Theorem dp41 1195

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). (4)=>(1) (Contributed by NM, 3-Apr-2012.)

Hypotheses

Ref	Expression
dp41.1	c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
dp41.2	c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
dp41.3	c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
dp41.4	p2 = ((a0 ∪ b0) ∩ (a1 ∪ b1))
dp41.5	p2 ≤ (a2 ∪ b2)

Assertion

Ref	Expression
dp41	c2 ≤ (c0 ∪ c1)

Proof of Theorem dp41

Step	Hyp	Ref	Expression
1		dp41.1	. 2 c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
2		dp41.2	. 2 c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
3		dp41.3	. 2 c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
4		id 59	. 2 (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∩ (a2 ∪ b2)) = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∩ (a2 ∪ b2))
5		dp41.4	. 2 p2 = ((a0 ∪ b0) ∩ (a1 ∪ b1))
6		dp41.5	. 2 p2 ≤ (a2 ∪ b2)
7	1, 2, 3, 4, 5, 6	dp41lemm 1194	1 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)