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Theorem oa4v3v 934
Description: 4-variable OA to 3-variable OA (Godowski/Greechie Eq. IV).
Hypotheses
Ref Expression
oa4v3v.1 d =< b'
oa4v3v.2 e =< c'
oa4v3v.3 ((d v b) ^ (e v c)) =< (b v (d ^ (e v ((d v e) ^ (b v c)))))
oa4v3v.4 d = (a ->2 b)'
oa4v3v.5 e = (a ->2 c)'
Assertion
Ref Expression
oa4v3v (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< ((b' ^ (a ->2 b)) v (c' ^ (a ->2 c)))
Proof of Theorem oa4v3v
StepHypRef Expression
1 oa4v3v.3 . . 3 ((d v b) ^ (e v c)) =< (b v (d ^ (e v ((d v e) ^ (b v c)))))
2 ax-a2 31 . . . . . 6 (d v b) = (b v d)
3 oa4v3v.4 . . . . . . 7 d = (a ->2 b)'
43lor 70 . . . . . 6 (b v d) = (b v (a ->2 b)')
5 oran1 91 . . . . . 6 (b v (a ->2 b)') = (b' ^ (a ->2 b))'
62, 4, 53tr 65 . . . . 5 (d v b) = (b' ^ (a ->2 b))'
7 ax-a2 31 . . . . . 6 (e v c) = (c v e)
8 oa4v3v.5 . . . . . . 7 e = (a ->2 c)'
98lor 70 . . . . . 6 (c v e) = (c v (a ->2 c)')
10 oran1 91 . . . . . 6 (c v (a ->2 c)') = (c' ^ (a ->2 c))'
117, 9, 103tr 65 . . . . 5 (e v c) = (c' ^ (a ->2 c))'
126, 112an 79 . . . 4 ((d v b) ^ (e v c)) = ((b' ^ (a ->2 b))' ^ (c' ^ (a ->2 c))')
13 anor3 90 . . . 4 ((b' ^ (a ->2 b))' ^ (c' ^ (a ->2 c))') = ((b' ^ (a ->2 b)) v (c' ^ (a ->2 c)))'
1412, 13ax-r2 36 . . 3 ((d v b) ^ (e v c)) = ((b' ^ (a ->2 b)) v (c' ^ (a ->2 c)))'
15 ancom 74 . . . . . . . . . 10 ((d v e) ^ (b v c)) = ((b v c) ^ (d v e))
163, 82or 72 . . . . . . . . . . . 12 (d v e) = ((a ->2 b)' v (a ->2 c)')
17 oran3 93 . . . . . . . . . . . 12 ((a ->2 b)' v (a ->2 c)') = ((a ->2 b) ^ (a ->2 c))'
1816, 17ax-r2 36 . . . . . . . . . . 11 (d v e) = ((a ->2 b) ^ (a ->2 c))'
1918lan 77 . . . . . . . . . 10 ((b v c) ^ (d v e)) = ((b v c) ^ ((a ->2 b) ^ (a ->2 c))')
20 anor1 88 . . . . . . . . . 10 ((b v c) ^ ((a ->2 b) ^ (a ->2 c))') = ((b v c)' v ((a ->2 b) ^ (a ->2 c)))'
2115, 19, 203tr 65 . . . . . . . . 9 ((d v e) ^ (b v c)) = ((b v c)' v ((a ->2 b) ^ (a ->2 c)))'
228, 212or 72 . . . . . . . 8 (e v ((d v e) ^ (b v c))) = ((a ->2 c)' v ((b v c)' v ((a ->2 b) ^ (a ->2 c)))')
23 oran3 93 . . . . . . . 8 ((a ->2 c)' v ((b v c)' v ((a ->2 b) ^ (a ->2 c)))') = ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))'
2422, 23ax-r2 36 . . . . . . 7 (e v ((d v e) ^ (b v c))) = ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))'
253, 242an 79 . . . . . 6 (d ^ (e v ((d v e) ^ (b v c)))) = ((a ->2 b)' ^ ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))')
26 anor3 90 . . . . . 6 ((a ->2 b)' ^ ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))') = ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))'
2725, 26ax-r2 36 . . . . 5 (d ^ (e v ((d v e) ^ (b v c)))) = ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))'
2827lor 70 . . . 4 (b v (d ^ (e v ((d v e) ^ (b v c))))) = (b v ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))')
29 oran1 91 . . . 4 (b v ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))') = (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))))'
3028, 29ax-r2 36 . . 3 (b v (d ^ (e v ((d v e) ^ (b v c))))) = (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))))'
311, 14, 30le3tr2 141 . 2 ((b' ^ (a ->2 b)) v (c' ^ (a ->2 c)))' =< (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))))'
3231lecon1 155 1 (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< ((b' ^ (a ->2 b)) v (c' ^ (a ->2 c)))
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v 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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