Theorem dp41leml 1193

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
dp41leml	((c0 ∪ (b2 ∩ (a0 ∪ a2))) ∪ (c1 ∪ (a2 ∩ (b1 ∪ b2)))) = (c0 ∪ c1)

Proof of Theorem dp41leml

Step	Hyp	Ref	Expression
1		or4 84	. 2 ((c0 ∪ (b2 ∩ (a0 ∪ a2))) ∪ (c1 ∪ (a2 ∩ (b1 ∪ b2)))) = ((c0 ∪ c1) ∪ ((b2 ∩ (a0 ∪ a2)) ∪ (a2 ∩ (b1 ∪ b2))))
2		orcom 73	. 2 ((c0 ∪ c1) ∪ ((b2 ∩ (a0 ∪ a2)) ∪ (a2 ∩ (b1 ∪ b2)))) = (((b2 ∩ (a0 ∪ a2)) ∪ (a2 ∩ (b1 ∪ b2))) ∪ (c0 ∪ c1))
3		ancom 74	. . . . . 6 (b2 ∩ (a0 ∪ a2)) = ((a0 ∪ a2) ∩ b2)
4		leor 159	. . . . . . 7 b2 ≤ (b0 ∪ b2)
5	4	lelan 167	. . . . . 6 ((a0 ∪ a2) ∩ b2) ≤ ((a0 ∪ a2) ∩ (b0 ∪ b2))
6	3, 5	bltr 138	. . . . 5 (b2 ∩ (a0 ∪ a2)) ≤ ((a0 ∪ a2) ∩ (b0 ∪ b2))
7		leor 159	. . . . . 6 a2 ≤ (a1 ∪ a2)
8	7	leran 153	. . . . 5 (a2 ∩ (b1 ∪ b2)) ≤ ((a1 ∪ a2) ∩ (b1 ∪ b2))
9	6, 8	le2or 168	. . . 4 ((b2 ∩ (a0 ∪ a2)) ∪ (a2 ∩ (b1 ∪ b2))) ≤ (((a0 ∪ a2) ∩ (b0 ∪ b2)) ∪ ((a1 ∪ a2) ∩ (b1 ∪ b2)))
10		dp41lem.2	. . . . . . 7 c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
11		dp41lem.1	. . . . . . 7 c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
12	10, 11	2or 72	. . . . . 6 (c1 ∪ c0) = (((a0 ∪ a2) ∩ (b0 ∪ b2)) ∪ ((a1 ∪ a2) ∩ (b1 ∪ b2)))
13	12	cm 61	. . . . 5 (((a0 ∪ a2) ∩ (b0 ∪ b2)) ∪ ((a1 ∪ a2) ∩ (b1 ∪ b2))) = (c1 ∪ c0)
14		orcom 73	. . . . 5 (c1 ∪ c0) = (c0 ∪ c1)
15	13, 14	tr 62	. . . 4 (((a0 ∪ a2) ∩ (b0 ∪ b2)) ∪ ((a1 ∪ a2) ∩ (b1 ∪ b2))) = (c0 ∪ c1)
16	9, 15	lbtr 139	. . 3 ((b2 ∩ (a0 ∪ a2)) ∪ (a2 ∩ (b1 ∪ b2))) ≤ (c0 ∪ c1)
17	16	df-le2 131	. 2 (((b2 ∩ (a0 ∪ a2)) ∪ (a2 ∩ (b1 ∪ b2))) ∪ (c0 ∪ c1)) = (c0 ∪ c1)
18	1, 2, 17	3tr 65	1 ((c0 ∪ (b2 ∩ (a0 ∪ a2))) ∪ (c1 ∪ (a2 ∩ (b1 ∪ b2)))) = (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-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