# 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 )`
` |-  ( ( 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 ffvelrni
` |-  ( 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 )`
` |-  ( ( A e. CC /\ B e. ~H ) -> ( ( A x. ( T ` B ) ) + 0 ) = ( A x. ( T ` B ) ) )`
` |-  ( ( A e. CC /\ B e. ~H ) -> ( ( A x. ( T ` B ) ) + ( T ` 0h ) ) = ( A x. ( T ` B ) ) )`
` |-  ( ( A e. CC /\ B e. ~H ) -> ( T ` ( A .h B ) ) = ( A x. ( T ` B ) ) )`