Theorem oadistd 1003

Description: OA distributive law.

Hypotheses

Ref	Expression
oadistd.1	d =< (a ->2 b)
oadistd.2	e =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
oadistd.3	f =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
oadistd.4	(d ^ (a ->2 c)) =< f

Assertion

Ref	Expression
oadistd	(d ^ (e v f)) = ((d ^ e) v (d ^ f))

Proof of Theorem oadistd

Step	Hyp	Ref	Expression
1		oadistd.2	. . . . . . . . . 10 e =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
2		oadistd.3	. . . . . . . . . 10 f =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
3	1, 2	le2or 160	. . . . . . . . 9 (e v f) =< (((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))))
4		oridm 102	. . . . . . . . 9 (((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) = ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
5	3, 4	lbtr 131	. . . . . . . 8 (e v f) =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
6	5	lelan 159	. . . . . . 7 (d ^ (e v f)) =< (d ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))))
7	6	df2le2 128	. . . . . 6 ((d ^ (e v f)) ^ (d ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))))) = (d ^ (e v f))
8	7	ax-r1 34	. . . . 5 (d ^ (e v f)) = ((d ^ (e v f)) ^ (d ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))))
9		df-i0 42	. . . . . . . 8 ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) = ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
10	9	lan 70	. . . . . . 7 (d ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) = (d ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))
11		oadistd.1	. . . . . . . 8 d =< (a ->2 b)
12		leo 150	. . . . . . . . 9 (b v c)_|_ =< ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
13	9	ax-r1 34	. . . . . . . . 9 ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))) = ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
14	12, 13	lbtr 131	. . . . . . . 8 (b v c)_|_ =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
15	11, 14	oagen1b 995	. . . . . . 7 (d ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) = (d ^ (a ->2 c))
16	10, 15	ax-r2 35	. . . . . 6 (d ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) = (d ^ (a ->2 c))
17	16	lan 70	. . . . 5 ((d ^ (e v f)) ^ (d ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))))) = ((d ^ (e v f)) ^ (d ^ (a ->2 c)))
18	8, 17	ax-r2 35	. . . 4 (d ^ (e v f)) = ((d ^ (e v f)) ^ (d ^ (a ->2 c)))
19		lear 153	. . . . 5 ((d ^ (e v 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 v f)) ^ (d ^ (a ->2 c))) =< (d ^ f)
28	18, 27	bltr 130	. . 3 (d ^ (e v f)) =< (d ^ f)
29		leor 151	. . 3 (d ^ f) =< ((d ^ e) v (d ^ f))
30	28, 29	letr 129	. 2 (d ^ (e v f)) =< ((d ^ e) v (d ^ f))
31		ledi 166	. 2 ((d ^ e) v (d ^ f)) =< (d ^ (e v f))
32	30, 31	lebi 137	1 (d ^ (e v f)) = ((d ^ e) v (d ^ f))

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   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

metamath.org