Theorem dp41leme 1187

Description: Part of proof (4)=>(1) in Day/Pickering 1982. (Contributed by NM, 3-Apr-2012.)

Hypotheses

Ref	Expression
dp41lem.1	c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
dp41lem.2	c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
dp41lem.3	c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
dp41lem.4	p = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∩ (a2 ∪ b2))
dp41lem.5	p2 = ((a0 ∪ b0) ∩ (a1 ∪ b1))
dp41lem.6	p2 ≤ (a2 ∪ b2)

Assertion

Ref	Expression
dp41leme	(c2 ∩ ((c0 ∪ c1) ∪ (c2 ∩ (a0 ∪ b1)))) ≤ ((c0 ∪ c1) ∪ ((a0 ∩ (b0 ∪ b1)) ∪ (b1 ∩ (a0 ∪ a1))))

Proof of Theorem dp41leme

Step	Hyp	Ref	Expression
1		mldual 1124	. . 3 (c2 ∩ ((c0 ∪ c1) ∪ (c2 ∩ (a0 ∪ b1)))) = ((c2 ∩ (c0 ∪ c1)) ∪ (c2 ∩ (a0 ∪ b1)))
2		dp41lem.3	. . . . . 6 c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
3	2	ran 78	. . . . 5 (c2 ∩ (a0 ∪ b1)) = (((a0 ∪ a1) ∩ (b0 ∪ b1)) ∩ (a0 ∪ b1))
4		anass 76	. . . . 5 (((a0 ∪ a1) ∩ (b0 ∪ b1)) ∩ (a0 ∪ b1)) = ((a0 ∪ a1) ∩ ((b0 ∪ b1) ∩ (a0 ∪ b1)))
5		leor 159	. . . . . . . . 9 b1 ≤ (b0 ∪ b1)
6	5	mldual2i 1127	. . . . . . . 8 ((b0 ∪ b1) ∩ (a0 ∪ b1)) = (((b0 ∪ b1) ∩ a0) ∪ b1)
7		orcom 73	. . . . . . . 8 (((b0 ∪ b1) ∩ a0) ∪ b1) = (b1 ∪ ((b0 ∪ b1) ∩ a0))
8		ancom 74	. . . . . . . . 9 ((b0 ∪ b1) ∩ a0) = (a0 ∩ (b0 ∪ b1))
9	8	lor 70	. . . . . . . 8 (b1 ∪ ((b0 ∪ b1) ∩ a0)) = (b1 ∪ (a0 ∩ (b0 ∪ b1)))
10	6, 7, 9	3tr 65	. . . . . . 7 ((b0 ∪ b1) ∩ (a0 ∪ b1)) = (b1 ∪ (a0 ∩ (b0 ∪ b1)))
11	10	lan 77	. . . . . 6 ((a0 ∪ a1) ∩ ((b0 ∪ b1) ∩ (a0 ∪ b1))) = ((a0 ∪ a1) ∩ (b1 ∪ (a0 ∩ (b0 ∪ b1))))
12		leao1 162	. . . . . . 7 (a0 ∩ (b0 ∪ b1)) ≤ (a0 ∪ a1)
13	12	mldual2i 1127	. . . . . 6 ((a0 ∪ a1) ∩ (b1 ∪ (a0 ∩ (b0 ∪ b1)))) = (((a0 ∪ a1) ∩ b1) ∪ (a0 ∩ (b0 ∪ b1)))
14		orcom 73	. . . . . . 7 (((a0 ∪ a1) ∩ b1) ∪ (a0 ∩ (b0 ∪ b1))) = ((a0 ∩ (b0 ∪ b1)) ∪ ((a0 ∪ a1) ∩ b1))
15		ancom 74	. . . . . . . 8 ((a0 ∪ a1) ∩ b1) = (b1 ∩ (a0 ∪ a1))
16	15	lor 70	. . . . . . 7 ((a0 ∩ (b0 ∪ b1)) ∪ ((a0 ∪ a1) ∩ b1)) = ((a0 ∩ (b0 ∪ b1)) ∪ (b1 ∩ (a0 ∪ a1)))
17	14, 16	tr 62	. . . . . 6 (((a0 ∪ a1) ∩ b1) ∪ (a0 ∩ (b0 ∪ b1))) = ((a0 ∩ (b0 ∪ b1)) ∪ (b1 ∩ (a0 ∪ a1)))
18	11, 13, 17	3tr 65	. . . . 5 ((a0 ∪ a1) ∩ ((b0 ∪ b1) ∩ (a0 ∪ b1))) = ((a0 ∩ (b0 ∪ b1)) ∪ (b1 ∩ (a0 ∪ a1)))
19	3, 4, 18	3tr 65	. . . 4 (c2 ∩ (a0 ∪ b1)) = ((a0 ∩ (b0 ∪ b1)) ∪ (b1 ∩ (a0 ∪ a1)))
20	19	lor 70	. . 3 ((c2 ∩ (c0 ∪ c1)) ∪ (c2 ∩ (a0 ∪ b1))) = ((c2 ∩ (c0 ∪ c1)) ∪ ((a0 ∩ (b0 ∪ b1)) ∪ (b1 ∩ (a0 ∪ a1))))
21	1, 20	tr 62	. 2 (c2 ∩ ((c0 ∪ c1) ∪ (c2 ∩ (a0 ∪ b1)))) = ((c2 ∩ (c0 ∪ c1)) ∪ ((a0 ∩ (b0 ∪ b1)) ∪ (b1 ∩ (a0 ∪ a1))))
22		lear 161	. . 3 (c2 ∩ (c0 ∪ c1)) ≤ (c0 ∪ c1)
23	22	leror 152	. 2 ((c2 ∩ (c0 ∪ c1)) ∪ ((a0 ∩ (b0 ∪ b1)) ∪ (b1 ∩ (a0 ∪ a1)))) ≤ ((c0 ∪ c1) ∪ ((a0 ∩ (b0 ∪ b1)) ∪ (b1 ∩ (a0 ∪ a1))))
24	21, 23	bltr 138	1 (c2 ∩ ((c0 ∪ c1) ∪ (c2 ∩ (a0 ∪ b1)))) ≤ ((c0 ∪ c1) ∪ ((a0 ∩ (b0 ∪ b1)) ∪ (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-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-ml 1122

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:  dp41lemm  1194