Theorem omln 428

Description: Orthomodular law.

Assertion

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

Proof of Theorem omln

Step	Hyp	Ref	Expression
1		ax-a1 29	. . . 4 a = a⊥ ⊥
2	1	ran 71	. . 3 (a ∩ (a⊥ ∪ b)) = (a⊥ ⊥ ∩ (a⊥ ∪ b))
3	2	lor 66	. 2 (a⊥ ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ (a⊥ ⊥ ∩ (a⊥ ∪ b)))
4		oml 427	. 2 (a⊥ ∪ (a⊥ ⊥ ∩ (a⊥ ∪ b))) = (a⊥ ∪ b)
5	3, 4	ax-r2 35	1 (a⊥ ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ b)

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

This theorem is referenced by:  omla 429  i3lem4 489  lem4 493  i3abs1 504  u3lemona 609  kb10iii 875

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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