Metamath Proof Explorer


Theorem pjige0i

Description: The inner product of a projection and its argument is nonnegative. (Contributed by NM, 2-Jun-2006) (New usage is discouraged.)

Ref Expression
Hypothesis pjadjt.1
|- H e. CH
Assertion pjige0i
|- ( A e. ~H -> 0 <_ ( ( ( projh ` H ) ` A ) .ih A ) )

Proof

Step Hyp Ref Expression
1 pjadjt.1
 |-  H e. CH
2 1 pjhcli
 |-  ( A e. ~H -> ( ( projh ` H ) ` A ) e. ~H )
3 normcl
 |-  ( ( ( projh ` H ) ` A ) e. ~H -> ( normh ` ( ( projh ` H ) ` A ) ) e. RR )
4 2 3 syl
 |-  ( A e. ~H -> ( normh ` ( ( projh ` H ) ` A ) ) e. RR )
5 4 sqge0d
 |-  ( A e. ~H -> 0 <_ ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) )
6 1 pjinormi
 |-  ( A e. ~H -> ( ( ( projh ` H ) ` A ) .ih A ) = ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) )
7 5 6 breqtrrd
 |-  ( A e. ~H -> 0 <_ ( ( ( projh ` H ) ` A ) .ih A ) )