Theorem anorabs2 216

Description: Absorption law for ortholattices.

Assertion

Ref	Expression
anorabs2	(a ^ (b v (a ^ (b v c)))) = (a ^ (b v c))

Proof of Theorem anorabs2

Step	Hyp	Ref	Expression
1		lea 152	. . 3 (a ^ (b v (a ^ (b v c)))) =< a
2		lear 153	. . . 4 (a ^ (b v (a ^ (b v c)))) =< (b v (a ^ (b v c)))
3		leo 150	. . . . 5 b =< (b v c)
4		lear 153	. . . . 5 (a ^ (b v c)) =< (b v c)
5	3, 4	lel2or 162	. . . 4 (b v (a ^ (b v c))) =< (b v c)
6	2, 5	letr 129	. . 3 (a ^ (b v (a ^ (b v c)))) =< (b v c)
7	1, 6	ler2an 165	. 2 (a ^ (b v (a ^ (b v c)))) =< (a ^ (b v c))
8		lea 152	. . 3 (a ^ (b v c)) =< a
9		leor 151	. . 3 (a ^ (b v c)) =< (b v (a ^ (b v c)))
10	8, 9	ler2an 165	. 2 (a ^ (b v c)) =< (a ^ (b v (a ^ (b v c))))
11	7, 10	lebi 137	1 (a ^ (b v (a ^ (b v c)))) = (a ^ (b v c))

Syntax hints:   = wb 1   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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