Theorem sge0cl 43919

Description: The arbitrary sum of nonnegative extended reals is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
sge0cl.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
sge0cl.f	⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))

Assertion

Ref	Expression
sge0cl	⊢ (𝜑 → (Σ^‘𝐹) ∈ (0[,]+∞))

Proof of Theorem sge0cl

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6774	. . . . 5 ⊢ (𝐹 = ∅ → (Σ^‘𝐹) = (Σ^‘∅))
2		sge00 43914	. . . . . 6 ⊢ (Σ^‘∅) = 0
3	2	a1i 11	. . . . 5 ⊢ (𝐹 = ∅ → (Σ^‘∅) = 0)
4	1, 3	eqtrd 2778	. . . 4 ⊢ (𝐹 = ∅ → (Σ^‘𝐹) = 0)
5		0e0iccpnf 13191	. . . . 5 ⊢ 0 ∈ (0[,]+∞)
6	5	a1i 11	. . . 4 ⊢ (𝐹 = ∅ → 0 ∈ (0[,]+∞))
7	4, 6	eqeltrd 2839	. . 3 ⊢ (𝐹 = ∅ → (Σ^‘𝐹) ∈ (0[,]+∞))
8	7	adantl 482	. 2 ⊢ ((𝜑 ∧ 𝐹 = ∅) → (Σ^‘𝐹) ∈ (0[,]+∞))
9		sge0cl.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
10	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
11		sge0cl.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))
12	11	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
13		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
14	10, 12, 13	sge0pnfval 43911	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = +∞)
15		pnfel0pnf 43066	. . . . . 6 ⊢ +∞ ∈ (0[,]+∞)
16	15	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ (0[,]+∞))
17	14, 16	eqeltrd 2839	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ (0[,]+∞))
18	17	adantlr 712	. . 3 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ (0[,]+∞))
19		simpll 764	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 𝜑)
20		neqne 2951	. . . . 5 ⊢ (¬ 𝐹 = ∅ → 𝐹 ≠ ∅)
21	20	ad2antlr 724	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹 ≠ ∅)
22		simpr 485	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
23		0xr 11022	. . . . . 6 ⊢ 0 ∈ ℝ*
24	23	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 0 ∈ ℝ*)
25		pnfxr 11029	. . . . . 6 ⊢ +∞ ∈ ℝ*
26	25	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → +∞ ∈ ℝ*)
27	9	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
28	11	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
29		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
30	28, 29	fge0iccico 43908	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
31	27, 30	sge0reval 43910	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
32		elinel2 4130	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
33	32	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
34	11	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝐹:𝑋⟶(0[,]+∞))
35		elinel1 4129	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
36		elpwi 4542	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
37	35, 36	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ⊆ 𝑋)
38	37	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ⊆ 𝑋)
39	38	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑥 ⊆ 𝑋)
40		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
41	39, 40	sseldd 3922	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
42	34, 41	ffvelrnd 6962	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,]+∞))
43	42	adantllr 716	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,]+∞))
44		nne 2947	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (𝐹‘𝑦) ≠ +∞ ↔ (𝐹‘𝑦) = +∞)
45	44	biimpi 215	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝐹‘𝑦) ≠ +∞ → (𝐹‘𝑦) = +∞)
46	45	eqcomd 2744	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝐹‘𝑦) ≠ +∞ → +∞ = (𝐹‘𝑦))
47	46	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ ¬ (𝐹‘𝑦) ≠ +∞) → +∞ = (𝐹‘𝑦))
48	11	ffund 6604	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Fun 𝐹)
49	48	3ad2ant1 1132	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → Fun 𝐹)
50	41	3impa 1109	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
51	11	fdmd 6611	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom 𝐹 = 𝑋)
52	51	eqcomd 2744	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑋 = dom 𝐹)
53	52	3ad2ant1 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑋 = dom 𝐹)
54	50, 53	eleqtrd 2841	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝐹)
55		fvelrn 6954	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ ran 𝐹)
56	49, 54, 55	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ran 𝐹)
57	56	ad5ant134 1366	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ ¬ (𝐹‘𝑦) ≠ +∞) → (𝐹‘𝑦) ∈ ran 𝐹)
58	47, 57	eqeltrd 2839	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ ¬ (𝐹‘𝑦) ≠ +∞) → +∞ ∈ ran 𝐹)
59	29	ad3antrrr 727	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ ¬ (𝐹‘𝑦) ≠ +∞) → ¬ +∞ ∈ ran 𝐹)
