Theorem womle2a 287

Description: An equivalent to the WOM law.

Hypothesis

Ref	Expression
womle2a.1	(a ^ (a ->2 b)) =< ((a ->2 b)_|_ v (a ->1 b))

Assertion

Ref	Expression
womle2a	((a ->2 b)_|_ v (a ->1 b)) = 1

Proof of Theorem womle2a

Step	Hyp	Ref	Expression
1		or4 77	. . 3 (((a ->2 b)_|_ v (a ->2 b)_|_) v ((a ->1 b) v a_|_)) = (((a ->2 b)_|_ v (a ->1 b)) v ((a ->2 b)_|_ v a_|_))
2		oridm 102	. . . 4 ((a ->2 b)_|_ v (a ->2 b)_|_) = (a ->2 b)_|_
3		df-i1 43	. . . . . 6 (a ->1 b) = (a_|_ v (a ^ b))
4	3	ax-r5 37	. . . . 5 ((a ->1 b) v a_|_) = ((a_|_ v (a ^ b)) v a_|_)
5		oridm 102	. . . . . . 7 (a_|_ v a_|_) = a_|_
6	5	ax-r5 37	. . . . . 6 ((a_|_ v a_|_) v (a ^ b)) = (a_|_ v (a ^ b))
7		or32 75	. . . . . 6 ((a_|_ v (a ^ b)) v a_|_) = ((a_|_ v a_|_) v (a ^ b))
8	6, 7, 3	3tr1 60	. . . . 5 ((a_|_ v (a ^ b)) v a_|_) = (a ->1 b)
9	4, 8	ax-r2 35	. . . 4 ((a ->1 b) v a_|_) = (a ->1 b)
10	2, 9	2or 67	. . 3 (((a ->2 b)_|_ v (a ->2 b)_|_) v ((a ->1 b) v a_|_)) = ((a ->2 b)_|_ v (a ->1 b))
11		ax-a2 30	. . . . 5 ((a ->2 b)_|_ v a_|_) = (a_|_ v (a ->2 b)_|_)
12		oran3 85	. . . . 5 (a_|_ v (a ->2 b)_|_) = (a ^ (a ->2 b))_|_
13	11, 12	ax-r2 35	. . . 4 ((a ->2 b)_|_ v a_|_) = (a ^ (a ->2 b))_|_
14	13	lor 66	. . 3 (((a ->2 b)_|_ v (a ->1 b)) v ((a ->2 b)_|_ v a_|_)) = (((a ->2 b)_|_ v (a ->1 b)) v (a ^ (a ->2 b))_|_)
15	1, 10, 14	3tr2 61	. 2 ((a ->2 b)_|_ v (a ->1 b)) = (((a ->2 b)_|_ v (a ->1 b)) v (a ^ (a ->2 b))_|_)
16		le1 138	. . 3 (((a ->2 b)_|_ v (a ->1 b)) v (a ^ (a ->2 b))_|_) =< 1
17		df-t 40	. . . 4 1 = ((a ^ (a ->2 b)) v (a ^ (a ->2 b))_|_)
18		womle2a.1	. . . . 5 (a ^ (a ->2 b)) =< ((a ->2 b)_|_ v (a ->1 b))
19	18	leror 144	. . . 4 ((a ^ (a ->2 b)) v (a ^ (a ->2 b))_|_) =< (((a ->2 b)_|_ v (a ->1 b)) v (a ^ (a ->2 b))_|_)
20	17, 19	bltr 130	. . 3 1 =< (((a ->2 b)_|_ v (a ->1 b)) v (a ^ (a ->2 b))_|_)
21	16, 20	lebi 137	. 2 (((a ->2 b)_|_ v (a ->1 b)) v (a ^ (a ->2 b))_|_) = 1
22	15, 21	ax-r2 35	1 ((a ->2 b)_|_ v (a ->1 b)) = 1

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->1 wi1 13   ->2 wi2 14

This theorem is referenced by:  womle 290

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

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

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