Theorem ortha 420

Description: Property of orthogonality

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 149	. . . 4 b ≤ a⊥
3	2	lelan 159	. . 3 (a ∩ b) ≤ (a ∩ a⊥ )
4		dff 93	. . . 4 0 = (a ∩ a⊥ )
5	4	ax-r1 34	. . 3 (a ∩ a⊥ ) = 0
6	3, 5	lbtr 131	. 2 (a ∩ b) ≤ 0
7		le0 139	. 2 0 ≤ (a ∩ b)
8	6, 7	lebi 137	1 (a ∩ b) = 0

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

This theorem is referenced by:  mhlemlem1 856  mhlem 858

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

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123

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