Theorem oa4cl 1007

Description: 4-variable OA closed equational form)

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 151	. . 3 a ≤ (b⊥ ∪ a)
2		oran2 84	. . 3 (b⊥ ∪ a) = (b ∩ a⊥ )⊥
3	1, 2	lbtr 131	. 2 a ≤ (b ∩ a⊥ )⊥
4		leor 151	. . 3 c ≤ (d⊥ ∪ c)
5		oran2 84	. . 3 (d⊥ ∪ c) = (d ∩ c⊥ )⊥
6	4, 5	lbtr 131	. 2 c ≤ (d ∩ c⊥ )⊥
7	3, 6	ax-oal4 1006	1 ((a ∪ (b ∩ a⊥ )) ∩ (c ∪ (d ∩ c⊥ ))) ≤ ((b ∩ a⊥ ) ∪ (a ∩ (c ∪ ((a ∪ c) ∩ ((b ∩ a⊥ ) ∪ (d ∩ c⊥ ))))))

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7

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  ax-oal4 1006

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

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