Theorem omlan 448

Description: Orthomodular law. (Contributed by NM, 7-Nov-1997.)

Assertion

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

Proof of Theorem omlan

Step	Hyp	Ref	Expression
1		ax-a1 30	. . . 4 a = a⊥ ⊥
2	1	ax-r5 38	. . 3 (a ∪ (a⊥ ∩ b)) = (a⊥ ⊥ ∪ (a⊥ ∩ b))
3	2	lan 77	. 2 (a⊥ ∩ (a ∪ (a⊥ ∩ b))) = (a⊥ ∩ (a⊥ ⊥ ∪ (a⊥ ∩ b)))
4		omla 447	. 2 (a⊥ ∩ (a⊥ ⊥ ∪ (a⊥ ∩ b))) = (a⊥ ∩ b)
5	3, 4	ax-r2 36	1 (a⊥ ∩ (a ∪ (a⊥ ∩ b))) = (a⊥ ∩ b)

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7

This proof depends on 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 proof depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42

This theorem is used by:  i3lem1  504  i3lem3  506  u1lem8  776  u3lem10  785  3vth1  804  1oaii  824  mlaconjolem  885  oatr  928  oalii  1002  oaliv  1003