Theorem dp34 1181

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

Hypotheses

Ref	Expression
dp34.1	c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
dp34.2	c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
dp34.3	c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
dp34.4	p = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∩ (a2 ∪ b2))

Assertion

Ref	Expression
dp34	p ≤ ((a0 ∪ b1) ∪ (c2 ∩ (c0 ∪ c1)))

Proof of Theorem dp34

Step	Hyp	Ref	Expression
1		dp34.1	. . . 4 c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
2		dp34.2	. . . 4 c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
3		dp34.3	. . . 4 c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
4		dp34.4	. . . 4 p = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∩ (a2 ∪ b2))
5	1, 2, 3, 4	dp53 1170	. . 3 p ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))
6		lear 161	. . . 4 (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))) ≤ (b1 ∪ (c2 ∩ (c0 ∪ c1)))
7	6	lelor 166	. . 3 (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))) ≤ (a0 ∪ (b1 ∪ (c2 ∩ (c0 ∪ c1))))
8	5, 7	letr 137	. 2 p ≤ (a0 ∪ (b1 ∪ (c2 ∩ (c0 ∪ c1))))
9		orass 75	. . 3 ((a0 ∪ b1) ∪ (c2 ∩ (c0 ∪ c1))) = (a0 ∪ (b1 ∪ (c2 ∩ (c0 ∪ c1))))
10	9	cm 61	. 2 (a0 ∪ (b1 ∪ (c2 ∩ (c0 ∪ c1)))) = ((a0 ∪ b1) ∪ (c2 ∩ (c0 ∪ c1)))
11	8, 10	lbtr 139	1 p ≤ ((a0 ∪ 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:  dp41lema  1182  xdp41  1198  xxdp41  1201  xdp45lem  1204  xdp43lem  1205  xdp45  1206  xdp43  1207  3dp43  1208