Theorem dp41lemk 1192

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
dp41lemk	(((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (b2 ∩ (a0 ∪ a2)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (a2 ∩ (b1 ∪ b2))))) = ((c0 ∪ (b2 ∩ (a0 ∪ a2))) ∪ (c1 ∪ (a2 ∩ (b1 ∪ b2))))

Proof of Theorem dp41lemk

Step	Hyp	Ref	Expression
1		leao3 164	. . . 4 (b2 ∩ (a0 ∪ a2)) ≤ (b1 ∪ b2)
2	1	mldual2i 1127	. . 3 ((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (b2 ∩ (a0 ∪ a2)))) = (((b1 ∪ b2) ∩ (a1 ∪ a2)) ∪ (b2 ∩ (a0 ∪ a2)))
3		dp41lem.1	. . . . . 6 c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
4		ancom 74	. . . . . 6 ((a1 ∪ a2) ∩ (b1 ∪ b2)) = ((b1 ∪ b2) ∩ (a1 ∪ a2))
5	3, 4	tr 62	. . . . 5 c0 = ((b1 ∪ b2) ∩ (a1 ∪ a2))
6	5	ror 71	. . . 4 (c0 ∪ (b2 ∩ (a0 ∪ a2))) = (((b1 ∪ b2) ∩ (a1 ∪ a2)) ∪ (b2 ∩ (a0 ∪ a2)))
7	6	cm 61	. . 3 (((b1 ∪ b2) ∩ (a1 ∪ a2)) ∪ (b2 ∩ (a0 ∪ a2))) = (c0 ∪ (b2 ∩ (a0 ∪ a2)))
8	2, 7	tr 62	. 2 ((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (b2 ∩ (a0 ∪ a2)))) = (c0 ∪ (b2 ∩ (a0 ∪ a2)))
9		leao3 164	. . . 4 (a2 ∩ (b1 ∪ b2)) ≤ (a0 ∪ a2)
10	9	mldual2i 1127	. . 3 ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (a2 ∩ (b1 ∪ b2)))) = (((a0 ∪ a2) ∩ (b0 ∪ b2)) ∪ (a2 ∩ (b1 ∪ b2)))
11		dp41lem.2	. . . . 5 c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
12	11	ror 71	. . . 4 (c1 ∪ (a2 ∩ (b1 ∪ b2))) = (((a0 ∪ a2) ∩ (b0 ∪ b2)) ∪ (a2 ∩ (b1 ∪ b2)))
13	12	cm 61	. . 3 (((a0 ∪ a2) ∩ (b0 ∪ b2)) ∪ (a2 ∩ (b1 ∪ b2))) = (c1 ∪ (a2 ∩ (b1 ∪ b2)))
14	10, 13	tr 62	. 2 ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (a2 ∩ (b1 ∪ b2)))) = (c1 ∪ (a2 ∩ (b1 ∪ b2)))
15	8, 14	2or 72	1 (((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (b2 ∩ (a0 ∪ a2)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (a2 ∩ (b1 ∪ b2))))) = ((c0 ∪ (b2 ∩ (a0 ∪ a2))) ∪ (c1 ∪ (a2 ∩ (b1 ∪ 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  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