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Theorem dp35lemf 1172
 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
dp35lemf (a0p) ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))

Proof of Theorem dp35lemf
StepHypRef Expression
1 leo 158 . 2 a0 ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))
2 dp35lem.1 . . 3 c0 = ((a1a2) ∩ (b1b2))
3 dp35lem.2 . . 3 c1 = ((a0a2) ∩ (b0b2))
4 dp35lem.3 . . 3 c2 = ((a0a1) ∩ (b0b1))
5 dp35lem.4 . . 3 p0 = ((a1b1) ∩ (a2b2))
6 dp35lem.5 . . 3 p = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
72, 3, 4, 5, 6dp35lemg 1171 . 2 p ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))
81, 7lel2or 170 1 (a0p) ≤ (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 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:  dp35leme  1173
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