Theorem oa3to4lem6 930

Description: Orthoarguesian law (Godowski/Greechie 3-variable to 4-variable). The first 2 hypotheses are those for 4-OA. The next 3 are variable substitutions into 3-OA. The last is the 3-OA. The proof uses OM logic only.

Hypotheses

Ref	Expression
oa3to4lem6.oa4.1	a =< b_|_
oa3to4lem6.oa4.2	c =< d_|_
oa3to4lem6.3	g = ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))
oa3to4lem6.4	e = a_|_
oa3to4lem6.5	f = c_|_
oa3to4lem6.oa3	(e ^ ((e ->1 g) v ((f ->1 g) ^ ((e ^ f) v ((e ->1 g) ^ (f ->1 g)))))) =< ((e ^ g) v (f ^ g))

Assertion

Ref	Expression
oa3to4lem6	((a v b) ^ (c v d)) =< (a v (b ^ (d v ((a v c) ^ (b v d)))))

Proof of Theorem oa3to4lem6

Step	Hyp	Ref	Expression
1		oa3to4lem6.oa4.1	. . . . . 6 a =< b_|_
2	1	lecon3 149	. . . . 5 b =< a_|_
3	2	lecon 146	. . . 4 a_|__|_ =< b_|_
4		oa3to4lem6.oa4.2	. . . . . 6 c =< d_|_
5	4	lecon3 149	. . . . 5 d =< c_|_
6	5	lecon 146	. . . 4 c_|__|_ =< d_|_
7		id 58	. . . 4 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_)) = ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))
8		oa3to4lem6.oa3	. . . . 5 (e ^ ((e ->1 g) v ((f ->1 g) ^ ((e ^ f) v ((e ->1 g) ^ (f ->1 g)))))) =< ((e ^ g) v (f ^ g))
9		oa3to4lem6.4	. . . . . 6 e = a_|_
10		oa3to4lem6.3	. . . . . . . 8 g = ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))
11	9, 10	ud1lem0ab 249	. . . . . . 7 (e ->1 g) = (a_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_)))
12		oa3to4lem6.5	. . . . . . . . 9 f = c_|_
13	12, 10	ud1lem0ab 249	. . . . . . . 8 (f ->1 g) = (c_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_)))
14	9, 12	2an 72	. . . . . . . . 9 (e ^ f) = (a_|_ ^ c_|_)
15	11, 13	2an 72	. . . . . . . . 9 ((e ->1 g) ^ (f ->1 g)) = ((a_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))) ^ (c_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))))
16	14, 15	2or 67	. . . . . . . 8 ((e ^ f) v ((e ->1 g) ^ (f ->1 g))) = ((a_|_ ^ c_|_) v ((a_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))) ^ (c_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_)))))
17	13, 16	2an 72	. . . . . . 7 ((f ->1 g) ^ ((e ^ f) v ((e ->1 g) ^ (f ->1 g)))) = ((c_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))) ^ ((a_|_ ^ c_|_) v ((a_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))) ^ (c_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))))))
18	11, 17	2or 67	. . . . . 6 ((e ->1 g) v ((f ->1 g) ^ ((e ^ f) v ((e ->1 g) ^ (f ->1 g))))) = ((a_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))) v ((c_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))) ^ ((a_|_ ^ c_|_) v ((a_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))) ^ (c_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_)))))))
19	9, 18	2an 72	. . . . 5 (e ^ ((e ->1 g) v ((f ->1 g) ^ ((e ^ f) v ((e ->1 g) ^ (f ->1 g)))))) = (a_|_ ^ ((a_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))) v ((c_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))) ^ ((a_|_ ^ c_|_) v ((a_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))) ^ (c_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))))))))
