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Theorem dp41lema 1180
 Description: Part of proof (4)=>(1) in Day/Pickering 1982.
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
dp41lema ((a0b0) ∩ (a1b1)) ≤ ((a0b1) ∪ (c2 ∩ (c0c1)))

Proof of Theorem dp41lema
StepHypRef Expression
1 dp41lem.5 . . . . . . 7 p2 = ((a0b0) ∩ (a1b1))
21cm 61 . . . . . 6 ((a0b0) ∩ (a1b1)) = p2
3 dp41lem.6 . . . . . 6 p2 ≤ (a2b2)
42, 3bltr 138 . . . . 5 ((a0b0) ∩ (a1b1)) ≤ (a2b2)
54df2le2 136 . . . 4 (((a0b0) ∩ (a1b1)) ∩ (a2b2)) = ((a0b0) ∩ (a1b1))
65cm 61 . . 3 ((a0b0) ∩ (a1b1)) = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
7 dp41lem.4 . . . 4 p = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
87cm 61 . . 3 (((a0b0) ∩ (a1b1)) ∩ (a2b2)) = p
96, 8tr 62 . 2 ((a0b0) ∩ (a1b1)) = p
10 dp41lem.1 . . 3 c0 = ((a1a2) ∩ (b1b2))
11 dp41lem.2 . . 3 c1 = ((a0a2) ∩ (b0b2))
12 dp41lem.3 . . 3 c2 = ((a0a1) ∩ (b0b1))
1310, 11, 12, 7dp34 1179 . 2 p ≤ ((a0b1) ∪ (c2 ∩ (c0c1)))
149, 13bltr 138 1 ((a0b0) ∩ (a1b1)) ≤ ((a0b1) ∪ (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:  dp41lemc  1183
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