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Metamath Proof Explorer

Theorem hoaddassi

Description: Associativity of sum of Hilbert space operators. (Contributed by NM, 26-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 hoaddassi 
|- ( ( R +op S ) +op T ) = ( R +op ( S +op T ) )

Proof

Step Hyp Ref Expression
1 hods.1 
 |-  R : ~H --> ~H
2 hods.2 
 |-  S : ~H --> ~H
3 hods.3 
 |-  T : ~H --> ~H
4 hosval 
 |-  ( ( R : ~H --> ~H /\ S : ~H --> ~H /\ x e. ~H ) -> ( ( R +op S ) ` x ) = ( ( R ` x ) +h ( S ` x ) ) )
5 1 2 4 mp3an12 
 |-  ( x e. ~H -> ( ( R +op S ) ` x ) = ( ( R ` x ) +h ( S ` x ) ) )
6 5 oveq1d 
 |-  ( x e. ~H -> ( ( ( R +op S ) ` x ) +h ( T ` x ) ) = ( ( ( R ` x ) +h ( S ` x ) ) +h ( T ` x ) ) )
7 1 2 hoaddcli 
 |-  ( R +op S ) : ~H --> ~H
8 hosval 
 |-  ( ( ( R +op S ) : ~H --> ~H /\ T : ~H --> ~H /\ x e. ~H ) -> ( ( ( R +op S ) +op T ) ` x ) = ( ( ( R +op S ) ` x ) +h ( T ` x ) ) )
9 7 3 8 mp3an12 
 |-  ( x e. ~H -> ( ( ( R +op S ) +op T ) ` x ) = ( ( ( R +op S ) ` x ) +h ( T ` x ) ) )
10 hosval 
 |-  ( ( S : ~H --> ~H /\ T : ~H --> ~H /\ x e. ~H ) -> ( ( S +op T ) ` x ) = ( ( S ` x ) +h ( T ` x ) ) )
11 2 3 10 mp3an12 
 |-  ( x e. ~H -> ( ( S +op T ) ` x ) = ( ( S ` x ) +h ( T ` x ) ) )
12 11 oveq2d 
 |-  ( x e. ~H -> ( ( R ` x ) +h ( ( S +op T ) ` x ) ) = ( ( R ` x ) +h ( ( S ` x ) +h ( T ` x ) ) ) )
13 2 3 hoaddcli 
 |-  ( S +op T ) : ~H --> ~H
14 hosval 
 |-  ( ( R : ~H --> ~H /\ ( S +op T ) : ~H --> ~H /\ x e. ~H ) -> ( ( R +op ( S +op T ) ) ` x ) = ( ( R ` x ) +h ( ( S +op T ) ` x ) ) )
15 1 13 14 mp3an12 
 |-  ( x e. ~H -> ( ( R +op ( S +op T ) ) ` x ) = ( ( R ` x ) +h ( ( S +op T ) ` x ) ) )
16 1 ffvelcdmi 
 |-  ( x e. ~H -> ( R ` x ) e. ~H )
17 2 ffvelcdmi 
 |-  ( x e. ~H -> ( S ` x ) e. ~H )
18 3 ffvelcdmi 
 |-  ( x e. ~H -> ( T ` x ) e. ~H )
19 ax-hvass 
 |-  ( ( ( R ` x ) e. ~H /\ ( S ` x ) e. ~H /\ ( T ` x ) e. ~H ) -> ( ( ( R ` x ) +h ( S ` x ) ) +h ( T ` x ) ) = ( ( R ` x ) +h ( ( S ` x ) +h ( T ` x ) ) ) )
20 16 17 18 19 syl3anc 
 |-  ( x e. ~H -> ( ( ( R ` x ) +h ( S ` x ) ) +h ( T ` x ) ) = ( ( R ` x ) +h ( ( S ` x ) +h ( T ` x ) ) ) )
21 12 15 20 3eqtr4d 
 |-  ( x e. ~H -> ( ( R +op ( S +op T ) ) ` x ) = ( ( ( R ` x ) +h ( S ` x ) ) +h ( T ` x ) ) )
22 6 9 21 3eqtr4d 
 |-  ( x e. ~H -> ( ( ( R +op S ) +op T ) ` x ) = ( ( R +op ( S +op T ) ) ` x ) )
23 22 rgen 
 |-  A. x e. ~H ( ( ( R +op S ) +op T ) ` x ) = ( ( R +op ( S +op T ) ) ` x )
24 7 3 hoaddcli 
 |-  ( ( R +op S ) +op T ) : ~H --> ~H
25 1 13 hoaddcli 
 |-  ( R +op ( S +op T ) ) : ~H --> ~H
26 24 25 hoeqi 
 |-  ( A. x e. ~H ( ( ( R +op S ) +op T ) ` x ) = ( ( R +op ( S +op T ) ) ` x ) <-> ( ( R +op S ) +op T ) = ( R +op ( S +op T ) ) )
27 23 26 mpbi 
 |-  ( ( R +op S ) +op T ) = ( R +op ( S +op T ) )