Theorem dp15lemf 1159

Description: Part of proof (1)=>(5) in Day/Pickering 1982. (Contributed by NM, 1-Apr-2012.)

Hypotheses

Ref	Expression
dp15lema.1	d = (a2 ∪ (a0 ∩ (a1 ∪ b1)))
dp15lema.2	p0 = ((a1 ∪ b1) ∩ (a2 ∪ b2))
dp15lema.3	e = (b0 ∩ (a0 ∪ p0))

Assertion

Ref	Expression
dp15lemf	(((a0 ∪ a2) ∩ ((b0 ∩ (a0 ∪ p0)) ∪ b2)) ∪ (((a1 ∪ a2) ∪ (b1 ∩ (a0 ∪ a1))) ∩ (b1 ∪ b2))) ≤ (((a1 ∪ a2) ∩ (b1 ∪ b2)) ∪ (((a0 ∪ a2) ∩ (b0 ∪ b2)) ∪ (b1 ∩ (a0 ∪ a1))))

Proof of Theorem dp15lemf

Step	Hyp	Ref	Expression
1		lea 160	. . . . 5 (b0 ∩ (a0 ∪ p0)) ≤ b0
2	1	leror 152	. . . 4 ((b0 ∩ (a0 ∪ p0)) ∪ b2) ≤ (b0 ∪ b2)
3	2	lelan 167	. . 3 ((a0 ∪ a2) ∩ ((b0 ∩ (a0 ∪ p0)) ∪ b2)) ≤ ((a0 ∪ a2) ∩ (b0 ∪ b2))
4		leao1 162	. . . . . 6 (b1 ∩ (a0 ∪ a1)) ≤ (b1 ∪ b2)
5	4	mldual2i 1127	. . . . 5 ((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (b1 ∩ (a0 ∪ a1)))) = (((b1 ∪ b2) ∩ (a1 ∪ a2)) ∪ (b1 ∩ (a0 ∪ a1)))
6		ancom 74	. . . . 5 ((b1 ∪ b2) ∩ ((a1 ∪ a2) ∪ (b1 ∩ (a0 ∪ a1)))) = (((a1 ∪ a2) ∪ (b1 ∩ (a0 ∪ a1))) ∩ (b1 ∪ b2))
7		ancom 74	. . . . . 6 ((b1 ∪ b2) ∩ (a1 ∪ a2)) = ((a1 ∪ a2) ∩ (b1 ∪ b2))
8	7	ror 71	. . . . 5 (((b1 ∪ b2) ∩ (a1 ∪ a2)) ∪ (b1 ∩ (a0 ∪ a1))) = (((a1 ∪ a2) ∩ (b1 ∪ b2)) ∪ (b1 ∩ (a0 ∪ a1)))
9	5, 6, 8	3tr2 64	. . . 4 (((a1 ∪ a2) ∪ (b1 ∩ (a0 ∪ a1))) ∩ (b1 ∪ b2)) = (((a1 ∪ a2) ∩ (b1 ∪ b2)) ∪ (b1 ∩ (a0 ∪ a1)))
10	9	bile 142	. . 3 (((a1 ∪ a2) ∪ (b1 ∩ (a0 ∪ a1))) ∩ (b1 ∪ b2)) ≤ (((a1 ∪ a2) ∩ (b1 ∪ b2)) ∪ (b1 ∩ (a0 ∪ a1)))
11	3, 10	le2or 168	. 2 (((a0 ∪ a2) ∩ ((b0 ∩ (a0 ∪ p0)) ∪ b2)) ∪ (((a1 ∪ a2) ∪ (b1 ∩ (a0 ∪ a1))) ∩ (b1 ∪ b2))) ≤ (((a0 ∪ a2) ∩ (b0 ∪ b2)) ∪ (((a1 ∪ a2) ∩ (b1 ∪ b2)) ∪ (b1 ∩ (a0 ∪ a1))))
12		or12 80	. 2 (((a0 ∪ a2) ∩ (b0 ∪ b2)) ∪ (((a1 ∪ a2) ∩ (b1 ∪ b2)) ∪ (b1 ∩ (a0 ∪ a1)))) = (((a1 ∪ a2) ∩ (b1 ∪ b2)) ∪ (((a0 ∪ a2) ∩ (b0 ∪ b2)) ∪ (b1 ∩ (a0 ∪ a1))))
13	11, 12	lbtr 139	1 (((a0 ∪ a2) ∩ ((b0 ∩ (a0 ∪ p0)) ∪ b2)) ∪ (((a1 ∪ a2) ∪ (b1 ∩ (a0 ∪ a1))) ∩ (b1 ∪ b2))) ≤ (((a1 ∪ a2) ∩ (b1 ∪ b2)) ∪ (((a0 ∪ a2) ∩ (b0 ∪ b2)) ∪ (b1 ∩ (a0 ∪ a1))))

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:  dp15lemh  1161