60	58, 59	condan 815	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ≠ +∞)
61		ge0xrre 43069	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑦) ∈ (0[,]+∞) ∧ (𝐹‘𝑦) ≠ +∞) → (𝐹‘𝑦) ∈ ℝ)
62	43, 60, 61	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℝ)
63	33, 62	fsumrecl 15446	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ ℝ)
64	63	ralrimiva 3103	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ ℝ)
65		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
66	65	rnmptss 6996	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ ℝ → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ)
67	64, 66	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ)
68		ressxr 11019	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ*
69	68	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ℝ ⊆ ℝ*)
70	67, 69	sstrd 3931	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ*)
71		supxrcl 13049	. . . . . . . 8 ⊢ (ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ* → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ) ∈ ℝ*)
72	70, 71	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ) ∈ ℝ*)
73	31, 72	eqeltrd 2839	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ ℝ*)
74	73	adantlr 712	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ ℝ*)
75	52	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → 𝑋 = dom 𝐹)
76		neneq 2949	. . . . . . . . . . . 12 ⊢ (𝐹 ≠ ∅ → ¬ 𝐹 = ∅)
77	76	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → ¬ 𝐹 = ∅)
78		frel 6605	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋⟶(0[,]+∞) → Rel 𝐹)
79	11, 78	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → Rel 𝐹)
80	79	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → Rel 𝐹)
81		reldm0 5837	. . . . . . . . . . . 12 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
82	80, 81	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
83	77, 82	mtbid 324	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → ¬ dom 𝐹 = ∅)
84	83	neqned 2950	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → dom 𝐹 ≠ ∅)
85	75, 84	eqnetrd 3011	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → 𝑋 ≠ ∅)
86		n0 4280	. . . . . . . 8 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝑋)
87	85, 86	sylib 217	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → ∃𝑧 𝑧 ∈ 𝑋)
88	87	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → ∃𝑧 𝑧 ∈ 𝑋)
89	23	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → 0 ∈ ℝ*)
90	11	ffvelrnda 6961	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ (0[,]+∞))
91	90	adantlr 712	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ (0[,]+∞))
92		nne 2947	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝐹‘𝑧) ≠ +∞ ↔ (𝐹‘𝑧) = +∞)
93	92	biimpi 215	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝐹‘𝑧) ≠ +∞ → (𝐹‘𝑧) = +∞)
94	93	eqcomd 2744	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝐹‘𝑧) ≠ +∞ → +∞ = (𝐹‘𝑧))
95	94	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) ∧ ¬ (𝐹‘𝑧) ≠ +∞) → +∞ = (𝐹‘𝑧))
96	11	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐹:𝑋⟶(0[,]+∞))
97	96	ffund 6604	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → Fun 𝐹)
98		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
99	52	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑋 = dom 𝐹)
100	98, 99	eleqtrd 2841	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ dom 𝐹)
101		fvelrn 6954	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ ran 𝐹)
102	97, 100, 101	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ran 𝐹)
103	102	adantlr 712	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ran 𝐹)
104	103	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) ∧ ¬ (𝐹‘𝑧) ≠ +∞) → (𝐹‘𝑧) ∈ ran 𝐹)
105	95, 104	eqeltrd 2839	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) ∧ ¬ (𝐹‘𝑧) ≠ +∞) → +∞ ∈ ran 𝐹)
106	29	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) ∧ ¬ (𝐹‘𝑧) ≠ +∞) → ¬ +∞ ∈ ran 𝐹)
107	105, 106	condan 815	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ≠ +∞)
108		ge0xrre 43069	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑧) ∈ (0[,]+∞) ∧ (𝐹‘𝑧) ≠ +∞) → (𝐹‘𝑧) ∈ ℝ)
109	91, 107, 108	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℝ)
110	109	rexrd 11025	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℝ*)
111	73	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (Σ^‘𝐹) ∈ ℝ*)
112	23	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 0 ∈ ℝ*)
113	25	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → +∞ ∈ ℝ*)
