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Metamath Proof Explorer

Theorem fsumreclf

Description: Closure of a finite sum of reals. (Contributed by Glauco Siliprandi, 21-Nov-2020)

Ref Expression
Hypotheses fsumreclf.k 
|- F/ k ph
fsumreclf.a 
|- ( ph -> A e. Fin )
fsumreclf.b 
|- ( ( ph /\ k e. A ) -> B e. RR )
Assertion fsumreclf 
|- ( ph -> sum_ k e. A B e. RR )

Proof

Step Hyp Ref Expression
1 fsumreclf.k 
 |-  F/ k ph
2 fsumreclf.a 
 |-  ( ph -> A e. Fin )
3 fsumreclf.b 
 |-  ( ( ph /\ k e. A ) -> B e. RR )
4 csbeq1a 
 |-  ( k = j -> B = [_ j / k ]_ B )
5 nfcv 
 |-  F/_ j B
6 nfcsb1v 
 |-  F/_ k [_ j / k ]_ B
7 4 5 6 cbvsum 
 |-  sum_ k e. A B = sum_ j e. A [_ j / k ]_ B
8 7 a1i 
 |-  ( ph -> sum_ k e. A B = sum_ j e. A [_ j / k ]_ B )
9 nfv 
 |-  F/ k j e. A
10 1 9 nfan 
 |-  F/ k ( ph /\ j e. A )
11 6 nfel1 
 |-  F/ k [_ j / k ]_ B e. RR
12 10 11 nfim 
 |-  F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. RR )
13 eleq1w 
 |-  ( k = j -> ( k e. A <-> j e. A ) )
14 13 anbi2d 
 |-  ( k = j -> ( ( ph /\ k e. A ) <-> ( ph /\ j e. A ) ) )
15 4 eleq1d 
 |-  ( k = j -> ( B e. RR <-> [_ j / k ]_ B e. RR ) )
16 14 15 imbi12d 
 |-  ( k = j -> ( ( ( ph /\ k e. A ) -> B e. RR ) <-> ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. RR ) ) )
17 12 16 3 chvarfv 
 |-  ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. RR )
18 2 17 fsumrecl 
 |-  ( ph -> sum_ j e. A [_ j / k ]_ B e. RR )
19 8 18 eqeltrd 
 |-  ( ph -> sum_ k e. A B e. RR )