Metamath Proof Explorer


Theorem fsuppmapnn0fiub

Description: If all functions of a finite set of functions over the nonnegative integers are finitely supported, then the support of all these functions is contained in a finite set of sequential integers starting at 0 and ending with the supremum of the union of the support of these functions. (Contributed by AV, 2-Oct-2019) (Proof shortened by JJ, 2-Aug-2021)

Ref Expression
Hypotheses fsuppmapnn0fiub.u U = f M supp Z f
fsuppmapnn0fiub.s S = sup U <
Assertion fsuppmapnn0fiub M R 0 M Fin Z V f M finSupp Z f U f M f supp Z 0 S

Proof

Step Hyp Ref Expression
1 fsuppmapnn0fiub.u U = f M supp Z f
2 fsuppmapnn0fiub.s S = sup U <
3 nfv f M R 0 M Fin Z V
4 nfra1 f f M finSupp Z f
5 nfv f U
6 4 5 nfan f f M finSupp Z f U
7 3 6 nfan f M R 0 M Fin Z V f M finSupp Z f U
8 suppssdm f supp Z dom f
9 ssel2 M R 0 f M f R 0
10 elmapfn f R 0 f Fn 0
11 fndm f Fn 0 dom f = 0
12 eqimss dom f = 0 dom f 0
13 9 10 11 12 4syl M R 0 f M dom f 0
14 13 3ad2antl1 M R 0 M Fin Z V f M dom f 0
15 8 14 sstrid M R 0 M Fin Z V f M f supp Z 0
16 15 sseld M R 0 M Fin Z V f M x supp Z f x 0
17 16 adantlr M R 0 M Fin Z V f M finSupp Z f U f M x supp Z f x 0
18 17 imp M R 0 M Fin Z V f M finSupp Z f U f M x supp Z f x 0
19 1 2 fsuppmapnn0fiublem M R 0 M Fin Z V f M finSupp Z f U S 0
20 19 imp M R 0 M Fin Z V f M finSupp Z f U S 0
21 20 ad2antrr M R 0 M Fin Z V f M finSupp Z f U f M x supp Z f S 0
22 9 10 11 3syl M R 0 f M dom f = 0
23 22 ex M R 0 f M dom f = 0
24 23 3ad2ant1 M R 0 M Fin Z V f M dom f = 0
25 24 adantr M R 0 M Fin Z V f M finSupp Z f U f M dom f = 0
26 25 imp M R 0 M Fin Z V f M finSupp Z f U f M dom f = 0
27 nn0ssre 0
28 26 27 eqsstrdi M R 0 M Fin Z V f M finSupp Z f U f M dom f
29 8 28 sstrid M R 0 M Fin Z V f M finSupp Z f U f M f supp Z
30 29 ex M R 0 M Fin Z V f M finSupp Z f U f M f supp Z
31 7 30 ralrimi M R 0 M Fin Z V f M finSupp Z f U f M f supp Z
32 31 ad2antrr M R 0 M Fin Z V f M finSupp Z f U f M x supp Z f f M f supp Z
33 iunss f M supp Z f f M f supp Z
34 32 33 sylibr M R 0 M Fin Z V f M finSupp Z f U f M x supp Z f f M supp Z f
35 1 34 eqsstrid M R 0 M Fin Z V f M finSupp Z f U f M x supp Z f U
36 simp2 M R 0 M Fin Z V M Fin
37 id finSupp Z f finSupp Z f
38 37 fsuppimpd finSupp Z f f supp Z Fin
39 38 ralimi f M finSupp Z f f M f supp Z Fin
40 39 adantr f M finSupp Z f U f M f supp Z Fin
41 36 40 anim12i M R 0 M Fin Z V f M finSupp Z f U M Fin f M f supp Z Fin
42 41 ad2antrr M R 0 M Fin Z V f M finSupp Z f U f M x supp Z f M Fin f M f supp Z Fin
43 iunfi M Fin f M f supp Z Fin f M supp Z f Fin
44 42 43 syl M R 0 M Fin Z V f M finSupp Z f U f M x supp Z f f M supp Z f Fin
45 1 44 eqeltrid M R 0 M Fin Z V f M finSupp Z f U f M x supp Z f U Fin
46 rspe f M x supp Z f f M x supp Z f
47 eliun x f M supp Z f f M x supp Z f
48 46 47 sylibr f M x supp Z f x f M supp Z f
49 48 1 eleqtrrdi f M x supp Z f x U
50 49 adantll M R 0 M Fin Z V f M finSupp Z f U f M x supp Z f x U
51 2 a1i M R 0 M Fin Z V f M finSupp Z f U f M x supp Z f S = sup U <
52 35 45 50 51 supfirege M R 0 M Fin Z V f M finSupp Z f U f M x supp Z f x S
53 elfz2nn0 x 0 S x 0 S 0 x S
54 18 21 52 53 syl3anbrc M R 0 M Fin Z V f M finSupp Z f U f M x supp Z f x 0 S
55 54 ex M R 0 M Fin Z V f M finSupp Z f U f M x supp Z f x 0 S
56 55 ssrdv M R 0 M Fin Z V f M finSupp Z f U f M f supp Z 0 S
57 56 ex M R 0 M Fin Z V f M finSupp Z f U f M f supp Z 0 S
58 7 57 ralrimi M R 0 M Fin Z V f M finSupp Z f U f M f supp Z 0 S
59 58 ex M R 0 M Fin Z V f M finSupp Z f U f M f supp Z 0 S