Theorem oa3to4 951

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. (Contributed by NM, 19-Dec-1998.)

Hypotheses

Ref	Expression
oa3to4.oa4.1	a ≤ b⊥
oa3to4.oa4.2	c ≤ d⊥
oa3to4.3	g = ((b⊥ ∩ a⊥ ) ∪ (d⊥ ∩ c⊥ ))
oa3to4.4	e = b⊥
oa3to4.5	f = d⊥
oa3to4.oa3	(e ∩ ((e →1 g) ∪ ((f →1 g) ∩ ((e ∩ f) ∪ ((e →1 g) ∩ (f →1 g)))))) ≤ ((e ∩ g) ∪ (f ∩ g))

Assertion

Ref	Expression
oa3to4	((a ∪ b) ∩ (c ∪ d)) ≤ (b ∪ (a ∩ (c ∪ ((a ∪ c) ∩ (b ∪ d)))))

Proof of Theorem oa3to4

Step	Hyp	Ref	Expression
1		oa3to4.oa4.1	. . . 4 a ≤ b⊥
2	1	lecon3 157	. . 3 b ≤ a⊥
3		oa3to4.oa4.2	. . . 4 c ≤ d⊥
4	3	lecon3 157	. . 3 d ≤ c⊥
5		oa3to4.3	. . 3 g = ((b⊥ ∩ a⊥ ) ∪ (d⊥ ∩ c⊥ ))
6		oa3to4.4	. . 3 e = b⊥
7		oa3to4.5	. . 3 f = d⊥
8		oa3to4.oa3	. . 3 (e ∩ ((e →1 g) ∪ ((f →1 g) ∩ ((e ∩ f) ∪ ((e →1 g) ∩ (f →1 g)))))) ≤ ((e ∩ g) ∪ (f ∩ g))
9	2, 4, 5, 6, 7, 8	oa3to4lem6 950	. 2 ((b ∪ a) ∩ (d ∪ c)) ≤ (b ∪ (a ∩ (c ∪ ((b ∪ d) ∩ (a ∪ c)))))
10	9	oa3to4lem5 949	1 ((a ∪ b) ∩ (c ∪ d)) ≤ (b ∪ (a ∩ (c ∪ ((a ∪ c) ∩ (b ∪ d)))))

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by: (None)