20	9, 10	2an 72	. . . . . 6 (e ^ g) = (a_|_ ^ ((a_|_ ^ b_|_) v (c_|_ ^ d_|_)))
21	12, 10	2an 72	. . . . . 6 (f ^ g) = (c_|_ ^ ((a_|_ ^ b_|_) v (c_|_ ^ d_|_)))
22	20, 21	2or 67	. . . . 5 ((e ^ g) v (f ^ g)) = ((a_|_ ^ ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))) v (c_|_ ^ ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))))
23	8, 19, 22	le3tr2 133	. . . 4 (a_|_ ^ ((a_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))) v ((c_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))) ^ ((a_|_ ^ c_|_) v ((a_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))) ^ (c_|_ ->1 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_)))))))) =< ((a_|_ ^ ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))) v (c_|_ ^ ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))))
24	3, 6, 7, 23	oa3to4lem4 928	. . 3 (a_|_ ^ (b_|_ v (d_|_ ^ ((a_|_ ^ c_|_) v (b_|_ ^ d_|_))))) =< ((a_|_ ^ b_|_) v (c_|_ ^ d_|_))
25		anor3 82	. . . . . . . . . . 11 (a_|_ ^ c_|_) = (a v c)_|_
26		anor3 82	. . . . . . . . . . 11 (b_|_ ^ d_|_) = (b v d)_|_
27	25, 26	2or 67	. . . . . . . . . 10 ((a_|_ ^ c_|_) v (b_|_ ^ d_|_)) = ((a v c)_|_ v (b v d)_|_)
28		oran3 85	. . . . . . . . . 10 ((a v c)_|_ v (b v d)_|_) = ((a v c) ^ (b v d))_|_
29	27, 28	ax-r2 35	. . . . . . . . 9 ((a_|_ ^ c_|_) v (b_|_ ^ d_|_)) = ((a v c) ^ (b v d))_|_
30	29	lan 70	. . . . . . . 8 (d_|_ ^ ((a_|_ ^ c_|_) v (b_|_ ^ d_|_))) = (d_|_ ^ ((a v c) ^ (b v d))_|_)
31		anor3 82	. . . . . . . 8 (d_|_ ^ ((a v c) ^ (b v d))_|_) = (d v ((a v c) ^ (b v d)))_|_
32	30, 31	ax-r2 35	. . . . . . 7 (d_|_ ^ ((a_|_ ^ c_|_) v (b_|_ ^ d_|_))) = (d v ((a v c) ^ (b v d)))_|_
33	32	lor 66	. . . . . 6 (b_|_ v (d_|_ ^ ((a_|_ ^ c_|_) v (b_|_ ^ d_|_)))) = (b_|_ v (d v ((a v c) ^ (b v d)))_|_)
34		oran3 85	. . . . . 6 (b_|_ v (d v ((a v c) ^ (b v d)))_|_) = (b ^ (d v ((a v c) ^ (b v d))))_|_
35	33, 34	ax-r2 35	. . . . 5 (b_|_ v (d_|_ ^ ((a_|_ ^ c_|_) v (b_|_ ^ d_|_)))) = (b ^ (d v ((a v c) ^ (b v d))))_|_
36	35	lan 70	. . . 4 (a_|_ ^ (b_|_ v (d_|_ ^ ((a_|_ ^ c_|_) v (b_|_ ^ d_|_))))) = (a_|_ ^ (b ^ (d v ((a v c) ^ (b v d))))_|_)
37		anor3 82	. . . 4 (a_|_ ^ (b ^ (d v ((a v c) ^ (b v d))))_|_) = (a v (b ^ (d v ((a v c) ^ (b v d)))))_|_
38	36, 37	ax-r2 35	. . 3 (a_|_ ^ (b_|_ v (d_|_ ^ ((a_|_ ^ c_|_) v (b_|_ ^ d_|_))))) = (a v (b ^ (d v ((a v c) ^ (b v d)))))_|_
39		anor3 82	. . . . 5 (a_|_ ^ b_|_) = (a v b)_|_
40		anor3 82	. . . . 5 (c_|_ ^ d_|_) = (c v d)_|_
41	39, 40	2or 67	. . . 4 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_)) = ((a v b)_|_ v (c v d)_|_)
42		oran3 85	. . . 4 ((a v b)_|_ v (c v d)_|_) = ((a v b) ^ (c v d))_|_
43	41, 42	ax-r2 35	. . 3 ((a_|_ ^ b_|_) v (c_|_ ^ d_|_)) = ((a v b) ^ (c v d))_|_
44	24, 38, 43	le3tr2 133	. 2 (a v (b ^ (d v ((a v c) ^ (b v d)))))_|_ =< ((a v b) ^ (c v d))_|_
45	44	lecon1 147	1 ((a v b) ^ (c v d)) =< (a v (b ^ (d v ((a v c) ^ (b v d)))))

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13

This theorem is referenced by:  oa3to4 931

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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