Theorem oml6 488

Description: Orthomodular law.

Assertion

Ref	Expression
oml6	(a ∪ (b ∩ (a⊥ ∪ b⊥ ))) = (a ∪ b)

Proof of Theorem oml6

Step	Hyp	Ref	Expression
1		comor1 461	. . . 4 (a⊥ ∪ b⊥ ) C a⊥
2	1	comcom7 460	. . 3 (a⊥ ∪ b⊥ ) C a
3		comor2 462	. . . 4 (a⊥ ∪ b⊥ ) C b⊥
4	3	comcom7 460	. . 3 (a⊥ ∪ b⊥ ) C b
5	2, 4	fh4c 478	. 2 (a ∪ (b ∩ (a⊥ ∪ b⊥ ))) = ((a ∪ b) ∩ (a ∪ (a⊥ ∪ b⊥ )))
6		df-t 41	. . . . . 6 1 = (a ∪ a⊥ )
7	6	ax-r5 38	. . . . 5 (1 ∪ b⊥ ) = ((a ∪ a⊥ ) ∪ b⊥ )
8		ax-a2 31	. . . . . 6 (1 ∪ b⊥ ) = (b⊥ ∪ 1)
9		or1 104	. . . . . 6 (b⊥ ∪ 1) = 1
10	8, 9	ax-r2 36	. . . . 5 (1 ∪ b⊥ ) = 1
11		ax-a3 32	. . . . 5 ((a ∪ a⊥ ) ∪ b⊥ ) = (a ∪ (a⊥ ∪ b⊥ ))
12	7, 10, 11	3tr2 64	. . . 4 1 = (a ∪ (a⊥ ∪ b⊥ ))
13	12	ax-r1 35	. . 3 (a ∪ (a⊥ ∪ b⊥ )) = 1
14	13	lan 77	. 2 ((a ∪ b) ∩ (a ∪ (a⊥ ∪ b⊥ ))) = ((a ∪ b) ∩ 1)
15		an1 106	. 2 ((a ∪ b) ∩ 1) = (a ∪ b)
16	5, 14, 15	3tr 65	1 (a ∪ (b ∩ (a⊥ ∪ b⊥ ))) = (a ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  sa5  836