Theorem dp41lemf 1188

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
dp41lemf	((c0 ∪ c1) ∪ ((a0 ∩ (b0 ∪ b1)) ∪ (b1 ∩ (a0 ∪ a1)))) = (((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (b1 ∩ (a0 ∪ a1)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (a0 ∩ (b0 ∪ b1)))))

Proof of Theorem dp41lemf

Step	Hyp	Ref	Expression
1		orcom 73	. . 3 ((a0 ∩ (b0 ∪ b1)) ∪ (b1 ∩ (a0 ∪ a1))) = ((b1 ∩ (a0 ∪ a1)) ∪ (a0 ∩ (b0 ∪ b1)))
2	1	lor 70	. 2 ((c0 ∪ c1) ∪ ((a0 ∩ (b0 ∪ b1)) ∪ (b1 ∩ (a0 ∪ a1)))) = ((c0 ∪ c1) ∪ ((b1 ∩ (a0 ∪ a1)) ∪ (a0 ∩ (b0 ∪ b1))))
3		or4 84	. . 3 ((c0 ∪ c1) ∪ ((b1 ∩ (a0 ∪ a1)) ∪ (a0 ∩ (b0 ∪ b1)))) = ((c0 ∪ (b1 ∩ (a0 ∪ a1))) ∪ (c1 ∪ (a0 ∩ (b0 ∪ b1))))
4		dp41lem.1	. . . . . 6 c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
5		ancom 74	. . . . . 6 ((a1 ∪ a2) ∩ (b1 ∪ b2)) = ((b1 ∪ b2) ∩ (a1 ∪ a2))
6	4, 5	tr 62	. . . . 5 c0 = ((b1 ∪ b2) ∩ (a1 ∪ a2))
7	6	ror 71	. . . 4 (c0 ∪ (b1 ∩ (a0 ∪ a1))) = (((b1 ∪ b2) ∩ (a1 ∪ a2)) ∪ (b1 ∩ (a0 ∪ a1)))
8		dp41lem.2	. . . . 5 c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
9	8	ror 71	. . . 4 (c1 ∪ (a0 ∩ (b0 ∪ b1))) = (((a0 ∪ a2) ∩ (b0 ∪ b2)) ∪ (a0 ∩ (b0 ∪ b1)))
10	7, 9	2or 72	. . 3 ((c0 ∪ (b1 ∩ (a0 ∪ a1))) ∪ (c1 ∪ (a0 ∩ (b0 ∪ b1)))) = ((((b1 ∪ b2) ∩ (a1 ∪ a2)) ∪ (b1 ∩ (a0 ∪ a1))) ∪ (((a0 ∪ a2) ∩ (b0 ∪ b2)) ∪ (a0 ∩ (b0 ∪ b1))))
11	3, 10	tr 62	. 2 ((c0 ∪ c1) ∪ ((b1 ∩ (a0 ∪ a1)) ∪ (a0 ∩ (b0 ∪ b1)))) = ((((b1 ∪ b2) ∩ (a1 ∪ a2)) ∪ (b1 ∩ (a0 ∪ a1))) ∪ (((a0 ∪ a2) ∩ (b0 ∪ b2)) ∪ (a0 ∩ (b0 ∪ b1))))
12		leao1 162	. . . 4 (b1 ∩ (a0 ∪ a1)) ≤ (b1 ∪ b2)
13	12	mli 1126	. . 3 (((b1 ∪ b2) ∩ (a1 ∪ a2)) ∪ (b1 ∩ (a0 ∪ a1))) = ((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (b1 ∩ (a0 ∪ a1))))
14		leao1 162	. . . 4 (a0 ∩ (b0 ∪ b1)) ≤ (a0 ∪ a2)
15	14	mli 1126	. . 3 (((a0 ∪ a2) ∩ (b0 ∪ b2)) ∪ (a0 ∩ (b0 ∪ b1))) = ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (a0 ∩ (b0 ∪ b1))))
16	13, 15	2or 72	. 2 ((((b1 ∪ b2) ∩ (a1 ∪ a2)) ∪ (b1 ∩ (a0 ∪ a1))) ∪ (((a0 ∪ a2) ∩ (b0 ∪ b2)) ∪ (a0 ∩ (b0 ∪ b1)))) = (((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (b1 ∩ (a0 ∪ a1)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (a0 ∩ (b0 ∪ b1)))))
17	2, 11, 16	3tr 65	1 ((c0 ∪ c1) ∪ ((a0 ∩ (b0 ∪ b1)) ∪ (b1 ∩ (a0 ∪ a1)))) = (((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (b1 ∩ (a0 ∪ a1)))) ∪ ((a0 ∪ a2) ∩ ((b0 ∪ b2) ∪ (a0 ∩ (b0 ∪ 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:  dp41lemm  1194