Metamath Proof Explorer

Theorem lnfnl

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

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

Proof

Step Hyp Ref Expression
1 ellnfn 
 |-  ( T e. LinFn <-> ( T : ~H --> CC /\ A. x e. CC A. y e. ~H A. z e. ~H ( T ` ( ( x .h y ) +h z ) ) = ( ( x x. ( T ` y ) ) + ( T ` z ) ) ) )
2 1 simprbi 
 |-  ( T e. LinFn -> A. x e. CC A. y e. ~H A. z e. ~H ( T ` ( ( x .h y ) +h z ) ) = ( ( x x. ( T ` y ) ) + ( T ` z ) ) )
3 oveq1 
 |-  ( x = A -> ( x .h y ) = ( A .h y ) )
4 3 fvoveq1d 
 |-  ( x = A -> ( T ` ( ( x .h y ) +h z ) ) = ( T ` ( ( A .h y ) +h z ) ) )
5 oveq1 
 |-  ( x = A -> ( x x. ( T ` y ) ) = ( A x. ( T ` y ) ) )
6 5 oveq1d 
 |-  ( x = A -> ( ( x x. ( T ` y ) ) + ( T ` z ) ) = ( ( A x. ( T ` y ) ) + ( T ` z ) ) )
7 4 6 eqeq12d 
 |-  ( x = A -> ( ( T ` ( ( x .h y ) +h z ) ) = ( ( x x. ( T ` y ) ) + ( T ` z ) ) <-> ( T ` ( ( A .h y ) +h z ) ) = ( ( A x. ( T ` y ) ) + ( T ` z ) ) ) )
8 oveq2 
 |-  ( y = B -> ( A .h y ) = ( A .h B ) )
9 8 fvoveq1d 
 |-  ( y = B -> ( T ` ( ( A .h y ) +h z ) ) = ( T ` ( ( A .h B ) +h z ) ) )
10 fveq2 
 |-  ( y = B -> ( T ` y ) = ( T ` B ) )
11 10 oveq2d 
 |-  ( y = B -> ( A x. ( T ` y ) ) = ( A x. ( T ` B ) ) )
12 11 oveq1d 
 |-  ( y = B -> ( ( A x. ( T ` y ) ) + ( T ` z ) ) = ( ( A x. ( T ` B ) ) + ( T ` z ) ) )
13 9 12 eqeq12d 
 |-  ( y = B -> ( ( T ` ( ( A .h y ) +h z ) ) = ( ( A x. ( T ` y ) ) + ( T ` z ) ) <-> ( T ` ( ( A .h B ) +h z ) ) = ( ( A x. ( T ` B ) ) + ( T ` z ) ) ) )
14 oveq2 
 |-  ( z = C -> ( ( A .h B ) +h z ) = ( ( A .h B ) +h C ) )
15 14 fveq2d 
 |-  ( z = C -> ( T ` ( ( A .h B ) +h z ) ) = ( T ` ( ( A .h B ) +h C ) ) )
16 fveq2 
 |-  ( z = C -> ( T ` z ) = ( T ` C ) )
17 16 oveq2d 
 |-  ( z = C -> ( ( A x. ( T ` B ) ) + ( T ` z ) ) = ( ( A x. ( T ` B ) ) + ( T ` C ) ) )
18 15 17 eqeq12d 
 |-  ( z = C -> ( ( T ` ( ( A .h B ) +h z ) ) = ( ( A x. ( T ` B ) ) + ( T ` z ) ) <-> ( T ` ( ( A .h B ) +h C ) ) = ( ( A x. ( T ` B ) ) + ( T ` C ) ) ) )
19 7 13 18 rspc3v 
 |-  ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( A. x e. CC A. y e. ~H A. z e. ~H ( T ` ( ( x .h y ) +h z ) ) = ( ( x x. ( T ` y ) ) + ( T ` z ) ) -> ( T ` ( ( A .h B ) +h C ) ) = ( ( A x. ( T ` B ) ) + ( T ` C ) ) ) )
20 2 19 syl5 
 |-  ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( T e. LinFn -> ( T ` ( ( A .h B ) +h C ) ) = ( ( A x. ( T ` B ) ) + ( T ` C ) ) ) )
21 20 3expb 
 |-  ( ( A e. CC /\ ( B e. ~H /\ C e. ~H ) ) -> ( T e. LinFn -> ( T ` ( ( A .h B ) +h C ) ) = ( ( A x. ( T ` B ) ) + ( T ` C ) ) ) )
22 21 impcom 
 |-  ( ( T e. LinFn /\ ( A e. CC /\ ( B e. ~H /\ C e. ~H ) ) ) -> ( T ` ( ( A .h B ) +h C ) ) = ( ( A x. ( T ` B ) ) + ( T ` C ) ) )
23 22 anassrs 
 |-  ( ( ( T e. LinFn /\ A e. CC ) /\ ( B e. ~H /\ C e. ~H ) ) -> ( T ` ( ( A .h B ) +h C ) ) = ( ( A x. ( T ` B ) ) + ( T ` C ) ) )