Metamath Proof Explorer


Theorem fsumcvg4

Description: A serie with finite support is a finite sum, and therefore converges. (Contributed by Thierry Arnoux, 6-Sep-2017) (Revised by Thierry Arnoux, 1-Sep-2019)

Ref Expression
Hypotheses fsumcvg4.s S = M
fsumcvg4.m φ M
fsumcvg4.c φ F : S
fsumcvg4.f φ F -1 0 Fin
Assertion fsumcvg4 φ seq M + F dom

Proof

Step Hyp Ref Expression
1 fsumcvg4.s S = M
2 fsumcvg4.m φ M
3 fsumcvg4.c φ F : S
4 fsumcvg4.f φ F -1 0 Fin
5 ffun F : S Fun F
6 difpreima Fun F F -1 0 = F -1 F -1 0
7 3 5 6 3syl φ F -1 0 = F -1 F -1 0
8 difss F -1 F -1 0 F -1
9 7 8 eqsstrdi φ F -1 0 F -1
10 fimacnv F : S F -1 = S
11 3 10 syl φ F -1 = S
12 9 11 sseqtrd φ F -1 0 S
13 exmidd φ k S k F -1 0 ¬ k F -1 0
14 eqid F k = F k
15 14 biantru k F -1 0 k F -1 0 F k = F k
16 15 a1i φ k S k F -1 0 k F -1 0 F k = F k
17 1 fvexi S V
18 17 a1i φ S V
19 0nn0 0 0
20 19 a1i φ 0 0
21 eqid 0 = 0
22 21 ffs2 S V 0 0 F : S F supp 0 = F -1 0
23 18 20 3 22 syl3anc φ F supp 0 = F -1 0
24 3 ffnd φ F Fn S
25 suppvalfn F Fn S S V 0 0 F supp 0 = k S | F k 0
26 24 18 20 25 syl3anc φ F supp 0 = k S | F k 0
27 23 26 eqtr3d φ F -1 0 = k S | F k 0
28 27 eleq2d φ k F -1 0 k k S | F k 0
29 rabid k k S | F k 0 k S F k 0
30 28 29 bitrdi φ k F -1 0 k S F k 0
31 30 baibd φ k S k F -1 0 F k 0
32 31 necon2bbid φ k S F k = 0 ¬ k F -1 0
33 32 biimprd φ k S ¬ k F -1 0 F k = 0
34 33 pm4.71d φ k S ¬ k F -1 0 ¬ k F -1 0 F k = 0
35 16 34 orbi12d φ k S k F -1 0 ¬ k F -1 0 k F -1 0 F k = F k ¬ k F -1 0 F k = 0
36 13 35 mpbid φ k S k F -1 0 F k = F k ¬ k F -1 0 F k = 0
37 eqif F k = if k F -1 0 F k 0 k F -1 0 F k = F k ¬ k F -1 0 F k = 0
38 36 37 sylibr φ k S F k = if k F -1 0 F k 0
39 12 sselda φ k F -1 0 k S
40 3 ffvelrnda φ k S F k
41 39 40 syldan φ k F -1 0 F k
42 1 2 4 12 38 41 fsumcvg3 φ seq M + F dom