Theorem oml2 451

Description: Orthomodular law. Kalmbach 83 p. 22. (Contributed by NM, 27-Aug-1997.)

Hypothesis

Ref	Expression
oml2.1	a ≤ b

Assertion

Ref	Expression
oml2	(a ∪ (a⊥ ∩ b)) = b

Proof of Theorem oml2

Step	Hyp	Ref	Expression
1		oml 445	. 2 (a ∪ (a⊥ ∩ (a ∪ b))) = (a ∪ b)
2		oml2.1	. . . . 5 a ≤ b
3	2	df-le2 131	. . . 4 (a ∪ b) = b
4	3	lan 77	. . 3 (a⊥ ∩ (a ∪ b)) = (a⊥ ∩ b)
5	4	lor 70	. 2 (a ∪ (a⊥ ∩ (a ∪ b))) = (a ∪ (a⊥ ∩ b))
6	1, 5, 3	3tr2 64	1 (a ∪ (a⊥ ∩ b)) = b

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ 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  df-le2 131

This theorem is referenced by:  oml3  452  comcom  453  com3i  467  lem4  511  lem4.6.6i4j2  1101