# Metamath Proof Explorer

## Theorem hodsi

Description: Relationship between Hilbert space operator difference and sum. (Contributed by NM, 17-Nov-2000) (New usage is discouraged.)

Ref Expression
Hypotheses hods.1
|- R : ~H --> ~H
hods.2
|- S : ~H --> ~H
hods.3
|- T : ~H --> ~H
Assertion hodsi
|- ( ( R -op S ) = T <-> ( S +op T ) = R )

### Proof

Step Hyp Ref Expression
1 hods.1
|-  R : ~H --> ~H
2 hods.2
|-  S : ~H --> ~H
3 hods.3
|-  T : ~H --> ~H
4 1 ffvelrni
|-  ( x e. ~H -> ( R ` x ) e. ~H )
5 2 ffvelrni
|-  ( x e. ~H -> ( S ` x ) e. ~H )
6 3 ffvelrni
|-  ( x e. ~H -> ( T ` x ) e. ~H )
|-  ( ( ( R ` x ) e. ~H /\ ( S ` x ) e. ~H /\ ( T ` x ) e. ~H ) -> ( ( ( R ` x ) -h ( S ` x ) ) = ( T ` x ) <-> ( ( S ` x ) +h ( T ` x ) ) = ( R ` x ) ) )
8 4 5 6 7 syl3anc
|-  ( x e. ~H -> ( ( ( R ` x ) -h ( S ` x ) ) = ( T ` x ) <-> ( ( S ` x ) +h ( T ` x ) ) = ( R ` x ) ) )
9 hodval
|-  ( ( R : ~H --> ~H /\ S : ~H --> ~H /\ x e. ~H ) -> ( ( R -op S ) ` x ) = ( ( R ` x ) -h ( S ` x ) ) )
10 1 2 9 mp3an12
|-  ( x e. ~H -> ( ( R -op S ) ` x ) = ( ( R ` x ) -h ( S ` x ) ) )
11 10 eqeq1d
|-  ( x e. ~H -> ( ( ( R -op S ) ` x ) = ( T ` x ) <-> ( ( R ` x ) -h ( S ` x ) ) = ( T ` x ) ) )
12 hosval
|-  ( ( S : ~H --> ~H /\ T : ~H --> ~H /\ x e. ~H ) -> ( ( S +op T ) ` x ) = ( ( S ` x ) +h ( T ` x ) ) )
13 2 3 12 mp3an12
|-  ( x e. ~H -> ( ( S +op T ) ` x ) = ( ( S ` x ) +h ( T ` x ) ) )
14 13 eqeq1d
|-  ( x e. ~H -> ( ( ( S +op T ) ` x ) = ( R ` x ) <-> ( ( S ` x ) +h ( T ` x ) ) = ( R ` x ) ) )
15 8 11 14 3bitr4d
|-  ( x e. ~H -> ( ( ( R -op S ) ` x ) = ( T ` x ) <-> ( ( S +op T ) ` x ) = ( R ` x ) ) )
16 15 ralbiia
|-  ( A. x e. ~H ( ( R -op S ) ` x ) = ( T ` x ) <-> A. x e. ~H ( ( S +op T ) ` x ) = ( R ` x ) )
17 1 2 hosubcli
|-  ( R -op S ) : ~H --> ~H
18 17 3 hoeqi
|-  ( A. x e. ~H ( ( R -op S ) ` x ) = ( T ` x ) <-> ( R -op S ) = T )