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Theorem dp41lemm 1194
 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
dp41lemm c2 ≤ (c0c1)

Proof of Theorem dp41lemm
StepHypRef Expression
1 dp41lem.1 . . . . . . . 8 c0 = ((a1a2) ∩ (b1b2))
2 dp41lem.2 . . . . . . . 8 c1 = ((a0a2) ∩ (b0b2))
3 dp41lem.3 . . . . . . . 8 c2 = ((a0a1) ∩ (b0b1))
4 dp41lem.4 . . . . . . . 8 p = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
5 dp41lem.5 . . . . . . . 8 p2 = ((a0b0) ∩ (a1b1))
6 dp41lem.6 . . . . . . . 8 p2 ≤ (a2b2)
71, 2, 3, 4, 5, 6dp41lemb 1183 . . . . . . 7 c2 = ((c2 ∩ ((a0b0) ∪ b1)) ∩ ((a0a1) ∪ b1))
81, 2, 3, 4, 5, 6dp41lemc 1185 . . . . . . 7 ((c2 ∩ ((a0b0) ∪ b1)) ∩ ((a0a1) ∪ b1)) ≤ (c2 ∩ ((a0b1) ∪ (c2 ∩ (c0c1))))
97, 8bltr 138 . . . . . 6 c2 ≤ (c2 ∩ ((a0b1) ∪ (c2 ∩ (c0c1))))
101, 2, 3, 4, 5, 6dp41lemd 1186 . . . . . 6 (c2 ∩ ((a0b1) ∪ (c2 ∩ (c0c1)))) = (c2 ∩ ((c0c1) ∪ (c2 ∩ (a0b1))))
119, 10lbtr 139 . . . . 5 c2 ≤ (c2 ∩ ((c0c1) ∪ (c2 ∩ (a0b1))))
121, 2, 3, 4, 5, 6dp41leme 1187 . . . . 5 (c2 ∩ ((c0c1) ∪ (c2 ∩ (a0b1)))) ≤ ((c0c1) ∪ ((a0 ∩ (b0b1)) ∪ (b1 ∩ (a0a1))))
1311, 12letr 137 . . . 4 c2 ≤ ((c0c1) ∪ ((a0 ∩ (b0b1)) ∪ (b1 ∩ (a0a1))))
141, 2, 3, 4, 5, 6dp41lemf 1188 . . . . 5 ((c0c1) ∪ ((a0 ∩ (b0b1)) ∪ (b1 ∩ (a0a1)))) = (((b1b2) ∩ ((a1a2) ∪ (b1 ∩ (a0a1)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (a0 ∩ (b0b1)))))
151, 2, 3, 4, 5, 6dp41lemg 1189 . . . . 5 (((b1b2) ∩ ((a1a2) ∪ (b1 ∩ (a0a1)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (a0 ∩ (b0b1))))) = (((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a1b1)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a0b0)))))
1614, 15tr 62 . . . 4 ((c0c1) ∪ ((a0 ∩ (b0b1)) ∪ (b1 ∩ (a0a1)))) = (((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a1b1)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a0b0)))))
1713, 16lbtr 139 . . 3 c2 ≤ (((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a1b1)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a0b0)))))
181, 2, 3, 4, 5, 6dp41lemh 1190 . . 3 (((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a1b1)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a0b0))))) ≤ (((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a2b2)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a2b2)))))
1917, 18letr 137 . 2 c2 ≤ (((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a2b2)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a2b2)))))
201, 2, 3, 4, 5, 6dp41lemj 1191 . . 3 (((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a2b2)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a2b2))))) = (((b1b2) ∩ ((a1a2) ∪ (b2 ∩ (a0a2)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (a2 ∩ (b1b2)))))
211, 2, 3, 4, 5, 6dp41lemk 1192 . . 3 (((b1b2) ∩ ((a1a2) ∪ (b2 ∩ (a0a2)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (a2 ∩ (b1b2))))) = ((c0 ∪ (b2 ∩ (a0a2))) ∪ (c1 ∪ (a2 ∩ (b1b2))))
221, 2, 3, 4, 5, 6dp41leml 1193 . . 3 ((c0 ∪ (b2 ∩ (a0a2))) ∪ (c1 ∪ (a2 ∩ (b1b2)))) = (c0c1)
2320, 21, 223tr 65 . 2 (((b1b2) ∩ ((a1a2) ∪ (a0 ∩ (a2b2)))) ∪ ((a0a2) ∩ ((b0b2) ∪ (b1 ∩ (a2b2))))) = (c0c1)
2419, 23lbtr 139 1 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:  dp41  1195
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