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Theorem dp32 1194
 Description: Part of theorem from Alan Day and Doug Pickering, "A note on the Arguesian lattice identity," Studia Sci. Math. Hungar. 19:303-305 (1982). (3)=>(2)
Hypotheses
Ref Expression
dp32.1 c0 = ((a1a2) ∩ (b1b2))
dp32.2 c1 = ((a0a2) ∩ (b0b2))
dp32.3 c2 = ((a0a1) ∩ (b0b1))
dp32.4 p = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
Assertion
Ref Expression
dp32 p ≤ ((a0 ∩ (a1 ∪ (c2 ∩ (c0c1)))) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))

Proof of Theorem dp32
StepHypRef Expression
1 dp32.1 . . . 4 c0 = ((a1a2) ∩ (b1b2))
2 dp32.2 . . . 4 c1 = ((a0a2) ∩ (b0b2))
3 dp32.3 . . . 4 c2 = ((a0a1) ∩ (b0b1))
4 dp32.4 . . . 4 p = (((a0b0) ∩ (a1b1)) ∩ (a2b2))
51, 2, 3, 4dp53 1168 . . 3 p ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))
6 ancom 74 . . . . 5 ((a1a2) ∩ (b1b2)) = ((b1b2) ∩ (a1a2))
71, 6tr 62 . . . 4 c0 = ((b1b2) ∩ (a1a2))
8 ancom 74 . . . . 5 ((a0a2) ∩ (b0b2)) = ((b0b2) ∩ (a0a2))
92, 8tr 62 . . . 4 c1 = ((b0b2) ∩ (a0a2))
10 ancom 74 . . . . 5 ((a0a1) ∩ (b0b1)) = ((b0b1) ∩ (a0a1))
113, 10tr 62 . . . 4 c2 = ((b0b1) ∩ (a0a1))
12 orcom 73 . . . . . . 7 (a0b0) = (b0a0)
13 orcom 73 . . . . . . 7 (a1b1) = (b1a1)
1412, 132an 79 . . . . . 6 ((a0b0) ∩ (a1b1)) = ((b0a0) ∩ (b1a1))
15 orcom 73 . . . . . 6 (a2b2) = (b2a2)
1614, 152an 79 . . . . 5 (((a0b0) ∩ (a1b1)) ∩ (a2b2)) = (((b0a0) ∩ (b1a1)) ∩ (b2a2))
174, 16tr 62 . . . 4 p = (((b0a0) ∩ (b1a1)) ∩ (b2a2))
187, 9, 11, 17dp53 1168 . . 3 p ≤ (b0 ∪ (a0 ∩ (a1 ∪ (c2 ∩ (c0c1)))))
195, 18ler2an 173 . 2 p ≤ ((a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))) ∩ (b0 ∪ (a0 ∩ (a1 ∪ (c2 ∩ (c0c1))))))
20 leao1 162 . . . 4 (a0 ∩ (a1 ∪ (c2 ∩ (c0c1)))) ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))
2120mldual2i 1125 . . 3 ((a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))) ∩ (b0 ∪ (a0 ∩ (a1 ∪ (c2 ∩ (c0c1)))))) = (((a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))) ∩ b0) ∪ (a0 ∩ (a1 ∪ (c2 ∩ (c0c1)))))
22 ancom 74 . . . . 5 ((a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))) ∩ b0) = (b0 ∩ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))))
23 mldual 1122 . . . . 5 (b0 ∩ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))) = ((b0a0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))
24 leao2 163 . . . . . . . . . . 11 (b0a0) ≤ (a0a1)
25 leao1 162 . . . . . . . . . . 11 (b0a0) ≤ (b0b1)
2624, 25ler2an 173 . . . . . . . . . 10 (b0a0) ≤ ((a0a1) ∩ (b0b1))
273cm 61 . . . . . . . . . 10 ((a0a1) ∩ (b0b1)) = c2
2826, 27lbtr 139 . . . . . . . . 9 (b0a0) ≤ c2
29 leao2 163 . . . . . . . . . . . 12 (b0a0) ≤ (a0a2)
30 leao1 162 . . . . . . . . . . . 12 (b0a0) ≤ (b0b2)
3129, 30ler2an 173 . . . . . . . . . . 11 (b0a0) ≤ ((a0a2) ∩ (b0b2))
322cm 61 . . . . . . . . . . 11 ((a0a2) ∩ (b0b2)) = c1
3331, 32lbtr 139 . . . . . . . . . 10 (b0a0) ≤ c1
3433lerr 150 . . . . . . . . 9 (b0a0) ≤ (c0c1)
3528, 34ler2an 173 . . . . . . . 8 (b0a0) ≤ (c2 ∩ (c0c1))
3635lerr 150 . . . . . . 7 (b0a0) ≤ (b1 ∪ (c2 ∩ (c0c1)))
3736ml2i 1123 . . . . . 6 ((b0a0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))) = (((b0a0) ∪ b0) ∩ (b1 ∪ (c2 ∩ (c0c1))))
38 lea 160 . . . . . . . 8 (b0a0) ≤ b0
3938df-le2 131 . . . . . . 7 ((b0a0) ∪ b0) = b0
4039ran 78 . . . . . 6 (((b0a0) ∪ b0) ∩ (b1 ∪ (c2 ∩ (c0c1)))) = (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))
4137, 40tr 62 . . . . 5 ((b0a0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))) = (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))
4222, 23, 413tr 65 . . . 4 ((a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))) ∩ b0) = (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))
4342ror 71 . . 3 (((a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))) ∩ b0) ∪ (a0 ∩ (a1 ∪ (c2 ∩ (c0c1))))) = ((b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))) ∪ (a0 ∩ (a1 ∪ (c2 ∩ (c0c1)))))
44 orcom 73 . . 3 ((b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))) ∪ (a0 ∩ (a1 ∪ (c2 ∩ (c0c1))))) = ((a0 ∩ (a1 ∪ (c2 ∩ (c0c1)))) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))
4521, 43, 443tr 65 . 2 ((a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1))))) ∩ (b0 ∪ (a0 ∩ (a1 ∪ (c2 ∩ (c0c1)))))) = ((a0 ∩ (a1 ∪ (c2 ∩ (c0c1)))) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0c1)))))
4619, 45lbtr 139 1 p ≤ ((a0 ∩ (a1 ∪ (c2 ∩ (c0c1)))) ∪ (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:  dp23  1195
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