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Metamath Proof Explorer

Theorem hoaddcomi

Description: Commutativity of sum of Hilbert space operators. (Contributed by NM, 15-Nov-2000) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 hoeq.1 
 |-  S : ~H --> ~H
2 hoeq.2 
 |-  T : ~H --> ~H
3 1 ffvelcdmi 
 |-  ( x e. ~H -> ( S ` x ) e. ~H )
4 2 ffvelcdmi 
 |-  ( x e. ~H -> ( T ` x ) e. ~H )
5 ax-hvcom 
 |-  ( ( ( S ` x ) e. ~H /\ ( T ` x ) e. ~H ) -> ( ( S ` x ) +h ( T ` x ) ) = ( ( T ` x ) +h ( S ` x ) ) )
6 3 4 5 syl2anc 
 |-  ( x e. ~H -> ( ( S ` x ) +h ( T ` x ) ) = ( ( T ` x ) +h ( S ` x ) ) )
7 hosval 
 |-  ( ( S : ~H --> ~H /\ T : ~H --> ~H /\ x e. ~H ) -> ( ( S +op T ) ` x ) = ( ( S ` x ) +h ( T ` x ) ) )
8 1 2 7 mp3an12 
 |-  ( x e. ~H -> ( ( S +op T ) ` x ) = ( ( S ` x ) +h ( T ` x ) ) )
9 hosval 
 |-  ( ( T : ~H --> ~H /\ S : ~H --> ~H /\ x e. ~H ) -> ( ( T +op S ) ` x ) = ( ( T ` x ) +h ( S ` x ) ) )
10 2 1 9 mp3an12 
 |-  ( x e. ~H -> ( ( T +op S ) ` x ) = ( ( T ` x ) +h ( S ` x ) ) )
11 6 8 10 3eqtr4d 
 |-  ( x e. ~H -> ( ( S +op T ) ` x ) = ( ( T +op S ) ` x ) )
12 11 rgen 
 |-  A. x e. ~H ( ( S +op T ) ` x ) = ( ( T +op S ) ` x )
13 1 2 hoaddcli 
 |-  ( S +op T ) : ~H --> ~H
14 2 1 hoaddcli 
 |-  ( T +op S ) : ~H --> ~H
15 13 14 hoeqi 
 |-  ( A. x e. ~H ( ( S +op T ) ` x ) = ( ( T +op S ) ` x ) <-> ( S +op T ) = ( T +op S ) )
16 12 15 mpbi 
 |-  ( S +op T ) = ( T +op S )