Theorem d6oa 997

Description: Derivation of 6-variable orthoarguesian law from OA distributive law. (Contributed by NM, 30-Dec-1998.)

Hypotheses

Ref	Expression
d6oa.1	a ≤ b⊥
d6oa.2	c ≤ d⊥
d6oa.3	e ≤ f⊥

Assertion

Ref	Expression
d6oa	(((a ∪ b) ∩ (c ∪ d)) ∩ (e ∪ f)) ≤ (b ∪ (a ∩ (c ∪ (((a ∪ c) ∩ (b ∪ d)) ∩ (((a ∪ e) ∩ (b ∪ f)) ∪ ((c ∪ e) ∩ (d ∪ f)))))))

Proof of Theorem d6oa

Step	Hyp	Ref	Expression
1		d6oa.1	. 2 a ≤ b⊥
2		d6oa.2	. 2 c ≤ d⊥
3		d6oa.3	. 2 e ≤ f⊥
4		id 59	. 2 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )) = (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))
5		id 59	. 2 a⊥ = a⊥
6		id 59	. 2 c⊥ = c⊥
7		id 59	. 2 e⊥ = e⊥
8		id 59	. . . . 5 (((c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))) ∪ (((c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ ((a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))))) = (((c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))) ∪ (((c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ ((a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))))
9		id 59	. . . . 5 ((((c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (e⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))) ∪ (((c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ ((e⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))))) ∩ (((a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (e⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))) ∪ (((a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ ((e⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))))) = ((((c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (e⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))) ∪ (((c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ ((e⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))))) ∩ (((a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (e⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))) ∪ (((a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ ((e⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))))))
10	8, 9	d4oa 996	. . . 4 (((c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ ((((c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))) ∪ (((c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ ((a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))))) ∪ ((((c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (e⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))) ∪ (((c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ ((e⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))))) ∩ (((a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (e⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))) ∪ (((a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ ((e⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))))))) ≤ ((a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))
11		id 59	. . . 4 (a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) = (a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))
12		id 59	. . . 4 (c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) = (c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))
13		id 59	. . . 4 (e⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) = (e⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))
14	10, 11, 12, 13	oa4gto4u 976	. . 3 ((a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ ((a⊥ ⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∪ ((c⊥ ⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ ((((a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))) ∪ ((a⊥ ⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (c⊥ ⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))))) ∪ ((((a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (e⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))) ∪ ((a⊥ ⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (e⊥ ⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))))) ∩ (((c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (e⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))) ∪ ((c⊥ ⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (e⊥ ⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))))))))) ≤ (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))
15	14	oa4uto4 977	. 2 ((a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (a⊥ ∪ (c⊥ ∩ (((a⊥ ∩ c⊥ ) ∪ ((a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))))) ∪ (((a⊥ ∩ e⊥ ) ∪ ((a⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (e⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))))) ∩ ((c⊥ ∩ e⊥ ) ∪ ((c⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))) ∩ (e⊥ →1 (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ )))))))))) ≤ (((a⊥ ∩ b⊥ ) ∪ (c⊥ ∩ d⊥ )) ∪ (e⊥ ∩ f⊥ ))
16	1, 2, 3, 4, 5, 6, 7, 15	oa4to6 965	1 (((a ∪ b) ∩ (c ∪ d)) ∩ (e ∪ f)) ≤ (b ∪ (a ∩ (c ∪ (((a ∪ c) ∩ (b ∪ d)) ∩ (((a ∪ e) ∩ (b ∪ f)) ∪ ((c ∪ e) ∩ (d ∪ f)))))))

This proof depends on syntax axioms:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12

This proof depends on 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  ax-oadist 994

This proof depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is used by: (None)