Theorem oaliv 983

Description: Orthoarguesian law. Godowski/Greechie, Eq. IV.

Assertion

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

Proof of Theorem oaliv

Step	Hyp	Ref	Expression
1		lea 152	. . . 4 (b_|_ ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))) =< b_|_
2		oalii 982	. . . 4 (b_|_ ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))) =< a_|_
3	1, 2	ler2an 165	. . 3 (b_|_ ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))) =< (b_|_ ^ a_|_)
4		df-i2 44	. . . . . . 7 (a ->2 b) = (b v (a_|_ ^ b_|_))
5		ancom 68	. . . . . . . 8 (a_|_ ^ b_|_) = (b_|_ ^ a_|_)
6	5	lor 66	. . . . . . 7 (b v (a_|_ ^ b_|_)) = (b v (b_|_ ^ a_|_))
7	4, 6	ax-r2 35	. . . . . 6 (a ->2 b) = (b v (b_|_ ^ a_|_))
8	7	lan 70	. . . . 5 (b_|_ ^ (a ->2 b)) = (b_|_ ^ (b v (b_|_ ^ a_|_)))
9		omlan 430	. . . . 5 (b_|_ ^ (b v (b_|_ ^ a_|_))) = (b_|_ ^ a_|_)
10	8, 9	ax-r2 35	. . . 4 (b_|_ ^ (a ->2 b)) = (b_|_ ^ a_|_)
11	10	ax-r1 34	. . 3 (b_|_ ^ a_|_) = (b_|_ ^ (a ->2 b))
12	3, 11	lbtr 131	. 2 (b_|_ ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))) =< (b_|_ ^ (a ->2 b))
13		leo 150	. 2 (b_|_ ^ (a ->2 b)) =< ((b_|_ ^ (a ->2 b)) v (c_|_ ^ (a ->2 c)))
14	12, 13	letr 129	1 (b_|_ ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))) =< ((b_|_ ^ (a ->2 b)) v (c_|_ ^ (a ->2 c)))

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->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-r3 421  ax-3oa 978

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

metamath.org