Theorem dp53leme 1167

Description: Part of proof (5)=>(3) in Day/Pickering 1982. (Contributed by NM, 3-Apr-2012.)

Hypotheses

Ref	Expression
dp53lem.1	c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
dp53lem.2	c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
dp53lem.3	c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
dp53lem.4	p0 = ((a1 ∪ b1) ∩ (a2 ∪ b2))
dp53lem.5	p = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∩ (a2 ∪ b2))

Assertion

Ref	Expression
dp53leme	(b0 ∩ (a0 ∪ p0)) ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))

Proof of Theorem dp53leme

Step	Hyp	Ref	Expression
1		dp53lem.1	. . 3 c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
2		dp53lem.2	. . 3 c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
3		dp53lem.3	. . 3 c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
4		dp53lem.4	. . 3 p0 = ((a1 ∪ b1) ∩ (a2 ∪ b2))
5		dp53lem.5	. . 3 p = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∩ (a2 ∪ b2))
6	1, 2, 3, 4, 5	dp53lemd 1166	. 2 (b0 ∩ (a0 ∪ p0)) ≤ (b0 ∩ (((a0 ∩ b0) ∪ b1) ∪ (c2 ∩ (c0 ∪ c1))))
7		orass 75	. . . . . 6 (((a0 ∩ b0) ∪ b1) ∪ (c2 ∩ (c0 ∪ c1))) = ((a0 ∩ b0) ∪ (b1 ∪ (c2 ∩ (c0 ∪ c1))))
8		orcom 73	. . . . . 6 ((a0 ∩ b0) ∪ (b1 ∪ (c2 ∩ (c0 ∪ c1)))) = ((b1 ∪ (c2 ∩ (c0 ∪ c1))) ∪ (a0 ∩ b0))
9	7, 8	tr 62	. . . . 5 (((a0 ∩ b0) ∪ b1) ∪ (c2 ∩ (c0 ∪ c1))) = ((b1 ∪ (c2 ∩ (c0 ∪ c1))) ∪ (a0 ∩ b0))
10	9	lan 77	. . . 4 (b0 ∩ (((a0 ∩ b0) ∪ b1) ∪ (c2 ∩ (c0 ∪ c1)))) = (b0 ∩ ((b1 ∪ (c2 ∩ (c0 ∪ c1))) ∪ (a0 ∩ b0)))
11		lear 161	. . . . 5 (a0 ∩ b0) ≤ b0
12	11	mldual2i 1127	. . . 4 (b0 ∩ ((b1 ∪ (c2 ∩ (c0 ∪ c1))) ∪ (a0 ∩ b0))) = ((b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))) ∪ (a0 ∩ b0))
13		orcom 73	. . . 4 ((b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))) ∪ (a0 ∩ b0)) = ((a0 ∩ b0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))
14	10, 12, 13	3tr 65	. . 3 (b0 ∩ (((a0 ∩ b0) ∪ b1) ∪ (c2 ∩ (c0 ∪ c1)))) = ((a0 ∩ b0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))
15		lea 160	. . . 4 (a0 ∩ b0) ≤ a0
16	15	leror 152	. . 3 ((a0 ∩ b0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))) ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))
17	14, 16	bltr 138	. 2 (b0 ∩ (((a0 ∩ b0) ∪ b1) ∪ (c2 ∩ (c0 ∪ c1)))) ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))
18	6, 17	letr 137	1 (b0 ∩ (a0 ∪ p0)) ≤ (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:  dp53lemf  1168