Theorem oadistb 1020

Description: Distributive law derived from OAL. (Contributed by NM, 20-Nov-1998.)

Hypotheses

Ref	Expression
oadistb.2	d ≤ (a →2 b)
oadistb.1	e ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))

Assertion

Ref	Expression
oadistb	(d ∩ (e ∪ ((a →2 b) ∩ (a →2 c)))) = ((d ∩ e) ∪ (d ∩ ((a →2 b) ∩ (a →2 c))))

Proof of Theorem oadistb

Step	Hyp	Ref	Expression
1		oadistb.2	. . . . . . 7 d ≤ (a →2 b)
2	1	df2le2 136	. . . . . 6 (d ∩ (a →2 b)) = d
3	2	ran 78	. . . . 5 ((d ∩ (a →2 b)) ∩ (e ∪ ((a →2 b) ∩ (a →2 c)))) = (d ∩ (e ∪ ((a →2 b) ∩ (a →2 c))))
4	3	ax-r1 35	. . . 4 (d ∩ (e ∪ ((a →2 b) ∩ (a →2 c)))) = ((d ∩ (a →2 b)) ∩ (e ∪ ((a →2 b) ∩ (a →2 c))))
5		anass 76	. . . . 5 ((d ∩ (a →2 b)) ∩ (e ∪ ((a →2 b) ∩ (a →2 c)))) = (d ∩ ((a →2 b) ∩ (e ∪ ((a →2 b) ∩ (a →2 c)))))
6		oadistb.1	. . . . . . 7 e ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
7	6	oagen1 1014	. . . . . 6 ((a →2 b) ∩ (e ∪ ((a →2 b) ∩ (a →2 c)))) = ((a →2 b) ∩ (a →2 c))
8	7	lan 77	. . . . 5 (d ∩ ((a →2 b) ∩ (e ∪ ((a →2 b) ∩ (a →2 c))))) = (d ∩ ((a →2 b) ∩ (a →2 c)))
9	5, 8	ax-r2 36	. . . 4 ((d ∩ (a →2 b)) ∩ (e ∪ ((a →2 b) ∩ (a →2 c)))) = (d ∩ ((a →2 b) ∩ (a →2 c)))
10	4, 9	ax-r2 36	. . 3 (d ∩ (e ∪ ((a →2 b) ∩ (a →2 c)))) = (d ∩ ((a →2 b) ∩ (a →2 c)))
11		leor 159	. . 3 (d ∩ ((a →2 b) ∩ (a →2 c))) ≤ ((d ∩ e) ∪ (d ∩ ((a →2 b) ∩ (a →2 c))))
12	10, 11	bltr 138	. 2 (d ∩ (e ∪ ((a →2 b) ∩ (a →2 c)))) ≤ ((d ∩ e) ∪ (d ∩ ((a →2 b) ∩ (a →2 c))))
13		ledi 174	. 2 ((d ∩ e) ∪ (d ∩ ((a →2 b) ∩ (a →2 c)))) ≤ (d ∩ (e ∪ ((a →2 b) ∩ (a →2 c))))
14	12, 13	lebi 145	1 (d ∩ (e ∪ ((a →2 b) ∩ (a →2 c)))) = ((d ∩ e) ∪ (d ∩ ((a →2 b) ∩ (a →2 c))))

Syntax hints:   = wb 1   ≤ wle 2   ∪ 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)