Theorem anorabs 217

Description: Absorption law for ortholattices.

Assertion

Ref	Expression
anorabs	(a_|_ ^ (b v (a_|_ ^ (a v b)))) = (a_|_ ^ (a v b))

Proof of Theorem anorabs

Step	Hyp	Ref	Expression
1		lea 152	. . 3 (a_|_ ^ (b v (a_|_ ^ (a v b)))) =< a_|_
2		lear 153	. . . 4 (a_|_ ^ (b v (a_|_ ^ (a v b)))) =< (b v (a_|_ ^ (a v b)))
3		leor 151	. . . . 5 b =< (a v b)
4		lear 153	. . . . 5 (a_|_ ^ (a v b)) =< (a v b)
5	3, 4	lel2or 162	. . . 4 (b v (a_|_ ^ (a v b))) =< (a v b)
6	2, 5	letr 129	. . 3 (a_|_ ^ (b v (a_|_ ^ (a v b)))) =< (a v b)
7	1, 6	ler2an 165	. 2 (a_|_ ^ (b v (a_|_ ^ (a v b)))) =< (a_|_ ^ (a v b))
8		lea 152	. . 3 (a_|_ ^ (a v b)) =< a_|_
9		leor 151	. . 3 (a_|_ ^ (a v b)) =< (b v (a_|_ ^ (a v b)))
10	8, 9	ler2an 165	. 2 (a_|_ ^ (a v b)) =< (a_|_ ^ (b v (a_|_ ^ (a v b))))
11	7, 10	lebi 137	1 (a_|_ ^ (b v (a_|_ ^ (a v b)))) = (a_|_ ^ (a v b))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7

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

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

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