Theorem dp32 1196

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)=>(2). (Contributed by NM, 4-Apr-2012.)

Hypotheses

Ref	Expression
dp32.1	c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
dp32.2	c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
dp32.3	c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
dp32.4	p = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∩ (a2 ∪ b2))

Assertion

Ref	Expression
dp32	p ≤ ((a0 ∩ (a1 ∪ (c2 ∩ (c0 ∪ c1)))) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))

Proof of Theorem dp32

Step	Hyp	Ref	Expression
1		dp32.1	. . . 4 c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
2		dp32.2	. . . 4 c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
3		dp32.3	. . . 4 c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
4		dp32.4	. . . 4 p = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∩ (a2 ∪ b2))
5	1, 2, 3, 4	dp53 1170	. . 3 p ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))
6		ancom 74	. . . . 5 ((a1 ∪ a2) ∩ (b1 ∪ b2)) = ((b1 ∪ b2) ∩ (a1 ∪ a2))
7	1, 6	tr 62	. . . 4 c0 = ((b1 ∪ b2) ∩ (a1 ∪ a2))
8		ancom 74	. . . . 5 ((a0 ∪ a2) ∩ (b0 ∪ b2)) = ((b0 ∪ b2) ∩ (a0 ∪ a2))
9	2, 8	tr 62	. . . 4 c1 = ((b0 ∪ b2) ∩ (a0 ∪ a2))
10		ancom 74	. . . . 5 ((a0 ∪ a1) ∩ (b0 ∪ b1)) = ((b0 ∪ b1) ∩ (a0 ∪ a1))
11	3, 10	tr 62	. . . 4 c2 = ((b0 ∪ b1) ∩ (a0 ∪ a1))
12		orcom 73	. . . . . . 7 (a0 ∪ b0) = (b0 ∪ a0)
13		orcom 73	. . . . . . 7 (a1 ∪ b1) = (b1 ∪ a1)
14	12, 13	2an 79	. . . . . 6 ((a0 ∪ b0) ∩ (a1 ∪ b1)) = ((b0 ∪ a0) ∩ (b1 ∪ a1))
15		orcom 73	. . . . . 6 (a2 ∪ b2) = (b2 ∪ a2)
16	14, 15	2an 79	. . . . 5 (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∩ (a2 ∪ b2)) = (((b0 ∪ a0) ∩ (b1 ∪ a1)) ∩ (b2 ∪ a2))
17	4, 16	tr 62	. . . 4 p = (((b0 ∪ a0) ∩ (b1 ∪ a1)) ∩ (b2 ∪ a2))
18	7, 9, 11, 17	dp53 1170	. . 3 p ≤ (b0 ∪ (a0 ∩ (a1 ∪ (c2 ∩ (c0 ∪ c1)))))
19	5, 18	ler2an 173	. 2 p ≤ ((a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))) ∩ (b0 ∪ (a0 ∩ (a1 ∪ (c2 ∩ (c0 ∪ c1))))))
20		leao1 162	. . . 4 (a0 ∩ (a1 ∪ (c2 ∩ (c0 ∪ c1)))) ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))
21	20	mldual2i 1127	. . 3 ((a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))) ∩ (b0 ∪ (a0 ∩ (a1 ∪ (c2 ∩ (c0 ∪ c1)))))) = (((a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))) ∩ b0) ∪ (a0 ∩ (a1 ∪ (c2 ∩ (c0 ∪ c1)))))
22		ancom 74	. . . . 5 ((a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))) ∩ b0) = (b0 ∩ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))))
23		mldual 1124	. . . . 5 (b0 ∩ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))) = ((b0 ∩ a0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))
24		leao2 163	. . . . . . . . . . 11 (b0 ∩ a0) ≤ (a0 ∪ a1)
25		leao1 162	. . . . . . . . . . 11 (b0 ∩ a0) ≤ (b0 ∪ b1)
26	24, 25	ler2an 173	. . . . . . . . . 10 (b0 ∩ a0) ≤ ((a0 ∪ a1) ∩ (b0 ∪ b1))
27	3	cm 61	. . . . . . . . . 10 ((a0 ∪ a1) ∩ (b0 ∪ b1)) = c2
28	26, 27	lbtr 139	. . . . . . . . 9 (b0 ∩ a0) ≤ c2
29		leao2 163	. . . . . . . . . . . 12 (b0 ∩ a0) ≤ (a0 ∪ a2)
30		leao1 162	. . . . . . . . . . . 12 (b0 ∩ a0) ≤ (b0 ∪ b2)
31	29, 30	ler2an 173	. . . . . . . . . . 11 (b0 ∩ a0) ≤ ((a0 ∪ a2) ∩ (b0 ∪ b2))
32	2	cm 61	. . . . . . . . . . 11 ((a0 ∪ a2) ∩ (b0 ∪ b2)) = c1
33	31, 32	lbtr 139	. . . . . . . . . 10 (b0 ∩ a0) ≤ c1
34	33	lerr 150	. . . . . . . . 9 (b0 ∩ a0) ≤ (c0 ∪ c1)
35	28, 34	ler2an 173	. . . . . . . 8 (b0 ∩ a0) ≤ (c2 ∩ (c0 ∪ c1))
36	35	lerr 150	. . . . . . 7 (b0 ∩ a0) ≤ (b1 ∪ (c2 ∩ (c0 ∪ c1)))
37	36	ml2i 1125	. . . . . 6 ((b0 ∩ a0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))) = (((b0 ∩ a0) ∪ b0) ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))
38		lea 160	. . . . . . . 8 (b0 ∩ a0) ≤ b0
39	38	df-le2 131	. . . . . . 7 ((b0 ∩ a0) ∪ b0) = b0
40	39	ran 78	. . . . . 6 (((b0 ∩ a0) ∪ b0) ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))) = (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))
41	37, 40	tr 62	. . . . 5 ((b0 ∩ a0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))) = (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))
42	22, 23, 41	3tr 65	. . . 4 ((a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))) ∩ b0) = (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))
43	42	ror 71	. . 3 (((a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))) ∩ b0) ∪ (a0 ∩ (a1 ∪ (c2 ∩ (c0 ∪ c1))))) = ((b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))) ∪ (a0 ∩ (a1 ∪ (c2 ∩ (c0 ∪ c1)))))
44		orcom 73	. . 3 ((b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))) ∪ (a0 ∩ (a1 ∪ (c2 ∩ (c0 ∪ c1))))) = ((a0 ∩ (a1 ∪ (c2 ∩ (c0 ∪ c1)))) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))
45	21, 43, 44	3tr 65	. 2 ((a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))) ∩ (b0 ∪ (a0 ∩ (a1 ∪ (c2 ∩ (c0 ∪ c1)))))) = ((a0 ∩ (a1 ∪ (c2 ∩ (c0 ∪ c1)))) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))
46	19, 45	lbtr 139	1 p ≤ ((a0 ∩ (a1 ∪ (c2 ∩ (c0 ∪ c1)))) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))

This proof depends on syntax axioms:   = wb 1   ≤ wle 2   ∪ wo 6   ∩ wa 7

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-ml 1122  ax-arg 1153

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

This theorem is used by:  dp23  1197