Theorem oadistd 1003

Description: OA distributive law.

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 160	. . . . . . . . 9 (e ∪ f) ≤ (((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))) ∪ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))))
4		oridm 102	. . . . . . . . 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 131	. . . . . . . 8 (e ∪ f) ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
6	5	lelan 159	. . . . . . 7 (d ∩ (e ∪ f)) ≤ (d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))))
7	6	df2le2 128	. . . . . 6 ((d ∩ (e ∪ f)) ∩ (d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))))) = (d ∩ (e ∪ f))
8	7	ax-r1 34	. . . . 5 (d ∩ (e ∪ f)) = ((d ∩ (e ∪ f)) ∩ (d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))))
9		df-i0 42	. . . . . . . 8 ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))) = ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))
10	9	lan 70	. . . . . . 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 150	. . . . . . . . 9 (b ∪ c)⊥ ≤ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))
13	9	ax-r1 34	. . . . . . . . 9 ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c))) = ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
14	12, 13	lbtr 131	. . . . . . . 8 (b ∪ c)⊥ ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
15	11, 14	oagen1b 995	. . . . . . 7 (d ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) = (d ∩ (a →2 c))
16	10, 15	ax-r2 35	. . . . . 6 (d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) = (d ∩ (a →2 c))
17	16	lan 70	. . . . 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 35	. . . 4 (d ∩ (e ∪ f)) = ((d ∩ (e ∪ f)) ∩ (d ∩ (a →2 c)))
19		lear 153	. . . . 5 ((d ∩ (e ∪ f)) ∩ (d ∩ (a →2 c))) ≤ (d ∩ (a →2 c))
20		oadistd.4	. . . . . . . . 9 (d ∩ (a →2 c)) ≤ f
21	20	df2le2 128	. . . . . . . 8 ((d ∩ (a →2 c)) ∩ f) = (d ∩ (a →2 c))
22	21	ax-r1 34	. . . . . . 7 (d ∩ (a →2 c)) = ((d ∩ (a →2 c)) ∩ f)
23		an32 76	. . . . . . 7 ((d ∩ (a →2 c)) ∩ f) = ((d ∩ f) ∩ (a →2 c))
24	22, 23	ax-r2 35	. . . . . 6 (d ∩ (a →2 c)) = ((d ∩ f) ∩ (a →2 c))
25		lea 152	. . . . . 6 ((d ∩ f) ∩ (a →2 c)) ≤ (d ∩ f)
26	24, 25	bltr 130	. . . . 5 (d ∩ (a →2 c)) ≤ (d ∩ f)
27	19, 26	letr 129	. . . 4 ((d ∩ (e ∪ f)) ∩ (d ∩ (a →2 c))) ≤ (d ∩ f)
28	18, 27	bltr 130	. . 3 (d ∩ (e ∪ f)) ≤ (d ∩ f)
29		leor 151	. . 3 (d ∩ f) ≤ ((d ∩ e) ∪ (d ∩ f))
30	28, 29	letr 129	. 2 (d ∩ (e ∪ f)) ≤ ((d ∩ e) ∪ (d ∩ f))
31		ledi 166	. 2 ((d ∩ e) ∪ (d ∩ f)) ≤ (d ∩ (e ∪ f))
32	30, 31	lebi 137	1 (d ∩ (e ∪ f)) = ((d ∩ e) ∪ (d ∩ f))

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₀ wi0 12   →₂ wi2 14

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-3oa 978

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i0 42  df-i1 43  df-i2 44  df-le1 122  df-le2 123

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