Theorem omlan 430

Description: Orthomodular law.

Assertion

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

Proof of Theorem omlan

Step	Hyp	Ref	Expression
1		ax-a1 29	. . . 4 a = a⊥ ⊥
2	1	ax-r5 37	. . 3 (a ∪ (a⊥ ∩ b)) = (a⊥ ⊥ ∪ (a⊥ ∩ b))
3	2	lan 70	. 2 (a⊥ ∩ (a ∪ (a⊥ ∩ b))) = (a⊥ ∩ (a⊥ ⊥ ∪ (a⊥ ∩ b)))
4		omla 429	. 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:  i3lem1 486  i3lem3 488  u1lem8 758  u3lem10 767  3vth1 786  1oaii 806  mlaconjolem 867  oatr 908  oalii 982  oaliv 983

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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