Metamath Proof Explorer


Theorem gsumge0cl

Description: Closure of group sum, for finitely supported nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses gsumge0cl.1 G = 𝑠 * 𝑠 0 +∞
gsumge0cl.2 φ X V
gsumge0cl.3 φ F : X 0 +∞
gsumge0cl.4 φ finSupp 0 F
Assertion gsumge0cl φ G F 0 +∞

Proof

Step Hyp Ref Expression
1 gsumge0cl.1 G = 𝑠 * 𝑠 0 +∞
2 gsumge0cl.2 φ X V
3 gsumge0cl.3 φ F : X 0 +∞
4 gsumge0cl.4 φ finSupp 0 F
5 iccssxr 0 +∞ *
6 df-ss 0 +∞ * 0 +∞ * = 0 +∞
7 5 6 mpbi 0 +∞ * = 0 +∞
8 7 eqcomi 0 +∞ = 0 +∞ *
9 ovex 0 +∞ V
10 xrsbas * = Base 𝑠 *
11 1 10 ressbas 0 +∞ V 0 +∞ * = Base G
12 9 11 ax-mp 0 +∞ * = Base G
13 8 12 eqtri 0 +∞ = Base G
14 eqid 𝑠 * 𝑠 * −∞ = 𝑠 * 𝑠 * −∞
15 14 xrs1cmn 𝑠 * 𝑠 * −∞ CMnd
16 cmnmnd 𝑠 * 𝑠 * −∞ CMnd 𝑠 * 𝑠 * −∞ Mnd
17 15 16 ax-mp 𝑠 * 𝑠 * −∞ Mnd
18 xrge0cmn 𝑠 * 𝑠 0 +∞ CMnd
19 1 18 eqeltri G CMnd
20 cmnmnd G CMnd G Mnd
21 19 20 ax-mp G Mnd
22 17 21 pm3.2i 𝑠 * 𝑠 * −∞ Mnd G Mnd
23 eliccxr x 0 +∞ x *
24 mnfxr −∞ *
25 24 a1i x 0 +∞ −∞ *
26 0xr 0 *
27 26 a1i x 0 +∞ 0 *
28 mnflt0 −∞ < 0
29 28 a1i x 0 +∞ −∞ < 0
30 pnfxr +∞ *
31 30 a1i x 0 +∞ +∞ *
32 id x 0 +∞ x 0 +∞
33 iccgelb 0 * +∞ * x 0 +∞ 0 x
34 27 31 32 33 syl3anc x 0 +∞ 0 x
35 25 27 23 29 34 xrltletrd x 0 +∞ −∞ < x
36 25 23 35 xrgtned x 0 +∞ x −∞
37 nelsn x −∞ ¬ x −∞
38 36 37 syl x 0 +∞ ¬ x −∞
39 23 38 eldifd x 0 +∞ x * −∞
40 39 rgen x 0 +∞ x * −∞
41 dfss3 0 +∞ * −∞ x 0 +∞ x * −∞
42 40 41 mpbir 0 +∞ * −∞
43 0e0iccpnf 0 0 +∞
44 42 43 pm3.2i 0 +∞ * −∞ 0 0 +∞
45 difss * −∞ *
46 14 10 ressbas2 * −∞ * * −∞ = Base 𝑠 * 𝑠 * −∞
47 45 46 ax-mp * −∞ = Base 𝑠 * 𝑠 * −∞
48 14 xrs10 0 = 0 𝑠 * 𝑠 * −∞
49 xrex * V
50 difexg * V * −∞ V
51 49 50 ax-mp * −∞ V
52 44 simpli 0 +∞ * −∞
53 ressabs * −∞ V 0 +∞ * −∞ 𝑠 * 𝑠 * −∞ 𝑠 0 +∞ = 𝑠 * 𝑠 0 +∞
54 51 52 53 mp2an 𝑠 * 𝑠 * −∞ 𝑠 0 +∞ = 𝑠 * 𝑠 0 +∞
55 1 eqcomi 𝑠 * 𝑠 0 +∞ = G
56 54 55 eqtr2i G = 𝑠 * 𝑠 * −∞ 𝑠 0 +∞
57 47 48 56 submnd0 𝑠 * 𝑠 * −∞ Mnd G Mnd 0 +∞ * −∞ 0 0 +∞ 0 = 0 G
58 22 44 57 mp2an 0 = 0 G
59 19 a1i φ G CMnd
60 13 58 59 2 3 4 gsumcl φ G F 0 +∞