Theorem omln 446

Description: Orthomodular law. (Contributed by NM, 2-Nov-1997.)

Assertion

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

Proof of Theorem omln

Step	Hyp	Ref	Expression
1		ax-a1 30	. . . 4 a = a⊥ ⊥
2	1	ran 78	. . 3 (a ∩ (a⊥ ∪ b)) = (a⊥ ⊥ ∩ (a⊥ ∪ b))
3	2	lor 70	. 2 (a⊥ ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ (a⊥ ⊥ ∩ (a⊥ ∪ b)))
4		oml 445	. 2 (a⊥ ∪ (a⊥ ⊥ ∩ (a⊥ ∪ b))) = (a⊥ ∪ b)
5	3, 4	ax-r2 36	1 (a⊥ ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ b)

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

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  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

This theorem is referenced by:  omla  447  i3lem4  507  lem4  511  i3abs1  522  u3lemona  627  kb10iii  893  lem4.6.6i3j0  1098  lem4.6.6i3j1  1099