Theorem oa4cl 1027

Description: 4-variable OA closed equational form. (Contributed by NM, 1-Dec-1998.)

Assertion

Ref	Expression
oa4cl	((a ∪ (b ∩ a⊥ )) ∩ (c ∪ (d ∩ c⊥ ))) ≤ ((b ∩ a⊥ ) ∪ (a ∩ (c ∪ ((a ∪ c) ∩ ((b ∩ a⊥ ) ∪ (d ∩ c⊥ ))))))

Proof of Theorem oa4cl

Step	Hyp	Ref	Expression
1		leor 159	. . 3 a ≤ (b⊥ ∪ a)
2		oran2 92	. . 3 (b⊥ ∪ a) = (b ∩ a⊥ )⊥
3	1, 2	lbtr 139	. 2 a ≤ (b ∩ a⊥ )⊥
4		leor 159	. . 3 c ≤ (d⊥ ∪ c)
5		oran2 92	. . 3 (d⊥ ∪ c) = (d ∩ c⊥ )⊥
6	4, 5	lbtr 139	. 2 c ≤ (d ∩ c⊥ )⊥
7	3, 6	ax-oal4 1026	1 ((a ∪ (b ∩ a⊥ )) ∩ (c ∪ (d ∩ c⊥ ))) ≤ ((b ∩ a⊥ ) ∪ (a ∩ (c ∪ ((a ∪ c) ∩ ((b ∩ a⊥ ) ∪ (d ∩ c⊥ ))))))

This proof depends on syntax axioms:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-oal4 1026

This proof depends on definitions:  df-a 40  df-le1 130  df-le2 131

This theorem is used by: (None)