This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem shftval4

Description: Value of a sequence shifted by -u A . (Contributed by NM, 18-Aug-2005) (Revised by Mario Carneiro, 5-Nov-2013)

Ref Expression
Hypothesis shftfval.1 
|- F e. _V
Assertion shftval4 
|- ( ( A e. CC /\ B e. CC ) -> ( ( F shift -u A ) ` B ) = ( F ` ( A + B ) ) )

Proof

Step Hyp Ref Expression
1 shftfval.1 
 |-  F e. _V
2 negcl 
 |-  ( A e. CC -> -u A e. CC )
3 1 shftval 
 |-  ( ( -u A e. CC /\ B e. CC ) -> ( ( F shift -u A ) ` B ) = ( F ` ( B - -u A ) ) )
4 2 3 sylan 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( F shift -u A ) ` B ) = ( F ` ( B - -u A ) ) )
5 subneg 
 |-  ( ( B e. CC /\ A e. CC ) -> ( B - -u A ) = ( B + A ) )
6 5 ancoms 
 |-  ( ( A e. CC /\ B e. CC ) -> ( B - -u A ) = ( B + A ) )
7 addcom 
 |-  ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
8 6 7 eqtr4d 
 |-  ( ( A e. CC /\ B e. CC ) -> ( B - -u A ) = ( A + B ) )
9 8 fveq2d 
 |-  ( ( A e. CC /\ B e. CC ) -> ( F ` ( B - -u A ) ) = ( F ` ( A + B ) ) )
10 4 9 eqtrd 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( F shift -u A ) ` B ) = ( F ` ( A + B ) ) )