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Theorem dp35lemd 1174
 Description: Part of proof (3)=>(5) in Day/Pickering 1982. (Contributed by NM, 12-Apr-2012.)
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
dp35lemd (b0 ∩ (a0p0)) ≤ (b0 ∩ (((a0b0) ∪ b1) ∪ (c2 ∩ (c0c1))))

Proof of Theorem dp35lemd
StepHypRef Expression
1 lea 160 . . 3 (b0 ∩ (a0p0)) ≤ b0
2 dp35lem.1 . . . 4 c0 = ((a1a2) ∩ (b1b2))
3 dp35lem.2 . . . 4 c1 = ((a0a2) ∩ (b0b2))
4 dp35lem.3 . . . 4 c2 = ((a0a1) ∩ (b0b1))
5 dp35lem.4 . . . 4 p0 = ((a1b1) ∩ (a2b2))
6 dp35lem.5 . . . 4 p = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
72, 3, 4, 5, 6dp35leme 1173 . . 3 (b0 ∩ (a0p0)) ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))
81, 7ler2an 173 . 2 (b0 ∩ (a0p0)) ≤ (b0 ∩ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))))
9 lea 160 . . . 4 (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))) ≤ b0
109mldual2i 1127 . . 3 (b0 ∩ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))) = ((b0a0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))
11 lea 160 . . . . 5 (b0a0) ≤ b0
1211, 9lel2or 170 . . . 4 ((b0a0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))) ≤ b0
13 ancom 74 . . . . . . 7 (b0a0) = (a0b0)
1413bile 142 . . . . . 6 (b0a0) ≤ (a0b0)
15 lear 161 . . . . . 6 (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))) ≤ (b1 ∪ (c2 ∩ (c0c1)))
1614, 15le2or 168 . . . . 5 ((b0a0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))) ≤ ((a0b0) ∪ (b1 ∪ (c2 ∩ (c0c1))))
17 orass 75 . . . . . 6 (((a0b0) ∪ b1) ∪ (c2 ∩ (c0c1))) = ((a0b0) ∪ (b1 ∪ (c2 ∩ (c0c1))))
1817cm 61 . . . . 5 ((a0b0) ∪ (b1 ∪ (c2 ∩ (c0c1)))) = (((a0b0) ∪ b1) ∪ (c2 ∩ (c0c1)))
1916, 18lbtr 139 . . . 4 ((b0a0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))) ≤ (((a0b0) ∪ b1) ∪ (c2 ∩ (c0c1)))
2012, 19ler2an 173 . . 3 ((b0a0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))) ≤ (b0 ∩ (((a0b0) ∪ b1) ∪ (c2 ∩ (c0c1))))
2110, 20bltr 138 . 2 (b0 ∩ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))) ≤ (b0 ∩ (((a0b0) ∪ b1) ∪ (c2 ∩ (c0c1))))
228, 21letr 137 1 (b0 ∩ (a0p0)) ≤ (b0 ∩ (((a0b0) ∪ 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 1122  ax-arg 1153 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:  dp35lembb  1177
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