Theorem ortha 438

Description: Property of orthogonality. (Contributed by NM, 10-Mar-2002.)

Hypothesis

Ref	Expression
ortha.1	a ≤ b⊥

Assertion

Ref	Expression
ortha	(a ∩ b) = 0

Proof of Theorem ortha

Step	Hyp	Ref	Expression
1		ortha.1	. . . . 5 a ≤ b⊥
2	1	lecon3 157	. . . 4 b ≤ a⊥
3	2	lelan 167	. . 3 (a ∩ b) ≤ (a ∩ a⊥ )
4		dff 101	. . . 4 0 = (a ∩ a⊥ )
5	4	ax-r1 35	. . 3 (a ∩ a⊥ ) = 0
6	3, 5	lbtr 139	. 2 (a ∩ b) ≤ 0
7		le0 147	. 2 0 ≤ (a ∩ b)
8	6, 7	lebi 145	1 (a ∩ b) = 0

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∩ 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

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131

This theorem is referenced by:  mhlemlem1  874  mhlem  876  e2astlem1  895  lem3.3.7i4e1  1069  lem3.3.7i5e1  1072