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Metamath Proof Explorer

Theorem lnfnaddmuli

Description: Sum/product property of a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006) (New usage is discouraged.)

Ref Expression
Hypothesis lnfnl.1 
|- T e. LinFn
Assertion lnfnaddmuli 
|- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( T ` ( B +h ( A .h C ) ) ) = ( ( T ` B ) + ( A x. ( T ` C ) ) ) )

Proof

Step Hyp Ref Expression
1 lnfnl.1 
 |-  T e. LinFn
2 hvmulcl 
 |-  ( ( A e. CC /\ C e. ~H ) -> ( A .h C ) e. ~H )
3 1 lnfnaddi 
 |-  ( ( B e. ~H /\ ( A .h C ) e. ~H ) -> ( T ` ( B +h ( A .h C ) ) ) = ( ( T ` B ) + ( T ` ( A .h C ) ) ) )
4 2 3 sylan2 
 |-  ( ( B e. ~H /\ ( A e. CC /\ C e. ~H ) ) -> ( T ` ( B +h ( A .h C ) ) ) = ( ( T ` B ) + ( T ` ( A .h C ) ) ) )
5 4 3impb 
 |-  ( ( B e. ~H /\ A e. CC /\ C e. ~H ) -> ( T ` ( B +h ( A .h C ) ) ) = ( ( T ` B ) + ( T ` ( A .h C ) ) ) )
6 5 3com12 
 |-  ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( T ` ( B +h ( A .h C ) ) ) = ( ( T ` B ) + ( T ` ( A .h C ) ) ) )
7 1 lnfnmuli 
 |-  ( ( A e. CC /\ C e. ~H ) -> ( T ` ( A .h C ) ) = ( A x. ( T ` C ) ) )
8 7 3adant2 
 |-  ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( T ` ( A .h C ) ) = ( A x. ( T ` C ) ) )
9 8 oveq2d 
 |-  ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( ( T ` B ) + ( T ` ( A .h C ) ) ) = ( ( T ` B ) + ( A x. ( T ` C ) ) ) )
10 6 9 eqtrd 
 |-  ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( T ` ( B +h ( A .h C ) ) ) = ( ( T ` B ) + ( A x. ( T ` C ) ) ) )