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Theorem oa4v3v 934
 Description: 4-variable OA to 3-variable OA (Godowski/Greechie Eq. IV). (Contributed by NM, 28-Nov-1998.)
Hypotheses
Ref Expression
oa4v3v.1 db
oa4v3v.2 ec
oa4v3v.3 ((db) ∩ (ec)) ≤ (b ∪ (d ∩ (e ∪ ((de) ∩ (bc)))))
oa4v3v.4 d = (a2 b)
oa4v3v.5 e = (a2 c)
Assertion
Ref Expression
oa4v3v (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ ((b ∩ (a2 b)) ∪ (c ∩ (a2 c)))

Proof of Theorem oa4v3v
StepHypRef Expression
1 oa4v3v.3 . . 3 ((db) ∩ (ec)) ≤ (b ∪ (d ∩ (e ∪ ((de) ∩ (bc)))))
2 ax-a2 31 . . . . . 6 (db) = (bd)
3 oa4v3v.4 . . . . . . 7 d = (a2 b)
43lor 70 . . . . . 6 (bd) = (b ∪ (a2 b) )
5 oran1 91 . . . . . 6 (b ∪ (a2 b) ) = (b ∩ (a2 b))
62, 4, 53tr 65 . . . . 5 (db) = (b ∩ (a2 b))
7 ax-a2 31 . . . . . 6 (ec) = (ce)
8 oa4v3v.5 . . . . . . 7 e = (a2 c)
98lor 70 . . . . . 6 (ce) = (c ∪ (a2 c) )
10 oran1 91 . . . . . 6 (c ∪ (a2 c) ) = (c ∩ (a2 c))
117, 9, 103tr 65 . . . . 5 (ec) = (c ∩ (a2 c))
126, 112an 79 . . . 4 ((db) ∩ (ec)) = ((b ∩ (a2 b)) ∩ (c ∩ (a2 c)) )
13 anor3 90 . . . 4 ((b ∩ (a2 b)) ∩ (c ∩ (a2 c)) ) = ((b ∩ (a2 b)) ∪ (c ∩ (a2 c)))
1412, 13ax-r2 36 . . 3 ((db) ∩ (ec)) = ((b ∩ (a2 b)) ∪ (c ∩ (a2 c)))
15 ancom 74 . . . . . . . . . 10 ((de) ∩ (bc)) = ((bc) ∩ (de))
163, 82or 72 . . . . . . . . . . . 12 (de) = ((a2 b) ∪ (a2 c) )
17 oran3 93 . . . . . . . . . . . 12 ((a2 b) ∪ (a2 c) ) = ((a2 b) ∩ (a2 c))
1816, 17ax-r2 36 . . . . . . . . . . 11 (de) = ((a2 b) ∩ (a2 c))
1918lan 77 . . . . . . . . . 10 ((bc) ∩ (de)) = ((bc) ∩ ((a2 b) ∩ (a2 c)) )
20 anor1 88 . . . . . . . . . 10 ((bc) ∩ ((a2 b) ∩ (a2 c)) ) = ((bc) ∪ ((a2 b) ∩ (a2 c)))
2115, 19, 203tr 65 . . . . . . . . 9 ((de) ∩ (bc)) = ((bc) ∪ ((a2 b) ∩ (a2 c)))
228, 212or 72 . . . . . . . 8 (e ∪ ((de) ∩ (bc))) = ((a2 c) ∪ ((bc) ∪ ((a2 b) ∩ (a2 c))) )
23 oran3 93 . . . . . . . 8 ((a2 c) ∪ ((bc) ∪ ((a2 b) ∩ (a2 c))) ) = ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))
2422, 23ax-r2 36 . . . . . . 7 (e ∪ ((de) ∩ (bc))) = ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))
253, 242an 79 . . . . . 6 (d ∩ (e ∪ ((de) ∩ (bc)))) = ((a2 b) ∩ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) )
26 anor3 90 . . . . . 6 ((a2 b) ∩ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ) = ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))
2725, 26ax-r2 36 . . . . 5 (d ∩ (e ∪ ((de) ∩ (bc)))) = ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))
2827lor 70 . . . 4 (b ∪ (d ∩ (e ∪ ((de) ∩ (bc))))) = (b ∪ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))) )
29 oran1 91 . . . 4 (b ∪ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))) ) = (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))))
3028, 29ax-r2 36 . . 3 (b ∪ (d ∩ (e ∪ ((de) ∩ (bc))))) = (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))))
311, 14, 30le3tr2 141 . 2 ((b ∩ (a2 b)) ∪ (c ∩ (a2 c))) ≤ (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))))
3231lecon1 155 1 (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ ((b ∩ (a2 b)) ∪ (c ∩ (a2 c)))
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →2 wi2 13 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-a 40  df-le1 130  df-le2 131 This theorem is referenced by:  oa43v  1028  oa63v  1032
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