Theorem omla 429

Description: Orthomodular law.

Assertion

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

Proof of Theorem omla

Step	Hyp	Ref	Expression
1		df-a 39	. . . . . . 7 (a ^ (a_|_ v b_|_)) = (a_|_ v (a_|_ v b_|_)_|_)_|_
2		df-a 39	. . . . . . . . . 10 (a ^ b) = (a_|_ v b_|_)_|_
3	2	ax-r1 34	. . . . . . . . 9 (a_|_ v b_|_)_|_ = (a ^ b)
4	3	lor 66	. . . . . . . 8 (a_|_ v (a_|_ v b_|_)_|_) = (a_|_ v (a ^ b))
5	4	ax-r4 36	. . . . . . 7 (a_|_ v (a_|_ v b_|_)_|_)_|_ = (a_|_ v (a ^ b))_|_
6	1, 5	ax-r2 35	. . . . . 6 (a ^ (a_|_ v b_|_)) = (a_|_ v (a ^ b))_|_
7	6	ax-r1 34	. . . . 5 (a_|_ v (a ^ b))_|_ = (a ^ (a_|_ v b_|_))
8	7	lor 66	. . . 4 (a_|_ v (a_|_ v (a ^ b))_|_) = (a_|_ v (a ^ (a_|_ v b_|_)))
9		omln 428	. . . 4 (a_|_ v (a ^ (a_|_ v b_|_))) = (a_|_ v b_|_)
10	8, 9	ax-r2 35	. . 3 (a_|_ v (a_|_ v (a ^ b))_|_) = (a_|_ v b_|_)
11		df-a 39	. . . 4 (a ^ (a_|_ v (a ^ b))) = (a_|_ v (a_|_ v (a ^ b))_|_)_|_
12	11	con2 64	. . 3 (a ^ (a_|_ v (a ^ b)))_|_ = (a_|_ v (a_|_ v (a ^ b))_|_)
13	2	con2 64	. . 3 (a ^ b)_|_ = (a_|_ v b_|_)
14	10, 12, 13	3tr1 60	. 2 (a ^ (a_|_ v (a ^ b)))_|_ = (a ^ b)_|_
15	14	con1 63	1 (a ^ (a_|_ v (a ^ b))) = (a ^ b)

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

This theorem is referenced by:  omlan 430  oml5a 432  gsth2 472  oa3-2to2s 970

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

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41

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