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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 ffvelcdmi 
 |-  ( x e. ~H -> ( R ` x ) e. ~H )
5 2 ffvelcdmi 
 |-  ( x e. ~H -> ( S ` x ) e. ~H )
6 3 ffvelcdmi 
 |-  ( x e. ~H -> ( T ` x ) e. ~H )
7 hvsubadd 
 |-  ( ( ( 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 )
19 2 3 hoaddcli 
 |-  ( S +op T ) : ~H --> ~H
20 19 1 hoeqi 
 |-  ( A. x e. ~H ( ( S +op T ) ` x ) = ( R ` x ) <-> ( S +op T ) = R )
21 16 18 20 3bitr3i 
 |-  ( ( R -op S ) = T <-> ( S +op T ) = R )