Metamath Proof Explorer

Theorem nmoprepnf

Description: The norm of a Hilbert space operator is either real or plus infinity. (Contributed by NM, 5-Feb-2006) (New usage is discouraged.)

Ref Expression
Assertion nmoprepnf 
|- ( T : ~H --> ~H -> ( ( normop ` T ) e. RR <-> ( normop ` T ) =/= +oo ) )

Proof

Step Hyp Ref Expression
1 nmopsetretHIL 
 |-  ( T : ~H --> ~H -> { x | E. y e. ~H ( ( normh ` y ) <_ 1 /\ x = ( normh ` ( T ` y ) ) ) } C_ RR )
2 nmopsetn0 
 |-  ( normh ` ( T ` 0h ) ) e. { x | E. y e. ~H ( ( normh ` y ) <_ 1 /\ x = ( normh ` ( T ` y ) ) ) }
3 2 ne0ii 
 |-  { x | E. y e. ~H ( ( normh ` y ) <_ 1 /\ x = ( normh ` ( T ` y ) ) ) } =/= (/)
4 supxrre2 
 |-  ( ( { x | E. y e. ~H ( ( normh ` y ) <_ 1 /\ x = ( normh ` ( T ` y ) ) ) } C_ RR /\ { x | E. y e. ~H ( ( normh ` y ) <_ 1 /\ x = ( normh ` ( T ` y ) ) ) } =/= (/) ) -> ( sup ( { x | E. y e. ~H ( ( normh ` y ) <_ 1 /\ x = ( normh ` ( T ` y ) ) ) } , RR* , < ) e. RR <-> sup ( { x | E. y e. ~H ( ( normh ` y ) <_ 1 /\ x = ( normh ` ( T ` y ) ) ) } , RR* , < ) =/= +oo ) )
5 1 3 4 sylancl 
 |-  ( T : ~H --> ~H -> ( sup ( { x | E. y e. ~H ( ( normh ` y ) <_ 1 /\ x = ( normh ` ( T ` y ) ) ) } , RR* , < ) e. RR <-> sup ( { x | E. y e. ~H ( ( normh ` y ) <_ 1 /\ x = ( normh ` ( T ` y ) ) ) } , RR* , < ) =/= +oo ) )
6 nmopval 
 |-  ( T : ~H --> ~H -> ( normop ` T ) = sup ( { x | E. y e. ~H ( ( normh ` y ) <_ 1 /\ x = ( normh ` ( T ` y ) ) ) } , RR* , < ) )
7 6 eleq1d 
 |-  ( T : ~H --> ~H -> ( ( normop ` T ) e. RR <-> sup ( { x | E. y e. ~H ( ( normh ` y ) <_ 1 /\ x = ( normh ` ( T ` y ) ) ) } , RR* , < ) e. RR ) )
8 6 neeq1d 
 |-  ( T : ~H --> ~H -> ( ( normop ` T ) =/= +oo <-> sup ( { x | E. y e. ~H ( ( normh ` y ) <_ 1 /\ x = ( normh ` ( T ` y ) ) ) } , RR* , < ) =/= +oo ) )
9 5 7 8 3bitr4d 
 |-  ( T : ~H --> ~H -> ( ( normop ` T ) e. RR <-> ( normop ` T ) =/= +oo ) )