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Metamath Proof Explorer

Theorem gsumzrsum

Description: Relate a group sum on ZZring to a finite sum on the complex numbers. See also gsumfsum . (Contributed by Thierry Arnoux, 5-Oct-2025)

Ref Expression
Hypotheses gsumzrsum.1 
|- ( ph -> A e. Fin )
gsumzrsum.2 
|- ( ( ph /\ k e. A ) -> B e. ZZ )
Assertion gsumzrsum 
|- ( ph -> ( ZZring gsum ( k e. A |-> B ) ) = sum_ k e. A B )

Proof

Step Hyp Ref Expression
1 gsumzrsum.1 
 |-  ( ph -> A e. Fin )
2 gsumzrsum.2 
 |-  ( ( ph /\ k e. A ) -> B e. ZZ )
3 cnfldbas 
 |-  CC = ( Base ` CCfld )
4 cnfldadd 
 |-  + = ( +g ` CCfld )
5 df-zring 
 |-  ZZring = ( CCfld |`s ZZ )
6 cnfldex 
 |-  CCfld e. _V
7 6 a1i 
 |-  ( ph -> CCfld e. _V )
8 zsscn 
 |-  ZZ C_ CC
9 8 a1i 
 |-  ( ph -> ZZ C_ CC )
10 2 fmpttd 
 |-  ( ph -> ( k e. A |-> B ) : A --> ZZ )
11 0zd 
 |-  ( ph -> 0 e. ZZ )
12 addlid 
 |-  ( k e. CC -> ( 0 + k ) = k )
13 addrid 
 |-  ( k e. CC -> ( k + 0 ) = k )
14 12 13 jca 
 |-  ( k e. CC -> ( ( 0 + k ) = k /\ ( k + 0 ) = k ) )
15 14 adantl 
 |-  ( ( ph /\ k e. CC ) -> ( ( 0 + k ) = k /\ ( k + 0 ) = k ) )
16 3 4 5 7 1 9 10 11 15 gsumress 
 |-  ( ph -> ( CCfld gsum ( k e. A |-> B ) ) = ( ZZring gsum ( k e. A |-> B ) ) )
17 2 zcnd 
 |-  ( ( ph /\ k e. A ) -> B e. CC )
18 1 17 gsumfsum 
 |-  ( ph -> ( CCfld gsum ( k e. A |-> B ) ) = sum_ k e. A B )
19 16 18 eqtr3d 
 |-  ( ph -> ( ZZring gsum ( k e. A |-> B ) ) = sum_ k e. A B )