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Metamath Proof Explorer

Theorem nmbdoplb

Description: A lower bound for the norm of a bounded linear Hilbert space operator. (Contributed by NM, 18-Feb-2006) (New usage is discouraged.)

Ref Expression
Assertion nmbdoplb 
|- ( ( T e. BndLinOp /\ A e. ~H ) -> ( normh ` ( T ` A ) ) <_ ( ( normop ` T ) x. ( normh ` A ) ) )

Proof

Step Hyp Ref Expression
1 fveq1 
 |-  ( T = if ( T e. BndLinOp , T , 0hop ) -> ( T ` A ) = ( if ( T e. BndLinOp , T , 0hop ) ` A ) )
2 1 fveq2d 
 |-  ( T = if ( T e. BndLinOp , T , 0hop ) -> ( normh ` ( T ` A ) ) = ( normh ` ( if ( T e. BndLinOp , T , 0hop ) ` A ) ) )
3 fveq2 
 |-  ( T = if ( T e. BndLinOp , T , 0hop ) -> ( normop ` T ) = ( normop ` if ( T e. BndLinOp , T , 0hop ) ) )
4 3 oveq1d 
 |-  ( T = if ( T e. BndLinOp , T , 0hop ) -> ( ( normop ` T ) x. ( normh ` A ) ) = ( ( normop ` if ( T e. BndLinOp , T , 0hop ) ) x. ( normh ` A ) ) )
5 2 4 breq12d 
 |-  ( T = if ( T e. BndLinOp , T , 0hop ) -> ( ( normh ` ( T ` A ) ) <_ ( ( normop ` T ) x. ( normh ` A ) ) <-> ( normh ` ( if ( T e. BndLinOp , T , 0hop ) ` A ) ) <_ ( ( normop ` if ( T e. BndLinOp , T , 0hop ) ) x. ( normh ` A ) ) ) )
6 5 imbi2d 
 |-  ( T = if ( T e. BndLinOp , T , 0hop ) -> ( ( A e. ~H -> ( normh ` ( T ` A ) ) <_ ( ( normop ` T ) x. ( normh ` A ) ) ) <-> ( A e. ~H -> ( normh ` ( if ( T e. BndLinOp , T , 0hop ) ` A ) ) <_ ( ( normop ` if ( T e. BndLinOp , T , 0hop ) ) x. ( normh ` A ) ) ) ) )
7 0bdop 
 |-  0hop e. BndLinOp
8 7 elimel 
 |-  if ( T e. BndLinOp , T , 0hop ) e. BndLinOp
9 8 nmbdoplbi 
 |-  ( A e. ~H -> ( normh ` ( if ( T e. BndLinOp , T , 0hop ) ` A ) ) <_ ( ( normop ` if ( T e. BndLinOp , T , 0hop ) ) x. ( normh ` A ) ) )
10 6 9 dedth 
 |-  ( T e. BndLinOp -> ( A e. ~H -> ( normh ` ( T ` A ) ) <_ ( ( normop ` T ) x. ( normh ` A ) ) ) )
11 10 imp 
 |-  ( ( T e. BndLinOp /\ A e. ~H ) -> ( normh ` ( T ` A ) ) <_ ( ( normop ` T ) x. ( normh ` A ) ) )