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Theorem dp53leme 1165
 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
dp53leme (b0 ∩ (a0p0)) ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))

Proof of Theorem dp53leme
StepHypRef Expression
1 dp53lem.1 . . 3 c0 = ((a1a2) ∩ (b1b2))
2 dp53lem.2 . . 3 c1 = ((a0a2) ∩ (b0b2))
3 dp53lem.3 . . 3 c2 = ((a0a1) ∩ (b0b1))
4 dp53lem.4 . . 3 p0 = ((a1b1) ∩ (a2b2))
5 dp53lem.5 . . 3 p = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
61, 2, 3, 4, 5dp53lemd 1164 . 2 (b0 ∩ (a0p0)) ≤ (b0 ∩ (((a0b0) ∪ b1) ∪ (c2 ∩ (c0c1))))
7 orass 75 . . . . . 6 (((a0b0) ∪ b1) ∪ (c2 ∩ (c0c1))) = ((a0b0) ∪ (b1 ∪ (c2 ∩ (c0c1))))
8 orcom 73 . . . . . 6 ((a0b0) ∪ (b1 ∪ (c2 ∩ (c0c1)))) = ((b1 ∪ (c2 ∩ (c0c1))) ∪ (a0b0))
97, 8tr 62 . . . . 5 (((a0b0) ∪ b1) ∪ (c2 ∩ (c0c1))) = ((b1 ∪ (c2 ∩ (c0c1))) ∪ (a0b0))
109lan 77 . . . 4 (b0 ∩ (((a0b0) ∪ b1) ∪ (c2 ∩ (c0c1)))) = (b0 ∩ ((b1 ∪ (c2 ∩ (c0c1))) ∪ (a0b0)))
11 lear 161 . . . . 5 (a0b0) ≤ b0
1211mldual2i 1125 . . . 4 (b0 ∩ ((b1 ∪ (c2 ∩ (c0c1))) ∪ (a0b0))) = ((b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))) ∪ (a0b0))
13 orcom 73 . . . 4 ((b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))) ∪ (a0b0)) = ((a0b0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))
1410, 12, 133tr 65 . . 3 (b0 ∩ (((a0b0) ∪ b1) ∪ (c2 ∩ (c0c1)))) = ((a0b0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))
15 lea 160 . . . 4 (a0b0) ≤ a0
1615leror 152 . . 3 ((a0b0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))) ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))
1714, 16bltr 138 . 2 (b0 ∩ (((a0b0) ∪ b1) ∪ (c2 ∩ (c0c1)))) ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))
186, 17letr 137 1 (b0 ∩ (a0p0)) ≤ (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 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:  dp53lemf  1166
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