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Metamath Proof Explorer

Theorem nmopgt0

Description: A linear Hilbert space operator that is not identically zero has a positive norm. (Contributed by NM, 9-Feb-2006) (New usage is discouraged.)

Ref Expression
Assertion nmopgt0 
|- ( T : ~H --> ~H -> ( ( normop ` T ) =/= 0 <-> 0 < ( normop ` T ) ) )

Proof

Step Hyp Ref Expression
1 nmopxr 
 |-  ( T : ~H --> ~H -> ( normop ` T ) e. RR* )
2 nmopge0 
 |-  ( T : ~H --> ~H -> 0 <_ ( normop ` T ) )
3 0xr 
 |-  0 e. RR*
4 xrleltne 
 |-  ( ( 0 e. RR* /\ ( normop ` T ) e. RR* /\ 0 <_ ( normop ` T ) ) -> ( 0 < ( normop ` T ) <-> ( normop ` T ) =/= 0 ) )
5 3 4 mp3an1 
 |-  ( ( ( normop ` T ) e. RR* /\ 0 <_ ( normop ` T ) ) -> ( 0 < ( normop ` T ) <-> ( normop ` T ) =/= 0 ) )
6 1 2 5 syl2anc 
 |-  ( T : ~H --> ~H -> ( 0 < ( normop ` T ) <-> ( normop ` T ) =/= 0 ) )
7 6 bicomd 
 |-  ( T : ~H --> ~H -> ( ( normop ` T ) =/= 0 <-> 0 < ( normop ` T ) ) )