Theorem oa4v3v 914

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

Step	Hyp	Ref	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 30	. . . . . 6 (d v b) = (b v d)
3		oa4v3v.4	. . . . . . 7 d = (a ->2 b)_|_
4	3	lor 66	. . . . . 6 (b v d) = (b v (a ->2 b)_|_)
5		oran1 83	. . . . . 6 (b v (a ->2 b)_|_) = (b_|_ ^ (a ->2 b))_|_
6	2, 4, 5	3tr 62	. . . . 5 (d v b) = (b_|_ ^ (a ->2 b))_|_
7		ax-a2 30	. . . . . 6 (e v c) = (c v e)
8		oa4v3v.5	. . . . . . 7 e = (a ->2 c)_|_
9	8	lor 66	. . . . . 6 (c v e) = (c v (a ->2 c)_|_)
10		oran1 83	. . . . . 6 (c v (a ->2 c)_|_) = (c_|_ ^ (a ->2 c))_|_
11	7, 9, 10	3tr 62	. . . . 5 (e v c) = (c_|_ ^ (a ->2 c))_|_
12	6, 11	2an 72	. . . 4 ((d v b) ^ (e v c)) = ((b_|_ ^ (a ->2 b))_|_ ^ (c_|_ ^ (a ->2 c))_|_)
13		anor3 82	. . . 4 ((b_|_ ^ (a ->2 b))_|_ ^ (c_|_ ^ (a ->2 c))_|_) = ((b_|_ ^ (a ->2 b)) v (c_|_ ^ (a ->2 c)))_|_
14	12, 13	ax-r2 35	. . 3 ((d v b) ^ (e v c)) = ((b_|_ ^ (a ->2 b)) v (c_|_ ^ (a ->2 c)))_|_
15		ancom 68	. . . . . . . . . 10 ((d v e) ^ (b v c)) = ((b v c) ^ (d v e))
16	3, 8	2or 67	. . . . . . . . . . . 12 (d v e) = ((a ->2 b)_|_ v (a ->2 c)_|_)
17		oran3 85	. . . . . . . . . . . 12 ((a ->2 b)_|_ v (a ->2 c)_|_) = ((a ->2 b) ^ (a ->2 c))_|_
18	16, 17	ax-r2 35	. . . . . . . . . . 11 (d v e) = ((a ->2 b) ^ (a ->2 c))_|_
19	18	lan 70	. . . . . . . . . 10 ((b v c) ^ (d v e)) = ((b v c) ^ ((a ->2 b) ^ (a ->2 c))_|_)
20		anor1 80	. . . . . . . . . 10 ((b v c) ^ ((a ->2 b) ^ (a ->2 c))_|_) = ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))_|_
21	15, 19, 20	3tr 62	. . . . . . . . 9 ((d v e) ^ (b v c)) = ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))_|_
22	8, 21	2or 67	. . . . . . . 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 85	. . . . . . . 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))))_|_
24	22, 23	ax-r2 35	. . . . . . 7 (e v ((d v e) ^ (b v c))) = ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))_|_
25	3, 24	2an 72	. . . . . 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 82	. . . . . 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)))))_|_
27	25, 26	ax-r2 35	. . . . 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)))))_|_
28	27	lor 66	. . . 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 83	. . . 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))))))_|_
30	28, 29	ax-r2 35	. . 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))))))_|_
31	1, 14, 30	le3tr2 133	. 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))))))_|_
32	31	lecon1 147	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)))

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->2 wi2 14

This theorem is referenced by:  oa43v 1008  oa63v 1011

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-le1 122  df-le2 123

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