Metamath Proof Explorer


Theorem nmopgtmnf

Description: The norm of a Hilbert space operator is not minus infinity. (Contributed by NM, 2-Feb-2006) (New usage is discouraged.)

Ref Expression
Assertion nmopgtmnf
|- ( T : ~H --> ~H -> -oo < ( normop ` T ) )

Proof

Step Hyp Ref Expression
1 nmoprepnf
 |-  ( T : ~H --> ~H -> ( ( normop ` T ) e. RR <-> ( normop ` T ) =/= +oo ) )
2 df-ne
 |-  ( ( normop ` T ) =/= +oo <-> -. ( normop ` T ) = +oo )
3 1 2 syl6bb
 |-  ( T : ~H --> ~H -> ( ( normop ` T ) e. RR <-> -. ( normop ` T ) = +oo ) )
4 xor3
 |-  ( -. ( ( normop ` T ) e. RR <-> ( normop ` T ) = +oo ) <-> ( ( normop ` T ) e. RR <-> -. ( normop ` T ) = +oo ) )
5 nbior
 |-  ( -. ( ( normop ` T ) e. RR <-> ( normop ` T ) = +oo ) -> ( ( normop ` T ) e. RR \/ ( normop ` T ) = +oo ) )
6 4 5 sylbir
 |-  ( ( ( normop ` T ) e. RR <-> -. ( normop ` T ) = +oo ) -> ( ( normop ` T ) e. RR \/ ( normop ` T ) = +oo ) )
7 mnfltxr
 |-  ( ( ( normop ` T ) e. RR \/ ( normop ` T ) = +oo ) -> -oo < ( normop ` T ) )
8 3 6 7 3syl
 |-  ( T : ~H --> ~H -> -oo < ( normop ` T ) )