Theorem oml3 452

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

Hypotheses

Ref	Expression
oml3.1	a ≤ b
oml3.2	(b ∩ a⊥ ) = 0

Assertion

Ref	Expression
oml3	a = b

Proof of Theorem oml3

Step	Hyp	Ref	Expression
1		oml3.2	. . . . 5 (b ∩ a⊥ ) = 0
2	1	ax-r1 35	. . . 4 0 = (b ∩ a⊥ )
3		ancom 74	. . . 4 (b ∩ a⊥ ) = (a⊥ ∩ b)
4	2, 3	ax-r2 36	. . 3 0 = (a⊥ ∩ b)
5	4	lor 70	. 2 (a ∪ 0) = (a ∪ (a⊥ ∩ b))
6		or0 102	. 2 (a ∪ 0) = a
7		oml3.1	. . 3 a ≤ b
8	7	oml2 451	. 2 (a ∪ (a⊥ ∩ b)) = b
9	5, 6, 8	3tr2 64	1 a = b

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9

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:  fh1  469  fh2  470  mh  879