Metamath Proof Explorer

Theorem shftval5

Description: Value of a shifted sequence. (Contributed by NM, 19-Aug-2005) (Revised by Mario Carneiro, 5-Nov-2013)

Ref Expression
Hypothesis shftfval.1 
|- F e. _V
Assertion shftval5 
|- ( ( A e. CC /\ B e. CC ) -> ( ( F shift A ) ` ( B + A ) ) = ( F ` B ) )

Proof

Step Hyp Ref Expression
1 shftfval.1 
 |-  F e. _V
2 simpr 
 |-  ( ( B e. CC /\ A e. CC ) -> A e. CC )
3 addcl 
 |-  ( ( B e. CC /\ A e. CC ) -> ( B + A ) e. CC )
4 1 shftval 
 |-  ( ( A e. CC /\ ( B + A ) e. CC ) -> ( ( F shift A ) ` ( B + A ) ) = ( F ` ( ( B + A ) - A ) ) )
5 2 3 4 syl2anc 
 |-  ( ( B e. CC /\ A e. CC ) -> ( ( F shift A ) ` ( B + A ) ) = ( F ` ( ( B + A ) - A ) ) )
6 pncan 
 |-  ( ( B e. CC /\ A e. CC ) -> ( ( B + A ) - A ) = B )
7 6 fveq2d 
 |-  ( ( B e. CC /\ A e. CC ) -> ( F ` ( ( B + A ) - A ) ) = ( F ` B ) )
8 5 7 eqtrd 
 |-  ( ( B e. CC /\ A e. CC ) -> ( ( F shift A ) ` ( B + A ) ) = ( F ` B ) )
9 8 ancoms 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( F shift A ) ` ( B + A ) ) = ( F ` B ) )