Theorem oadistb 1000

Description: Distributive law derived from OAL.

Hypotheses

Ref	Expression
oadistb.2	d =< (a ->2 b)
oadistb.1	e =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))

Assertion

Ref	Expression
oadistb	(d ^ (e v ((a ->2 b) ^ (a ->2 c)))) = ((d ^ e) v (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 128	. . . . . 6 (d ^ (a ->2 b)) = d
3	2	ran 71	. . . . 5 ((d ^ (a ->2 b)) ^ (e v ((a ->2 b) ^ (a ->2 c)))) = (d ^ (e v ((a ->2 b) ^ (a ->2 c))))
4	3	ax-r1 34	. . . 4 (d ^ (e v ((a ->2 b) ^ (a ->2 c)))) = ((d ^ (a ->2 b)) ^ (e v ((a ->2 b) ^ (a ->2 c))))
5		anass 69	. . . . 5 ((d ^ (a ->2 b)) ^ (e v ((a ->2 b) ^ (a ->2 c)))) = (d ^ ((a ->2 b) ^ (e v ((a ->2 b) ^ (a ->2 c)))))
6		oadistb.1	. . . . . . 7 e =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
7	6	oagen1 994	. . . . . 6 ((a ->2 b) ^ (e v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
8	7	lan 70	. . . . 5 (d ^ ((a ->2 b) ^ (e v ((a ->2 b) ^ (a ->2 c))))) = (d ^ ((a ->2 b) ^ (a ->2 c)))
9	5, 8	ax-r2 35	. . . 4 ((d ^ (a ->2 b)) ^ (e v ((a ->2 b) ^ (a ->2 c)))) = (d ^ ((a ->2 b) ^ (a ->2 c)))
10	4, 9	ax-r2 35	. . 3 (d ^ (e v ((a ->2 b) ^ (a ->2 c)))) = (d ^ ((a ->2 b) ^ (a ->2 c)))
11		leor 151	. . 3 (d ^ ((a ->2 b) ^ (a ->2 c))) =< ((d ^ e) v (d ^ ((a ->2 b) ^ (a ->2 c))))
12	10, 11	bltr 130	. 2 (d ^ (e v ((a ->2 b) ^ (a ->2 c)))) =< ((d ^ e) v (d ^ ((a ->2 b) ^ (a ->2 c))))
13		ledi 166	. 2 ((d ^ e) v (d ^ ((a ->2 b) ^ (a ->2 c)))) =< (d ^ (e v ((a ->2 b) ^ (a ->2 c))))
14	12, 13	lebi 137	1 (d ^ (e v ((a ->2 b) ^ (a ->2 c)))) = ((d ^ e) v (d ^ ((a ->2 b) ^ (a ->2 c))))

Syntax hints:   = wb 1   =< wle 2   v 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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