Theorem oml2 433

Description: Orthomodular law. Kalmbach 83 p. 22.

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 427	. 2 (a ∪ (a⊥ ∩ (a ∪ b))) = (a ∪ b)
2		oml2.1	. . . . 5 a ≤ b
3	2	df-le2 123	. . . 4 (a ∪ b) = b
4	3	lan 70	. . 3 (a⊥ ∩ (a ∪ b)) = (a⊥ ∩ b)
5	4	lor 66	. 2 (a ∪ (a⊥ ∩ (a ∪ b))) = (a ∪ (a⊥ ∩ b))
6	1, 5, 3	3tr2 61	1 (a ∪ (a⊥ ∩ b)) = b

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7

This theorem is referenced by:  oml3 434  comcom 435  com3i 449  lem4 493

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  df-le2 123

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