Theorem dp15 1162

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). (1)=>(5). (Contributed by NM, 1-Apr-2012.)

Hypotheses

Ref	Expression
dp15.1	c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
dp15.2	c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
dp15.3	p0 = ((a1 ∪ b1) ∩ (a2 ∪ b2))

Assertion

Ref	Expression
dp15	((a0 ∪ a1) ∩ ((b0 ∩ (a0 ∪ p0)) ∪ b1)) ≤ ((c0 ∪ c1) ∪ (b1 ∩ (a0 ∪ a1)))

Proof of Theorem dp15

Step	Hyp	Ref	Expression
1		id 59	. 2 (a2 ∪ (a0 ∩ (a1 ∪ b1))) = (a2 ∪ (a0 ∩ (a1 ∪ b1)))
2		dp15.3	. 2 p0 = ((a1 ∪ b1) ∩ (a2 ∪ b2))
3		id 59	. 2 (b0 ∩ (a0 ∪ p0)) = (b0 ∩ (a0 ∪ p0))
4		dp15.1	. 2 c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
5		dp15.2	. 2 c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
6	1, 2, 3, 4, 5	dp15lemh 1161	1 ((a0 ∪ a1) ∩ ((b0 ∩ (a0 ∪ p0)) ∪ b1)) ≤ ((c0 ∪ c1) ∪ (b1 ∩ (a0 ∪ a1)))

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:  dp53lema  1163  xdp53  1200  xxdp53  1203