Theorem dp35lemb 1176

Description: Part of proof (3)=>(5) in Day/Pickering 1982. (Contributed by NM, 2-Apr-2012.)

Hypotheses

Ref	Expression
dp35lem.1	c0 = ((a1 ∪ a2) ∩ (b1 ∪ b2))
dp35lem.2	c1 = ((a0 ∪ a2) ∩ (b0 ∪ b2))
dp35lem.3	c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
dp35lem.4	p0 = ((a1 ∪ b1) ∩ (a2 ∪ b2))
dp35lem.5	p = (((a0 ∪ b0) ∩ (a1 ∪ b1)) ∩ (a2 ∪ b2))

Assertion

Ref	Expression
dp35lemb	(b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))) = (b0 ∩ (b1 ∪ ((a0 ∪ a1) ∩ (c0 ∪ c1))))

Proof of Theorem dp35lemb

Step	Hyp	Ref	Expression
1		dp35lem.3	. . . . . . 7 c2 = ((a0 ∪ a1) ∩ (b0 ∪ b1))
2	1	ran 78	. . . . . 6 (c2 ∩ (c0 ∪ c1)) = (((a0 ∪ a1) ∩ (b0 ∪ b1)) ∩ (c0 ∪ c1))
3		an32 83	. . . . . 6 (((a0 ∪ a1) ∩ (b0 ∪ b1)) ∩ (c0 ∪ c1)) = (((a0 ∪ a1) ∩ (c0 ∪ c1)) ∩ (b0 ∪ b1))
4	2, 3	tr 62	. . . . 5 (c2 ∩ (c0 ∪ c1)) = (((a0 ∪ a1) ∩ (c0 ∪ c1)) ∩ (b0 ∪ b1))
5	4	lor 70	. . . 4 (b1 ∪ (c2 ∩ (c0 ∪ c1))) = (b1 ∪ (((a0 ∪ a1) ∩ (c0 ∪ c1)) ∩ (b0 ∪ b1)))
6		leor 159	. . . . 5 b1 ≤ (b0 ∪ b1)
7	6	ml2i 1125	. . . 4 (b1 ∪ (((a0 ∪ a1) ∩ (c0 ∪ c1)) ∩ (b0 ∪ b1))) = ((b1 ∪ ((a0 ∪ a1) ∩ (c0 ∪ c1))) ∩ (b0 ∪ b1))
8		ancom 74	. . . 4 ((b1 ∪ ((a0 ∪ a1) ∩ (c0 ∪ c1))) ∩ (b0 ∪ b1)) = ((b0 ∪ b1) ∩ (b1 ∪ ((a0 ∪ a1) ∩ (c0 ∪ c1))))
9	5, 7, 8	3tr 65	. . 3 (b1 ∪ (c2 ∩ (c0 ∪ c1))) = ((b0 ∪ b1) ∩ (b1 ∪ ((a0 ∪ a1) ∩ (c0 ∪ c1))))
10	9	lan 77	. 2 (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))) = (b0 ∩ ((b0 ∪ b1) ∩ (b1 ∪ ((a0 ∪ a1) ∩ (c0 ∪ c1)))))
11		anass 76	. . 3 ((b0 ∩ (b0 ∪ b1)) ∩ (b1 ∪ ((a0 ∪ a1) ∩ (c0 ∪ c1)))) = (b0 ∩ ((b0 ∪ b1) ∩ (b1 ∪ ((a0 ∪ a1) ∩ (c0 ∪ c1)))))
12	11	cm 61	. 2 (b0 ∩ ((b0 ∪ b1) ∩ (b1 ∪ ((a0 ∪ a1) ∩ (c0 ∪ c1))))) = ((b0 ∩ (b0 ∪ b1)) ∩ (b1 ∪ ((a0 ∪ a1) ∩ (c0 ∪ c1))))
13		anabs 121	. . 3 (b0 ∩ (b0 ∪ b1)) = b0
14	13	ran 78	. 2 ((b0 ∩ (b0 ∪ b1)) ∩ (b1 ∪ ((a0 ∪ a1) ∩ (c0 ∪ c1)))) = (b0 ∩ (b1 ∪ ((a0 ∪ a1) ∩ (c0 ∪ c1))))
15	10, 12, 14	3tr 65	1 (b0 ∩ (b1 ∪ (c2 ∩ (c0 ∪ c1)))) = (b0 ∩ (b1 ∪ ((a0 ∪ a1) ∩ (c0 ∪ c1))))

Syntax hints:   = wb 1   ∪ 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:  dp35lembb  1177