Theorem oadistd 1023

Description: OA distributive law. (Contributed by NM, 21-Nov-1998.)

Hypotheses

Ref	Expression
oadistd.1	d ≤ (a →2 b)
oadistd.2	e ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
oadistd.3	f ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
oadistd.4	(d ∩ (a →2 c)) ≤ f

Assertion

Ref	Expression
oadistd	(d ∩ (e ∪ f)) = ((d ∩ e) ∪ (d ∩ f))

Proof of Theorem oadistd

Step	Hyp	Ref	Expression
1		oadistd.2	. . . . . . . . . 10 e ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
2		oadistd.3	. . . . . . . . . 10 f ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
3	1, 2	le2or 168	. . . . . . . . 9 (e ∪ f) ≤ (((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))) ∪ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))))
4		oridm 110	. . . . . . . . 9 (((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))) ∪ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) = ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
5	3, 4	lbtr 139	. . . . . . . 8 (e ∪ f) ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
6	5	lelan 167	. . . . . . 7 (d ∩ (e ∪ f)) ≤ (d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))))
7	6	df2le2 136	. . . . . 6 ((d ∩ (e ∪ f)) ∩ (d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))))) = (d ∩ (e ∪ f))
8	7	ax-r1 35	. . . . 5 (d ∩ (e ∪ f)) = ((d ∩ (e ∪ f)) ∩ (d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))))
9		df-i0 43	. . . . . . . 8 ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))) = ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))
10	9	lan 77	. . . . . . 7 (d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) = (d ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c))))
11		oadistd.1	. . . . . . . 8 d ≤ (a →2 b)
12		leo 158	. . . . . . . . 9 (b ∪ c)⊥ ≤ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))
13	9	ax-r1 35	. . . . . . . . 9 ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c))) = ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
14	12, 13	lbtr 139	. . . . . . . 8 (b ∪ c)⊥ ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
15	11, 14	oagen1b 1015	. . . . . . 7 (d ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) = (d ∩ (a →2 c))
16	10, 15	ax-r2 36	. . . . . 6 (d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) = (d ∩ (a →2 c))
17	16	lan 77	. . . . 5 ((d ∩ (e ∪ f)) ∩ (d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))))) = ((d ∩ (e ∪ f)) ∩ (d ∩ (a →2 c)))
18	8, 17	ax-r2 36	. . . 4 (d ∩ (e ∪ f)) = ((d ∩ (e ∪ f)) ∩ (d ∩ (a →2 c)))
19		lear 161	. . . . 5 ((d ∩ (e ∪ f)) ∩ (d ∩ (a →2 c))) ≤ (d ∩ (a →2 c))
20		oadistd.4	. . . . . . . . 9 (d ∩ (a →2 c)) ≤ f
21	20	df2le2 136	. . . . . . . 8 ((d ∩ (a →2 c)) ∩ f) = (d ∩ (a →2 c))
22	21	ax-r1 35	. . . . . . 7 (d ∩ (a →2 c)) = ((d ∩ (a →2 c)) ∩ f)
23		an32 83	. . . . . . 7 ((d ∩ (a →2 c)) ∩ f) = ((d ∩ f) ∩ (a →2 c))
24	22, 23	ax-r2 36	. . . . . 6 (d ∩ (a →2 c)) = ((d ∩ f) ∩ (a →2 c))
25		lea 160	. . . . . 6 ((d ∩ f) ∩ (a →2 c)) ≤ (d ∩ f)
26	24, 25	bltr 138	. . . . 5 (d ∩ (a →2 c)) ≤ (d ∩ f)
27	19, 26	letr 137	. . . 4 ((d ∩ (e ∪ f)) ∩ (d ∩ (a →2 c))) ≤ (d ∩ f)
28	18, 27	bltr 138	. . 3 (d ∩ (e ∪ f)) ≤ (d ∩ f)
29		leor 159	. . 3 (d ∩ f) ≤ ((d ∩ e) ∪ (d ∩ f))
30	28, 29	letr 137	. 2 (d ∩ (e ∪ f)) ≤ ((d ∩ e) ∪ (d ∩ f))
31		ledi 174	. 2 ((d ∩ e) ∪ (d ∩ f)) ≤ (d ∩ (e ∪ f))
32	30, 31	lebi 145	1 (d ∩ (e ∪ f)) = ((d ∩ e) ∪ (d ∩ f))

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₀ wi0 11   →₂ wi2 13

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-3oa 998

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131

This theorem is referenced by: (None)