Theorem dp41lemc0 1184

Description: Part of proof (4)=>(1) in Day/Pickering 1982. (Contributed by NM, 4-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
dp41lemc0	(((a0 ∪ b0) ∪ b1) ∩ ((a0 ∪ a1) ∪ b1)) = ((a0 ∪ b1) ∪ ((a0 ∪ b0) ∩ (a1 ∪ b1)))

Proof of Theorem dp41lemc0

Step	Hyp	Ref	Expression
1		ax-a2 31	. . . . . . 7 (a0 ∪ a1) = (a1 ∪ a0)
2	1	ror 71	. . . . . 6 ((a0 ∪ a1) ∪ b1) = ((a1 ∪ a0) ∪ b1)
3		or32 82	. . . . . 6 ((a1 ∪ a0) ∪ b1) = ((a1 ∪ b1) ∪ a0)
4	2, 3	tr 62	. . . . 5 ((a0 ∪ a1) ∪ b1) = ((a1 ∪ b1) ∪ a0)
5	4	lan 77	. . . 4 (((a0 ∪ b0) ∪ b1) ∩ ((a0 ∪ a1) ∪ b1)) = (((a0 ∪ b0) ∪ b1) ∩ ((a1 ∪ b1) ∪ a0))
6		ancom 74	. . . 4 (((a0 ∪ b0) ∪ b1) ∩ ((a1 ∪ b1) ∪ a0)) = (((a1 ∪ b1) ∪ a0) ∩ ((a0 ∪ b0) ∪ b1))
7	5, 6	tr 62	. . 3 (((a0 ∪ b0) ∪ b1) ∩ ((a0 ∪ a1) ∪ b1)) = (((a1 ∪ b1) ∪ a0) ∩ ((a0 ∪ b0) ∪ b1))
8		leor 159	. . . . 5 b1 ≤ (a1 ∪ b1)
9	8	ler 149	. . . 4 b1 ≤ ((a1 ∪ b1) ∪ a0)
10	9	mldual2i 1127	. . 3 (((a1 ∪ b1) ∪ a0) ∩ ((a0 ∪ b0) ∪ b1)) = ((((a1 ∪ b1) ∪ a0) ∩ (a0 ∪ b0)) ∪ b1)
11		ancom 74	. . . . 5 (((a1 ∪ b1) ∪ a0) ∩ (a0 ∪ b0)) = ((a0 ∪ b0) ∩ ((a1 ∪ b1) ∪ a0))
12		leo 158	. . . . . 6 a0 ≤ (a0 ∪ b0)
13	12	mldual2i 1127	. . . . 5 ((a0 ∪ b0) ∩ ((a1 ∪ b1) ∪ a0)) = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∪ a0)
14	11, 13	tr 62	. . . 4 (((a1 ∪ b1) ∪ a0) ∩ (a0 ∪ b0)) = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∪ a0)
15	14	ror 71	. . 3 ((((a1 ∪ b1) ∪ a0) ∩ (a0 ∪ b0)) ∪ b1) = ((((a0 ∪ b0) ∩ (a1 ∪ b1)) ∪ a0) ∪ b1)
16	7, 10, 15	3tr 65	. 2 (((a0 ∪ b0) ∪ b1) ∩ ((a0 ∪ a1) ∪ b1)) = ((((a0 ∪ b0) ∩ (a1 ∪ b1)) ∪ a0) ∪ b1)
17		orass 75	. 2 ((((a0 ∪ b0) ∩ (a1 ∪ b1)) ∪ a0) ∪ b1) = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∪ (a0 ∪ b1))
18		orcom 73	. 2 (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∪ (a0 ∪ b1)) = ((a0 ∪ b1) ∪ ((a0 ∪ b0) ∩ (a1 ∪ b1)))
19	16, 17, 18	3tr 65	1 (((a0 ∪ b0) ∪ b1) ∩ ((a0 ∪ a1) ∪ b1)) = ((a0 ∪ b1) ∪ ((a0 ∪ b0) ∩ (a1 ∪ b1)))

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:  dp41lemc  1185