Theorem distoa 924

Description: Derivation in OM of OA, assuming OA distributive law oadistd 1003.

Hypotheses

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

Assertion

Ref	Expression
distoa	((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)

Proof of Theorem distoa

Step	Hyp	Ref	Expression
1		1oa 802	. . 3 ((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)
2		2oath1 808	. . . 4 ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
3		lear 153	. . . 4 ((a ->2 b) ^ (a ->2 c)) =< (a ->2 c)
4	2, 3	bltr 130	. . 3 ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)
5	1, 4	le2or 160	. 2 (((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) v ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))) =< ((a ->2 c) v (a ->2 c))
6		distoa.4	. . . . 5 (d ^ (e v f)) = ((d ^ e) v (d ^ f))
7		distoa.1	. . . . . 6 d = (a ->2 b)
8		distoa.2	. . . . . . 7 e = ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))
9		distoa.3	. . . . . . 7 f = ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))
10	8, 9	2or 67	. . . . . 6 (e v f) = (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))
11	7, 10	2an 72	. . . . 5 (d ^ (e v f)) = ((a ->2 b) ^ (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))))
12	7, 8	2an 72	. . . . . 6 (d ^ e) = ((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c))))
13	7, 9	2an 72	. . . . . 6 (d ^ f) = ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))
14	12, 13	2or 67	. . . . 5 ((d ^ e) v (d ^ f)) = (((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) v ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))))
15	6, 11, 14	3tr2 61	. . . 4 ((a ->2 b) ^ (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))) = (((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) v ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))))
16	15	ax-r1 34	. . 3 (((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) v ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))) = ((a ->2 b) ^ (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))))
17		u12lem 753	. . . . 5 (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) = ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
18		df-i0 42	. . . . 5 ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) = ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
19	17, 18	ax-r2 35	. . . 4 (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) = ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
20	19	lan 70	. . 3 ((a ->2 b) ^ (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))) = ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))
21	16, 20	ax-r2 35	. 2 (((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) v ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))) = ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))
22		oridm 102	. 2 ((a ->2 c) v (a ->2 c)) = (a ->2 c)
23	5, 21, 22	le3tr2 133	1 ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->0 wi0 12   ->1 wi1 13   ->2 wi2 14

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i0 42  df-i1 43  df-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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