Theorem oalii 982

Description: Orthoarguesian law. Godowski/Greechie, Eq. II. This proof references oaliii 981 only.

Assertion

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

Proof of Theorem oalii

Step	Hyp	Ref	Expression
1		a5b 112	. . . . 5 ((a ->2 b) v ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))) = (a ->2 b)
2		oaliii 981	. . . . . 6 ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))
3	2	lor 66	. . . . 5 ((a ->2 b) v ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))) = ((a ->2 b) v ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))
4		df-i2 44	. . . . . 6 (a ->2 b) = (b v (a_|_ ^ b_|_))
5		ancom 68	. . . . . . 7 (a_|_ ^ b_|_) = (b_|_ ^ a_|_)
6	5	lor 66	. . . . . 6 (b v (a_|_ ^ b_|_)) = (b v (b_|_ ^ a_|_))
7	4, 6	ax-r2 35	. . . . 5 (a ->2 b) = (b v (b_|_ ^ a_|_))
8	1, 3, 7	3tr2 61	. . . 4 ((a ->2 b) v ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))) = (b v (b_|_ ^ a_|_))
9	8	lan 70	. . 3 (b_|_ ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))) = (b_|_ ^ (b v (b_|_ ^ a_|_)))
10		omlan 430	. . 3 (b_|_ ^ (b v (b_|_ ^ a_|_))) = (b_|_ ^ a_|_)
11	9, 10	ax-r2 35	. 2 (b_|_ ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))) = (b_|_ ^ a_|_)
12		lear 153	. 2 (b_|_ ^ a_|_) =< a_|_
13	11, 12	bltr 130	1 (b_|_ ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))) =< a_|_

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->2 wi2 14

This theorem is referenced by:  oaliv 983  oalem1 985

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

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