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Metamath Proof Explorer

Theorem regsumfsum

Description: Relate a group sum on ` ( CCfld |``s RR ) ` to a finite sum on the reals. Cf. gsumfsum . (Contributed by Thierry Arnoux, 7-Sep-2018)

Ref Expression
Hypotheses regsumfsum.1 
|- ( ph -> A e. Fin )
regsumfsum.2 
|- ( ( ph /\ k e. A ) -> B e. RR )
Assertion regsumfsum 
|- ( ph -> ( ( CCfld |`s RR ) gsum ( k e. A |-> B ) ) = sum_ k e. A B )

Proof

Step Hyp Ref Expression
1 regsumfsum.1 
 |-  ( ph -> A e. Fin )
2 regsumfsum.2 
 |-  ( ( ph /\ k e. A ) -> B e. RR )
3 cnfldbas 
 |-  CC = ( Base ` CCfld )
4 cnfldadd 
 |-  + = ( +g ` CCfld )
5 eqid 
 |-  ( CCfld |`s RR ) = ( CCfld |`s RR )
6 cnfldex 
 |-  CCfld e. _V
7 6 a1i 
 |-  ( ph -> CCfld e. _V )
8 ax-resscn 
 |-  RR C_ CC
9 8 a1i 
 |-  ( ph -> RR C_ CC )
10 2 fmpttd 
 |-  ( ph -> ( k e. A |-> B ) : A --> RR )
11 0red 
 |-  ( ph -> 0 e. RR )
12 simpr 
 |-  ( ( ph /\ x e. CC ) -> x e. CC )
13 12 addlidd 
 |-  ( ( ph /\ x e. CC ) -> ( 0 + x ) = x )
14 12 addridd 
 |-  ( ( ph /\ x e. CC ) -> ( x + 0 ) = x )
15 13 14 jca 
 |-  ( ( ph /\ x e. CC ) -> ( ( 0 + x ) = x /\ ( x + 0 ) = x ) )
16 3 4 5 7 1 9 10 11 15 gsumress 
 |-  ( ph -> ( CCfld gsum ( k e. A |-> B ) ) = ( ( CCfld |`s RR ) gsum ( k e. A |-> B ) ) )
17 2 recnd 
 |-  ( ( 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 -> ( ( CCfld |`s RR ) gsum ( k e. A |-> B ) ) = sum_ k e. A B )