Theorem dp23 1197

Description: Part of theorem from Alan Day and Doug Pickering, "A note on the Arguesian lattice identity", Studia Sci. Math. Hungar. 19:303-305 (1982). (2)=>(3). (Contributed by NM, 4-Apr-2012.)

Hypotheses

Ref	Expression
dp23.1	c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
dp23.2	c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
dp23.3	c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
dp23.4	p = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∩ (a2 ∪ b2))

Assertion

Ref	Expression
dp23	p ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))

Proof of Theorem dp23

Step	Hyp	Ref	Expression
1		dp23.1	. . 3 c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
2		dp23.2	. . 3 c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
3		dp23.3	. . 3 c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
4		dp23.4	. . 3 p = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∩ (a2 ∪ b2))
5	1, 2, 3, 4	dp32 1196	. 2 p ≤ ((a0 ∩ (a1 ∪ (c2 ∩ (c0 ∪ c1)))) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))
6		lea 160	. . 3 (a0 ∩ (a1 ∪ (c2 ∩ (c0 ∪ c1)))) ≤ a0
7	6	leror 152	. 2 ((a0 ∩ (a1 ∪ (c2 ∩ (c0 ∪ c1)))) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))) ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))
8	5, 7	letr 137	1 p ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))

Syntax hints:   = wb 1   ≤ wle 2   ∪ wo 6   ∩ wa 7

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-ml 1122  ax-arg 1153

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131

This theorem is referenced by: (None)