Theorem oaidlem2g 932

Description: Lemma for identity-like OA law (generalized). (Contributed by NM, 18-Feb-2002.)

Hypothesis

Ref	Expression
oaidlem2g.1	((c ∪ (a ∩ b))⊥ ∪ (a →1 b)) = 1

Assertion

Ref	Expression
oaidlem2g	(a ∩ (c ∪ (a ∩ b))) ≤ b

Proof of Theorem oaidlem2g

Step	Hyp	Ref	Expression
1		anidm 111	. . . . . . . . . 10 (a ∩ a) = a
2	1	ax-r1 35	. . . . . . . . 9 a = (a ∩ a)
3	2	ran 78	. . . . . . . 8 (a ∩ b) = ((a ∩ a) ∩ b)
4		anass 76	. . . . . . . 8 ((a ∩ a) ∩ b) = (a ∩ (a ∩ b))
5	3, 4	ax-r2 36	. . . . . . 7 (a ∩ b) = (a ∩ (a ∩ b))
6		leor 159	. . . . . . . 8 (a ∩ b) ≤ (c ∪ (a ∩ b))
7	6	lelan 167	. . . . . . 7 (a ∩ (a ∩ b)) ≤ (a ∩ (c ∪ (a ∩ b)))
8	5, 7	bltr 138	. . . . . 6 (a ∩ b) ≤ (a ∩ (c ∪ (a ∩ b)))
9	8	df-le2 131	. . . . 5 ((a ∩ b) ∪ (a ∩ (c ∪ (a ∩ b)))) = (a ∩ (c ∪ (a ∩ b)))
10		ax-a3 32	. . . . . 6 (((c ∪ (a ∩ b))⊥ ∪ a⊥ ) ∪ (a ∩ b)) = ((c ∪ (a ∩ b))⊥ ∪ (a⊥ ∪ (a ∩ b)))
11		ax-a2 31	. . . . . . . 8 ((c ∪ (a ∩ b))⊥ ∪ a⊥ ) = (a⊥ ∪ (c ∪ (a ∩ b))⊥ )
12		oran3 93	. . . . . . . 8 (a⊥ ∪ (c ∪ (a ∩ b))⊥ ) = (a ∩ (c ∪ (a ∩ b)))⊥
13	11, 12	ax-r2 36	. . . . . . 7 ((c ∪ (a ∩ b))⊥ ∪ a⊥ ) = (a ∩ (c ∪ (a ∩ b)))⊥
14	13	ax-r5 38	. . . . . 6 (((c ∪ (a ∩ b))⊥ ∪ a⊥ ) ∪ (a ∩ b)) = ((a ∩ (c ∪ (a ∩ b)))⊥ ∪ (a ∩ b))
15		df-i1 44	. . . . . . . . 9 (a →1 b) = (a⊥ ∪ (a ∩ b))
16	15	lor 70	. . . . . . . 8 ((c ∪ (a ∩ b))⊥ ∪ (a →1 b)) = ((c ∪ (a ∩ b))⊥ ∪ (a⊥ ∪ (a ∩ b)))
17	16	ax-r1 35	. . . . . . 7 ((c ∪ (a ∩ b))⊥ ∪ (a⊥ ∪ (a ∩ b))) = ((c ∪ (a ∩ b))⊥ ∪ (a →1 b))
18		oaidlem2g.1	. . . . . . 7 ((c ∪ (a ∩ b))⊥ ∪ (a →1 b)) = 1
19	17, 18	ax-r2 36	. . . . . 6 ((c ∪ (a ∩ b))⊥ ∪ (a⊥ ∪ (a ∩ b))) = 1
20	10, 14, 19	3tr2 64	. . . . 5 ((a ∩ (c ∪ (a ∩ b)))⊥ ∪ (a ∩ b)) = 1
21	9, 20	lem3.1 443	. . . 4 (a ∩ b) = (a ∩ (c ∪ (a ∩ b)))
22	21	ax-r1 35	. . 3 (a ∩ (c ∪ (a ∩ b))) = (a ∩ b)
23	22	bile 142	. 2 (a ∩ (c ∪ (a ∩ b))) ≤ (a ∩ b)
24		lear 161	. 2 (a ∩ b) ≤ b
25	23, 24	letr 137	1 (a ∩ (c ∪ (a ∩ b))) ≤ b

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₁ wi1 12

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-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131

This theorem is referenced by: (None)