Theorem dp35lemd 1174

Description: Part of proof (3)=>(5) in Day/Pickering 1982. (Contributed by NM, 12-Apr-2012.)

Hypotheses

Ref	Expression
dp35lem.1	c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
dp35lem.2	c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
dp35lem.3	c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
dp35lem.4	p0 = ((a1 ∪ b1) ∩ (a2 ∪ b2))
dp35lem.5	p = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∩ (a2 ∪ b2))

Assertion

Ref	Expression
dp35lemd	(b0 ∩ (a0 ∪ p0)) ≤ (b0 ∩ (((a0 ∩ b0) ∪ b1) ∪ (c2 ∩ (c0 ∪ c1))))

Proof of Theorem dp35lemd

Step	Hyp	Ref	Expression
1		lea 160	. . 3 (b0 ∩ (a0 ∪ p0)) ≤ b0
2		dp35lem.1	. . . 4 c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
3		dp35lem.2	. . . 4 c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
4		dp35lem.3	. . . 4 c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
5		dp35lem.4	. . . 4 p0 = ((a1 ∪ b1) ∩ (a2 ∪ b2))
6		dp35lem.5	. . . 4 p = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∩ (a2 ∪ b2))
7	2, 3, 4, 5, 6	dp35leme 1173	. . 3 (b0 ∩ (a0 ∪ p0)) ≤ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))
8	1, 7	ler2an 173	. 2 (b0 ∩ (a0 ∪ p0)) ≤ (b0 ∩ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))))
9		lea 160	. . . 4 (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))) ≤ b0
10	9	mldual2i 1127	. . 3 (b0 ∩ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))) = ((b0 ∩ a0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))
11		lea 160	. . . . 5 (b0 ∩ a0) ≤ b0
12	11, 9	lel2or 170	. . . 4 ((b0 ∩ a0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))) ≤ b0
13		ancom 74	. . . . . . 7 (b0 ∩ a0) = (a0 ∩ b0)
14	13	bile 142	. . . . . 6 (b0 ∩ a0) ≤ (a0 ∩ b0)
15		lear 161	. . . . . 6 (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))) ≤ (b1 ∪ (c2 ∩ (c0 ∪ c1)))
16	14, 15	le2or 168	. . . . 5 ((b0 ∩ a0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))) ≤ ((a0 ∩ b0) ∪ (b1 ∪ (c2 ∩ (c0 ∪ c1))))
17		orass 75	. . . . . 6 (((a0 ∩ b0) ∪ b1) ∪ (c2 ∩ (c0 ∪ c1))) = ((a0 ∩ b0) ∪ (b1 ∪ (c2 ∩ (c0 ∪ c1))))
18	17	cm 61	. . . . 5 ((a0 ∩ b0) ∪ (b1 ∪ (c2 ∩ (c0 ∪ c1)))) = (((a0 ∩ b0) ∪ b1) ∪ (c2 ∩ (c0 ∪ c1)))
19	16, 18	lbtr 139	. . . 4 ((b0 ∩ a0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))) ≤ (((a0 ∩ b0) ∪ b1) ∪ (c2 ∩ (c0 ∪ c1)))
20	12, 19	ler2an 173	. . 3 ((b0 ∩ a0) ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1))))) ≤ (b0 ∩ (((a0 ∩ b0) ∪ b1) ∪ (c2 ∩ (c0 ∪ c1))))
21	10, 20	bltr 138	. 2 (b0 ∩ (a0 ∪ (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))))) ≤ (b0 ∩ (((a0 ∩ b0) ∪ b1) ∪ (c2 ∩ (c0 ∪ c1))))
22	8, 21	letr 137	1 (b0 ∩ (a0 ∪ p0)) ≤ (b0 ∩ (((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:  dp35lembb  1177