Theorem dp41lemh 1190

Description: Part of proof (4)=>(1) in Day/Pickering 1982. "By CP(a,b)". (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
dp41lemh	(((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (a0 ∩ (a1 ∪ b1)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (b1 ∩ (a0 ∪ b0))))) ≤ (((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (a0 ∩ (a2 ∪ b2)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (b1 ∩ (a2 ∪ b2)))))

Proof of Theorem dp41lemh

Step	Hyp	Ref	Expression
1		lea 160	. . . . 5 (a0 ∩ (a1 ∪ b1)) ≤ a0
2		leo 158	. . . . . . 7 a0 ≤ (a0 ∪ b0)
3	2	leran 153	. . . . . 6 (a0 ∩ (a1 ∪ b1)) ≤ ((a0 ∪ b0) ∩ (a1 ∪ b1))
4		dp41lem.5	. . . . . . . 8 p2 = ((a0 ∪ b0) ∩ (a1 ∪ b1))
5	4	cm 61	. . . . . . 7 ((a0 ∪ b0) ∩ (a1 ∪ b1)) = p2
6		dp41lem.6	. . . . . . 7 p2 ≤ (a2 ∪ b2)
7	5, 6	bltr 138	. . . . . 6 ((a0 ∪ b0) ∩ (a1 ∪ b1)) ≤ (a2 ∪ b2)
8	3, 7	letr 137	. . . . 5 (a0 ∩ (a1 ∪ b1)) ≤ (a2 ∪ b2)
9	1, 8	ler2an 173	. . . 4 (a0 ∩ (a1 ∪ b1)) ≤ (a0 ∩ (a2 ∪ b2))
10	9	lelor 166	. . 3 ((a1 ∪ a2) ∪ (a0 ∩ (a1 ∪ b1))) ≤ ((a1 ∪ a2) ∪ (a0 ∩ (a2 ∪ b2)))
11	10	lelan 167	. 2 ((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (a0 ∩ (a1 ∪ b1)))) ≤ ((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (a0 ∩ (a2 ∪ b2))))
12		lea 160	. . . . 5 (b1 ∩ (a0 ∪ b0)) ≤ b1
13		lear 161	. . . . . . 7 (b1 ∩ (a0 ∪ b0)) ≤ (a0 ∪ b0)
14		leao3 164	. . . . . . 7 (b1 ∩ (a0 ∪ b0)) ≤ (a1 ∪ b1)
15	13, 14	ler2an 173	. . . . . 6 (b1 ∩ (a0 ∪ b0)) ≤ ((a0 ∪ b0) ∩ (a1 ∪ b1))
16	15, 7	letr 137	. . . . 5 (b1 ∩ (a0 ∪ b0)) ≤ (a2 ∪ b2)
17	12, 16	ler2an 173	. . . 4 (b1 ∩ (a0 ∪ b0)) ≤ (b1 ∩ (a2 ∪ b2))
18	17	lelor 166	. . 3 ((b0 ∪ b2) ∪ (b1 ∩ (a0 ∪ b0))) ≤ ((b0 ∪ b2) ∪ (b1 ∩ (a2 ∪ b2)))
19	18	lelan 167	. 2 ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (b1 ∩ (a0 ∪ b0)))) ≤ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (b1 ∩ (a2 ∪ b2))))
20	11, 19	le2or 168	1 (((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (a0 ∩ (a1 ∪ b1)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (b1 ∩ (a0 ∪ b0))))) ≤ (((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (a0 ∩ (a2 ∪ b2)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (b1 ∩ (a2 ∪ b2)))))

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

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