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Metamath Proof Explorer

Theorem hosubcli

Description: Mapping of difference of Hilbert space operators. (Contributed by NM, 14-Nov-2000) (Revised by Mario Carneiro, 16-Nov-2013) (New usage is discouraged.)

Ref Expression
Hypotheses hoeq.1 
|- S : ~H --> ~H
hoeq.2 
|- T : ~H --> ~H
Assertion hosubcli 
|- ( S -op T ) : ~H --> ~H

Proof

Step Hyp Ref Expression
1 hoeq.1 
 |-  S : ~H --> ~H
2 hoeq.2 
 |-  T : ~H --> ~H
3 hodmval 
 |-  ( ( S : ~H --> ~H /\ T : ~H --> ~H ) -> ( S -op T ) = ( x e. ~H |-> ( ( S ` x ) -h ( T ` x ) ) ) )
4 1 2 3 mp2an 
 |-  ( S -op T ) = ( x e. ~H |-> ( ( S ` x ) -h ( T ` x ) ) )
5 1 ffvelcdmi 
 |-  ( x e. ~H -> ( S ` x ) e. ~H )
6 2 ffvelcdmi 
 |-  ( x e. ~H -> ( T ` x ) e. ~H )
7 hvsubcl 
 |-  ( ( ( S ` x ) e. ~H /\ ( T ` x ) e. ~H ) -> ( ( S ` x ) -h ( T ` x ) ) e. ~H )
8 5 6 7 syl2anc 
 |-  ( x e. ~H -> ( ( S ` x ) -h ( T ` x ) ) e. ~H )
9 4 8 fmpti 
 |-  ( S -op T ) : ~H --> ~H