Theorem oml3 434

Description: Orthomodular law. Kalmbach 83 p. 22.

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 34	. . . 4 0 = (b ∩ a⊥ )
3		ancom 68	. . . 4 (b ∩ a⊥ ) = (a⊥ ∩ b)
4	2, 3	ax-r2 35	. . 3 0 = (a⊥ ∩ b)
5	4	lor 66	. 2 (a ∪ 0) = (a ∪ (a⊥ ∩ b))
6		or0 94	. 2 (a ∪ 0) = a
7		oml3.1	. . 3 a ≤ b
8	7	oml2 433	. 2 (a ∪ (a⊥ ∩ b)) = b
9	5, 6, 8	3tr2 61	1 a = b

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 10

This theorem is referenced by:  fh1 451  fh2 452  mh 861

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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