Metamath Proof Explorer

Theorem ho2times

Description: Two times a Hilbert space operator. (Contributed by NM, 26-Aug-2006) (New usage is discouraged.)

Ref Expression
Assertion ho2times 
|- ( T : ~H --> ~H -> ( 2 .op T ) = ( T +op T ) )

Proof

Step Hyp Ref Expression
1 df-2 
 |-  2 = ( 1 + 1 )
2 1 oveq1i 
 |-  ( 2 .op T ) = ( ( 1 + 1 ) .op T )
3 ax-1cn 
 |-  1 e. CC
4 hoadddir 
 |-  ( ( 1 e. CC /\ 1 e. CC /\ T : ~H --> ~H ) -> ( ( 1 + 1 ) .op T ) = ( ( 1 .op T ) +op ( 1 .op T ) ) )
5 3 3 4 mp3an12 
 |-  ( T : ~H --> ~H -> ( ( 1 + 1 ) .op T ) = ( ( 1 .op T ) +op ( 1 .op T ) ) )
6 2 5 syl5eq 
 |-  ( T : ~H --> ~H -> ( 2 .op T ) = ( ( 1 .op T ) +op ( 1 .op T ) ) )
7 hoadddi 
 |-  ( ( 1 e. CC /\ T : ~H --> ~H /\ T : ~H --> ~H ) -> ( 1 .op ( T +op T ) ) = ( ( 1 .op T ) +op ( 1 .op T ) ) )
8 3 7 mp3an1 
 |-  ( ( T : ~H --> ~H /\ T : ~H --> ~H ) -> ( 1 .op ( T +op T ) ) = ( ( 1 .op T ) +op ( 1 .op T ) ) )
9 8 anidms 
 |-  ( T : ~H --> ~H -> ( 1 .op ( T +op T ) ) = ( ( 1 .op T ) +op ( 1 .op T ) ) )
10 hoaddcl 
 |-  ( ( T : ~H --> ~H /\ T : ~H --> ~H ) -> ( T +op T ) : ~H --> ~H )
11 10 anidms 
 |-  ( T : ~H --> ~H -> ( T +op T ) : ~H --> ~H )
12 homulid2 
 |-  ( ( T +op T ) : ~H --> ~H -> ( 1 .op ( T +op T ) ) = ( T +op T ) )
13 11 12 syl 
 |-  ( T : ~H --> ~H -> ( 1 .op ( T +op T ) ) = ( T +op T ) )
14 6 9 13 3eqtr2d 
 |-  ( T : ~H --> ~H -> ( 2 .op T ) = ( T +op T ) )