Theorem dp41lemm 1194

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
dp41lemm	c2 ≤ (c0 ∪ c1)

Proof of Theorem dp41lemm

Step	Hyp	Ref	Expression
1		dp41lem.1	. . . . . . . 8 c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
2		dp41lem.2	. . . . . . . 8 c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
3		dp41lem.3	. . . . . . . 8 c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
4		dp41lem.4	. . . . . . . 8 p = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∩ (a2 ∪ b2))
5		dp41lem.5	. . . . . . . 8 p2 = ((a0 ∪ b0) ∩ (a1 ∪ b1))
6		dp41lem.6	. . . . . . . 8 p2 ≤ (a2 ∪ b2)
7	1, 2, 3, 4, 5, 6	dp41lemb 1183	. . . . . . 7 c2 = ((c2 ∩ ((a0 ∪ b0) ∪ b1)) ∩ ((a0 ∪ a1) ∪ b1))
8	1, 2, 3, 4, 5, 6	dp41lemc 1185	. . . . . . 7 ((c2 ∩ ((a0 ∪ b0) ∪ b1)) ∩ ((a0 ∪ a1) ∪ b1)) ≤ (c2 ∩ ((a0 ∪ b1) ∪ (c2 ∩ (c0 ∪ c1))))
9	7, 8	bltr 138	. . . . . 6 c2 ≤ (c2 ∩ ((a0 ∪ b1) ∪ (c2 ∩ (c0 ∪ c1))))
10	1, 2, 3, 4, 5, 6	dp41lemd 1186	. . . . . 6 (c2 ∩ ((a0 ∪ b1) ∪ (c2 ∩ (c0 ∪ c1)))) = (c2 ∩ ((c0 ∪ c1) ∪ (c2 ∩ (a0 ∪ b1))))
11	9, 10	lbtr 139	. . . . 5 c2 ≤ (c2 ∩ ((c0 ∪ c1) ∪ (c2 ∩ (a0 ∪ b1))))
12	1, 2, 3, 4, 5, 6	dp41leme 1187	. . . . 5 (c2 ∩ ((c0 ∪ c1) ∪ (c2 ∩ (a0 ∪ b1)))) ≤ ((c0 ∪ c1) ∪ ((a0 ∩ (b0 ∪ b1)) ∪ (b1 ∩ (a0 ∪ a1))))
13	11, 12	letr 137	. . . 4 c2 ≤ ((c0 ∪ c1) ∪ ((a0 ∩ (b0 ∪ b1)) ∪ (b1 ∩ (a0 ∪ a1))))
14	1, 2, 3, 4, 5, 6	dp41lemf 1188	. . . . 5 ((c0 ∪ c1) ∪ ((a0 ∩ (b0 ∪ b1)) ∪ (b1 ∩ (a0 ∪ a1)))) = (((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (b1 ∩ (a0 ∪ a1)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (a0 ∩ (b0 ∪ b1)))))
15	1, 2, 3, 4, 5, 6	dp41lemg 1189	. . . . 5 (((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (b1 ∩ (a0 ∪ a1)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (a0 ∩ (b0 ∪ b1))))) = (((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (a0 ∩ (a1 ∪ b1)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (b1 ∩ (a0 ∪ b0)))))
16	14, 15	tr 62	. . . 4 ((c0 ∪ c1) ∪ ((a0 ∩ (b0 ∪ b1)) ∪ (b1 ∩ (a0 ∪ a1)))) = (((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (a0 ∩ (a1 ∪ b1)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (b1 ∩ (a0 ∪ b0)))))
17	13, 16	lbtr 139	. . 3 c2 ≤ (((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (a0 ∩ (a1 ∪ b1)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (b1 ∩ (a0 ∪ b0)))))
18	1, 2, 3, 4, 5, 6	dp41lemh 1190	. . 3 (((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)))))
19	17, 18	letr 137	. 2 c2 ≤ (((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (a0 ∩ (a2 ∪ b2)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (b1 ∩ (a2 ∪ b2)))))
20	1, 2, 3, 4, 5, 6	dp41lemj 1191	. . 3 (((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (a0 ∩ (a2 ∪ b2)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (b1 ∩ (a2 ∪ b2))))) = (((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (b2 ∩ (a0 ∪ a2)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (a2 ∩ (b1 ∪ b2)))))
21	1, 2, 3, 4, 5, 6	dp41lemk 1192	. . 3 (((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (b2 ∩ (a0 ∪ a2)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (a2 ∩ (b1 ∪ b2))))) = ((c0 ∪ (b2 ∩ (a0 ∪ a2))) ∪ (c1 ∪ (a2 ∩ (b1 ∪ b2))))
22	1, 2, 3, 4, 5, 6	dp41leml 1193	. . 3 ((c0 ∪ (b2 ∩ (a0 ∪ a2))) ∪ (c1 ∪ (a2 ∩ (b1 ∪ b2)))) = (c0 ∪ c1)
23	20, 21, 22	3tr 65	. 2 (((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (a0 ∩ (a2 ∪ b2)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (b1 ∩ (a2 ∪ b2))))) = (c0 ∪ c1)
24	19, 23	lbtr 139	1 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:  dp41  1195