Theorem oml 445

Description: Orthomodular law. Compare Thm. 1 of Pavicic 1987. (Contributed by NM, 12-Aug-1997.)

Assertion

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

Proof of Theorem oml

Step	Hyp	Ref	Expression
1		omlem1 127	. 2 ((a ∪ (a⊥ ∩ (a ∪ b))) ∪ (a ∪ b)) = (a ∪ b)
2		omlem2 128	. 2 ((a ∪ b)⊥ ∪ (a ∪ (a⊥ ∩ (a ∪ b)))) = 1
3	1, 2	lem3.1 443	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:  omln  446  oml5  449  oml2  451  ud1lem2  561  ud2lem2  564  ud3lem2  571  ud4lem2  582  ud5lem3  594  test  802  2oalem1  825  oas  925  oat  927  lem4.6.6i2j4  1097