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Metamath Proof Explorer

Theorem ho2coi

Description: Double composition of Hilbert space operators. (Contributed by NM, 1-Dec-2000) (New usage is discouraged.)

Ref Expression
Hypotheses hods.1 
|- R : ~H --> ~H
hods.2 
|- S : ~H --> ~H
hods.3 
|- T : ~H --> ~H
Assertion ho2coi 
|- ( A e. ~H -> ( ( ( R o. S ) o. T ) ` A ) = ( R ` ( S ` ( T ` A ) ) ) )

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 2 hocofi 
 |-  ( R o. S ) : ~H --> ~H
5 4 3 hocoi 
 |-  ( A e. ~H -> ( ( ( R o. S ) o. T ) ` A ) = ( ( R o. S ) ` ( T ` A ) ) )
6 3 ffvelcdmi 
 |-  ( A e. ~H -> ( T ` A ) e. ~H )
7 1 2 hocoi 
 |-  ( ( T ` A ) e. ~H -> ( ( R o. S ) ` ( T ` A ) ) = ( R ` ( S ` ( T ` A ) ) ) )
8 6 7 syl 
 |-  ( A e. ~H -> ( ( R o. S ) ` ( T ` A ) ) = ( R ` ( S ` ( T ` A ) ) ) )
9 5 8 eqtrd 
 |-  ( A e. ~H -> ( ( ( R o. S ) o. T ) ` A ) = ( R ` ( S ` ( T ` A ) ) ) )