Theorem oa3to4 931

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
oa3to4.oa4.1	a =< b_|_
oa3to4.oa4.2	c =< d_|_
oa3to4.3	g = ((b_|_ ^ a_|_) v (d_|_ ^ c_|_))
oa3to4.4	e = b_|_
oa3to4.5	f = d_|_
oa3to4.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
oa3to4	((a v b) ^ (c v d)) =< (b v (a ^ (c v ((a v c) ^ (b v d)))))

Proof of Theorem oa3to4

Step	Hyp	Ref	Expression
1		oa3to4.oa4.1	. . . 4 a =< b_|_
2	1	lecon3 149	. . 3 b =< a_|_
3		oa3to4.oa4.2	. . . 4 c =< d_|_
4	3	lecon3 149	. . 3 d =< c_|_
5		oa3to4.3	. . 3 g = ((b_|_ ^ a_|_) v (d_|_ ^ c_|_))
6		oa3to4.4	. . 3 e = b_|_
7		oa3to4.5	. . 3 f = d_|_
8		oa3to4.oa3	. . 3 (e ^ ((e ->1 g) v ((f ->1 g) ^ ((e ^ f) v ((e ->1 g) ^ (f ->1 g)))))) =< ((e ^ g) v (f ^ g))
9	2, 4, 5, 6, 7, 8	oa3to4lem6 930	. 2 ((b v a) ^ (d v c)) =< (b v (a ^ (c v ((b v d) ^ (a v c)))))
10	9	oa3to4lem5 929	1 ((a v b) ^ (c v d)) =< (b v (a ^ (c 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 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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