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 k φ
fsumreclf.a φ A Fin
fsumreclf.b φ k A B
Assertion fsumreclf φ k A B

Proof

Step Hyp Ref Expression
1 fsumreclf.k k φ
2 fsumreclf.a φ A Fin
3 fsumreclf.b φ k A B
4 csbeq1a k = j B = j / k B
5 nfcv _ j A
6 nfcv _ k A
7 nfcv _ j B
8 nfcsb1v _ k j / k B
9 4 5 6 7 8 cbvsum k A B = j A j / k B
10 9 a1i φ k A B = j A j / k B
11 nfv k j A
12 1 11 nfan k φ j A
13 8 nfel1 k j / k B
14 12 13 nfim k φ j A j / k B
15 eleq1w k = j k A j A
16 15 anbi2d k = j φ k A φ j A
17 4 eleq1d k = j B j / k B
18 16 17 imbi12d k = j φ k A B φ j A j / k B
19 14 18 3 chvarfv φ j A j / k B
20 2 19 fsumrecl φ j A j / k B
21 10 20 eqeltrd φ k A B