114		iccgelb 13135	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹‘𝑧) ∈ (0[,]+∞)) → 0 ≤ (𝐹‘𝑧))
115	112, 113, 90, 114	syl3anc 1370	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 0 ≤ (𝐹‘𝑧))
116	115	adantlr 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → 0 ≤ (𝐹‘𝑧))
117	70	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ*)
118		snelpwi 5360	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑋 → {𝑧} ∈ 𝒫 𝑋)
119		snfi 8834	. . . . . . . . . . . . . . . . 17 ⊢ {𝑧} ∈ Fin
120	119	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑋 → {𝑧} ∈ Fin)
121	118, 120	elind 4128	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑋 → {𝑧} ∈ (𝒫 𝑋 ∩ Fin))
122	121	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → {𝑧} ∈ (𝒫 𝑋 ∩ Fin))
123		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
124	109	recnd 11003	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℂ)
125		fveq2 6774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
126	125	sumsn 15458	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ ℂ) → Σ𝑦 ∈ {𝑧} (𝐹‘𝑦) = (𝐹‘𝑧))
127	123, 124, 126	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → Σ𝑦 ∈ {𝑧} (𝐹‘𝑦) = (𝐹‘𝑧))
128	127	eqcomd 2744	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) = Σ𝑦 ∈ {𝑧} (𝐹‘𝑦))
129		sumeq1 15400	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = {𝑧} → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) = Σ𝑦 ∈ {𝑧} (𝐹‘𝑦))
130	129	rspceeqv 3575	. . . . . . . . . . . . . 14 ⊢ (({𝑧} ∈ (𝒫 𝑋 ∩ Fin) ∧ (𝐹‘𝑧) = Σ𝑦 ∈ {𝑧} (𝐹‘𝑦)) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹‘𝑧) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
131	122, 128, 130	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹‘𝑧) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
132	65	elrnmpt 5865	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑧) ∈ (0[,]+∞) → ((𝐹‘𝑧) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹‘𝑧) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))
133	91, 132	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹‘𝑧) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))
134	131, 133	mpbird 256	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))
135		supxrub 13058	. . . . . . . . . . . 12 ⊢ ((ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ* ∧ (𝐹‘𝑧) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))) → (𝐹‘𝑧) ≤ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
136	117, 134, 135	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ≤ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
137	31	eqcomd 2744	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ) = (Σ^‘𝐹))
138	137	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ) = (Σ^‘𝐹))
139	136, 138	breqtrd 5100	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ≤ (Σ^‘𝐹))
140	89, 110, 111, 116, 139	xrletrd 12896	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → 0 ≤ (Σ^‘𝐹))
141	140	ex 413	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑧 ∈ 𝑋 → 0 ≤ (Σ^‘𝐹)))
142	141	adantlr 712	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (𝑧 ∈ 𝑋 → 0 ≤ (Σ^‘𝐹)))
143	142	exlimdv 1936	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (∃𝑧 𝑧 ∈ 𝑋 → 0 ≤ (Σ^‘𝐹)))
144	88, 143	mpd 15	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 0 ≤ (Σ^‘𝐹))
145		pnfge 12866	. . . . . . 7 ⊢ ((Σ^‘𝐹) ∈ ℝ* → (Σ^‘𝐹) ≤ +∞)
146	73, 145	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ≤ +∞)
147	146	adantlr 712	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ≤ +∞)
148	24, 26, 74, 144, 147	eliccxrd 43065	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ (0[,]+∞))
149	19, 21, 22, 148	syl21anc 835	. . 3 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ (0[,]+∞))
150	18, 149	pm2.61dan 810	. 2 ⊢ ((𝜑 ∧ ¬ 𝐹 = ∅) → (Σ^‘𝐹) ∈ (0[,]+∞))
151	8, 150	pm2.61dan 810	1 ⊢ (𝜑 → (Σ^‘𝐹) ∈ (0[,]+∞))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561   class class class wbr 5074   ↦ cmpt 5157  dom cdm 5589  ran crn 5590  Rel wrel 5594  Fun wfun 6427  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  Fincfn 8733  supcsup 9199  ℂcc 10869  ℝcr 10870  0cc0 10871  +∞cpnf 11006  ℝ^*cxr 11008   < clt 11009   ≤ cle 11010  [,]cicc 13082  Σcsu 15397  Σ^{^}csumge0 43900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-sum 15398  df-sumge0 43901

This theorem is referenced by:  sge0ge0  43922  sge0xrcl  43923  sge0split  43947  sge0iunmptlemre  43953  sge0iunmpt  43956  sge0nemnf  43958  sge0clmpt  43963  sge0isum  43965  psmeasure  44009  ovnsupge0  44095  ovnsubaddlem1  44108  sge0hsphoire  44127  hoidmvlelem1  44133  hspmbllem2  44165