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Theorem dp53lemd 1164
 Description: Part of proof (5)=>(3) in Day/Pickering 1982.
Hypotheses
Ref Expression
dp53lem.1 c0 = ((a1a2) ∩ (b1b2))
dp53lem.2 c1 = ((a0a2) ∩ (b0b2))
dp53lem.3 c2 = ((a0a1) ∩ (b0b1))
dp53lem.4 p0 = ((a1b1) ∩ (a2b2))
dp53lem.5 p = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
Assertion
Ref Expression
dp53lemd (b0 ∩ (a0p0)) ≤ (b0 ∩ (((a0b0) ∪ b1) ∪ (c2 ∩ (c0c1))))

Proof of Theorem dp53lemd
StepHypRef Expression
1 lea 160 . . 3 (b0 ∩ (a0p0)) ≤ b0
2 leor 159 . . . 4 (b0 ∩ (a0p0)) ≤ (b1 ∪ (b0 ∩ (a0p0)))
3 dp53lem.1 . . . . 5 c0 = ((a1a2) ∩ (b1b2))
4 dp53lem.2 . . . . 5 c1 = ((a0a2) ∩ (b0b2))
5 dp53lem.3 . . . . 5 c2 = ((a0a1) ∩ (b0b1))
6 dp53lem.4 . . . . 5 p0 = ((a1b1) ∩ (a2b2))
7 dp53lem.5 . . . . 5 p = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
83, 4, 5, 6, 7dp53lema 1161 . . . 4 (b1 ∪ (b0 ∩ (a0p0))) ≤ (b1 ∪ ((a0a1) ∩ (c0c1)))
92, 8letr 137 . . 3 (b0 ∩ (a0p0)) ≤ (b1 ∪ ((a0a1) ∩ (c0c1)))
101, 9ler2an 173 . 2 (b0 ∩ (a0p0)) ≤ (b0 ∩ (b1 ∪ ((a0a1) ∩ (c0c1))))
113, 4, 5, 6, 7dp53lemc 1163 . . . 4 (b0 ∩ (((a0b0) ∪ b1) ∪ (c2 ∩ (c0c1)))) = (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))
123, 4, 5, 6, 7dp53lemb 1162 . . . 4 (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))) = (b0 ∩ (b1 ∪ ((a0a1) ∩ (c0c1))))
1311, 12tr 62 . . 3 (b0 ∩ (((a0b0) ∪ b1) ∪ (c2 ∩ (c0c1)))) = (b0 ∩ (b1 ∪ ((a0a1) ∩ (c0c1))))
1413cm 61 . 2 (b0 ∩ (b1 ∪ ((a0a1) ∩ (c0c1)))) = (b0 ∩ (((a0b0) ∪ b1) ∪ (c2 ∩ (c0c1))))
1510, 14lbtr 139 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 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:  dp53leme  1165
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