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Metamath Proof Explorer

Theorem nn0gsumfz0

Description: Replacing a finitely supported function over the nonnegative integers by a function over a finite set of sequential integers in a finite group sum. (Contributed by AV, 9-Oct-2019)

Ref Expression
Hypotheses nn0gsumfz.b 
|- B = ( Base ` G )
nn0gsumfz.0 
|- .0. = ( 0g ` G )
nn0gsumfz.g 
|- ( ph -> G e. CMnd )
nn0gsumfz.f 
|- ( ph -> F e. ( B ^m NN0 ) )
nn0gsumfz.y 
|- ( ph -> F finSupp .0. )
Assertion nn0gsumfz0 
|- ( ph -> E. s e. NN0 E. f e. ( B ^m ( 0 ... s ) ) ( G gsum F ) = ( G gsum f ) )

Proof

Step Hyp Ref Expression
1 nn0gsumfz.b 
 |-  B = ( Base ` G )
2 nn0gsumfz.0 
 |-  .0. = ( 0g ` G )
3 nn0gsumfz.g 
 |-  ( ph -> G e. CMnd )
4 nn0gsumfz.f 
 |-  ( ph -> F e. ( B ^m NN0 ) )
5 nn0gsumfz.y 
 |-  ( ph -> F finSupp .0. )
6 1 2 3 4 5 nn0gsumfz 
 |-  ( ph -> E. s e. NN0 E. f e. ( B ^m ( 0 ... s ) ) ( f = ( F |` ( 0 ... s ) ) /\ A. x e. NN0 ( s < x -> ( F ` x ) = .0. ) /\ ( G gsum F ) = ( G gsum f ) ) )
7 simp3 
 |-  ( ( f = ( F |` ( 0 ... s ) ) /\ A. x e. NN0 ( s < x -> ( F ` x ) = .0. ) /\ ( G gsum F ) = ( G gsum f ) ) -> ( G gsum F ) = ( G gsum f ) )
8 7 reximi 
 |-  ( E. f e. ( B ^m ( 0 ... s ) ) ( f = ( F |` ( 0 ... s ) ) /\ A. x e. NN0 ( s < x -> ( F ` x ) = .0. ) /\ ( G gsum F ) = ( G gsum f ) ) -> E. f e. ( B ^m ( 0 ... s ) ) ( G gsum F ) = ( G gsum f ) )
9 8 reximi 
 |-  ( E. s e. NN0 E. f e. ( B ^m ( 0 ... s ) ) ( f = ( F |` ( 0 ... s ) ) /\ A. x e. NN0 ( s < x -> ( F ` x ) = .0. ) /\ ( G gsum F ) = ( G gsum f ) ) -> E. s e. NN0 E. f e. ( B ^m ( 0 ... s ) ) ( G gsum F ) = ( G gsum f ) )
10 6 9 syl 
 |-  ( ph -> E. s e. NN0 E. f e. ( B ^m ( 0 ... s ) ) ( G gsum F ) = ( G gsum f ) )