ (a0a1)))) ≤ ((c0c1) ∪ ((a0 ∩ (b0b1)) ∪ (b1 ∩ (a0a1))))
2421, 23bltr 138 1 (c2 ∩ ((c0c1) ∪ (c2 ∩ (a0b1)))) ≤ ((c0c1) ∪ ((a0 ∩ (b0b1)) ∪ (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-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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