Metamath Proof Explorer

Theorem fsumdifsnconst

Description: The sum of constant terms ( k is not free in C ) over an index set excluding a singleton. (Contributed by AV, 7-Jan-2022)

Ref Expression
Assertion fsumdifsnconst 
|- ( ( A e. Fin /\ B e. A /\ C e. CC ) -> sum_ k e. ( A \ { B } ) C = ( ( ( # ` A ) - 1 ) x. C ) )

Proof

Step Hyp Ref Expression
1 diffi 
 |-  ( A e. Fin -> ( A \ { B } ) e. Fin )
2 1 anim1i 
 |-  ( ( A e. Fin /\ C e. CC ) -> ( ( A \ { B } ) e. Fin /\ C e. CC ) )
3 2 3adant2 
 |-  ( ( A e. Fin /\ B e. A /\ C e. CC ) -> ( ( A \ { B } ) e. Fin /\ C e. CC ) )
4 fsumconst 
 |-  ( ( ( A \ { B } ) e. Fin /\ C e. CC ) -> sum_ k e. ( A \ { B } ) C = ( ( # ` ( A \ { B } ) ) x. C ) )
5 3 4 syl 
 |-  ( ( A e. Fin /\ B e. A /\ C e. CC ) -> sum_ k e. ( A \ { B } ) C = ( ( # ` ( A \ { B } ) ) x. C ) )
6 hashdifsn 
 |-  ( ( A e. Fin /\ B e. A ) -> ( # ` ( A \ { B } ) ) = ( ( # ` A ) - 1 ) )
7 6 3adant3 
 |-  ( ( A e. Fin /\ B e. A /\ C e. CC ) -> ( # ` ( A \ { B } ) ) = ( ( # ` A ) - 1 ) )
8 7 oveq1d 
 |-  ( ( A e. Fin /\ B e. A /\ C e. CC ) -> ( ( # ` ( A \ { B } ) ) x. C ) = ( ( ( # ` A ) - 1 ) x. C ) )
9 5 8 eqtrd 
 |-  ( ( A e. Fin /\ B e. A /\ C e. CC ) -> sum_ k e. ( A \ { B } ) C = ( ( ( # ` A ) - 1 ) x. C ) )