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Metamath Proof Explorer

Theorem lnfnmuli

Description: Multiplicative property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 lnfnl.1 
 |-  T e. LinFn
2 ax-hv0cl 
 |-  0h e. ~H
3 1 lnfnli 
 |-  ( ( A e. CC /\ B e. ~H /\ 0h e. ~H ) -> ( T ` ( ( A .h B ) +h 0h ) ) = ( ( A x. ( T ` B ) ) + ( T ` 0h ) ) )
4 2 3 mp3an3 
 |-  ( ( A e. CC /\ B e. ~H ) -> ( T ` ( ( A .h B ) +h 0h ) ) = ( ( A x. ( T ` B ) ) + ( T ` 0h ) ) )
5 hvmulcl 
 |-  ( ( A e. CC /\ B e. ~H ) -> ( A .h B ) e. ~H )
6 ax-hvaddid 
 |-  ( ( A .h B ) e. ~H -> ( ( A .h B ) +h 0h ) = ( A .h B ) )
7 5 6 syl 
 |-  ( ( A e. CC /\ B e. ~H ) -> ( ( A .h B ) +h 0h ) = ( A .h B ) )
8 7 fveq2d 
 |-  ( ( A e. CC /\ B e. ~H ) -> ( T ` ( ( A .h B ) +h 0h ) ) = ( T ` ( A .h B ) ) )
9 1 lnfn0i 
 |-  ( T ` 0h ) = 0
10 9 oveq2i 
 |-  ( ( A x. ( T ` B ) ) + ( T ` 0h ) ) = ( ( A x. ( T ` B ) ) + 0 )
11 1 lnfnfi 
 |-  T : ~H --> CC
12 11 ffvelcdmi 
 |-  ( B e. ~H -> ( T ` B ) e. CC )
13 mulcl 
 |-  ( ( A e. CC /\ ( T ` B ) e. CC ) -> ( A x. ( T ` B ) ) e. CC )
14 12 13 sylan2 
 |-  ( ( A e. CC /\ B e. ~H ) -> ( A x. ( T ` B ) ) e. CC )
15 14 addridd 
 |-  ( ( A e. CC /\ B e. ~H ) -> ( ( A x. ( T ` B ) ) + 0 ) = ( A x. ( T ` B ) ) )
16 10 15 eqtrid 
 |-  ( ( A e. CC /\ B e. ~H ) -> ( ( A x. ( T ` B ) ) + ( T ` 0h ) ) = ( A x. ( T ` B ) ) )
17 4 8 16 3eqtr3d 
 |-  ( ( A e. CC /\ B e. ~H ) -> ( T ` ( A .h B ) ) = ( A x. ( T ` B ) ) )