Theorem oml6 470

Description: Orthomodular law.

Assertion

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

Proof of Theorem oml6

Step	Hyp	Ref	Expression
1		comor1 443	. . . 4 (a⊥ ∪ b⊥ ) C a⊥
2	1	comcom7 442	. . 3 (a⊥ ∪ b⊥ ) C a
3		comor2 444	. . . 4 (a⊥ ∪ b⊥ ) C b⊥
4	3	comcom7 442	. . 3 (a⊥ ∪ b⊥ ) C b
5	2, 4	fh4c 460	. 2 (a ∪ (b ∩ (a⊥ ∪ b⊥ ))) = ((a ∪ b) ∩ (a ∪ (a⊥ ∪ b⊥ )))
6		df-t 40	. . . . . 6 1 = (a ∪ a⊥ )
7	6	ax-r5 37	. . . . 5 (1 ∪ b⊥ ) = ((a ∪ a⊥ ) ∪ b⊥ )
8		ax-a2 30	. . . . . 6 (1 ∪ b⊥ ) = (b⊥ ∪ 1)
9		or1 96	. . . . . 6 (b⊥ ∪ 1) = 1
10	8, 9	ax-r2 35	. . . . 5 (1 ∪ b⊥ ) = 1
11		ax-a3 31	. . . . 5 ((a ∪ a⊥ ) ∪ b⊥ ) = (a ∪ (a⊥ ∪ b⊥ ))
12	7, 10, 11	3tr2 61	. . . 4 1 = (a ∪ (a⊥ ∪ b⊥ ))
13	12	ax-r1 34	. . 3 (a ∪ (a⊥ ∪ b⊥ )) = 1
14	13	lan 70	. 2 ((a ∪ b) ∩ (a ∪ (a⊥ ∪ b⊥ ))) = ((a ∪ b) ∩ 1)
15		an1 98	. 2 ((a ∪ b) ∩ 1) = (a ∪ b)
16	5, 14, 15	3tr 62	1 (a ∪ (b ∩ (a⊥ ∪ b⊥ ))) = (a ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9

This theorem is referenced by:  sa5 818

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  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  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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