Theorem sge0cl 46830

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 6835	. . . . 5 ⊢ (𝐹 = ∅ → (Σ^‘𝐹) = (Σ^‘∅))
2		sge00 46825	. . . . . 6 ⊢ (Σ^‘∅) = 0
3	2	a1i 11	. . . . 5 ⊢ (𝐹 = ∅ → (Σ^‘∅) = 0)
4	1, 3	eqtrd 2772	. . . 4 ⊢ (𝐹 = ∅ → (Σ^‘𝐹) = 0)
5		0e0iccpnf 13406	. . . . 5 ⊢ 0 ∈ (0[,]+∞)
6	5	a1i 11	. . . 4 ⊢ (𝐹 = ∅ → 0 ∈ (0[,]+∞))
7	4, 6	eqeltrd 2837	. . 3 ⊢ (𝐹 = ∅ → (Σ^‘𝐹) ∈ (0[,]+∞))
8	7	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝐹 = ∅) → (Σ^‘𝐹) ∈ (0[,]+∞))
9		sge0cl.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
10	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
11		sge0cl.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))
12	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
13		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
14	10, 12, 13	sge0pnfval 46822	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = +∞)
15		pnfel0pnf 45979	. . . . . 6 ⊢ +∞ ∈ (0[,]+∞)
16	15	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ (0[,]+∞))
17	14, 16	eqeltrd 2837	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ (0[,]+∞))
18	17	adantlr 716	. . 3 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ (0[,]+∞))
19		simpll 767	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 𝜑)
20		neqne 2941	. . . . 5 ⊢ (¬ 𝐹 = ∅ → 𝐹 ≠ ∅)
21	20	ad2antlr 728	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹 ≠ ∅)
22		simpr 484	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
23		0xr 11186	. . . . . 6 ⊢ 0 ∈ ℝ*
24	23	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 0 ∈ ℝ*)
25		pnfxr 11193	. . . . . 6 ⊢ +∞ ∈ ℝ*
26	25	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → +∞ ∈ ℝ*)
27	9	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
28	11	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
29		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
30	28, 29	fge0iccico 46819	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
31	27, 30	sge0reval 46821	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
32		elinel2 4143	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
33	32	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
34	11	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝐹:𝑋⟶(0[,]+∞))
35		elinel1 4142	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
36		elpwi 4549	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
37	35, 36	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ⊆ 𝑋)
38	37	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ⊆ 𝑋)
39	38	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑥 ⊆ 𝑋)
40		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
41	39, 40	sseldd 3923	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
42	34, 41	ffvelcdmd 7032	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,]+∞))
43	42	adantllr 720	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,]+∞))
44		nne 2937	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (𝐹‘𝑦) ≠ +∞ ↔ (𝐹‘𝑦) = +∞)
45	44	biimpi 216	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝐹‘𝑦) ≠ +∞ → (𝐹‘𝑦) = +∞)
46	45	eqcomd 2743	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝐹‘𝑦) ≠ +∞ → +∞ = (𝐹‘𝑦))
47	46	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ ¬ (𝐹‘𝑦) ≠ +∞) → +∞ = (𝐹‘𝑦))
48	11	ffund 6667	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Fun 𝐹)
49	48	3ad2ant1 1134	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → Fun 𝐹)
50	41	3impa 1110	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
51	11	fdmd 6673	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom 𝐹 = 𝑋)
52	51	eqcomd 2743	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑋 = dom 𝐹)
53	52	3ad2ant1 1134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑋 = dom 𝐹)
54	50, 53	eleqtrd 2839	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝐹)
55		fvelrn 7023	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ ran 𝐹)
56	49, 54, 55	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ran 𝐹)
57	56	ad5ant134 1370	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ ¬ (𝐹‘𝑦) ≠ +∞) → (𝐹‘𝑦) ∈ ran 𝐹)
58	47, 57	eqeltrd 2837	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ ¬ (𝐹‘𝑦) ≠ +∞) → +∞ ∈ ran 𝐹)
59	29	ad3antrrr 731	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ ¬ (𝐹‘𝑦) ≠ +∞) → ¬ +∞ ∈ ran 𝐹)
60	58, 59	condan 818	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ≠ +∞)
61		ge0xrre 45982	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑦) ∈ (0[,]+∞) ∧ (𝐹‘𝑦) ≠ +∞) → (𝐹‘𝑦) ∈ ℝ)
62	43, 60, 61	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℝ)
63	33, 62	fsumrecl 15690	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ ℝ)
64	63	ralrimiva 3130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ ℝ)
65		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
66	65	rnmptss 7070	. . . . . . . . . 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 11183	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ*
69	68	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ℝ ⊆ ℝ*)
70	67, 69	sstrd 3933	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ*)
71		supxrcl 13261	. . . . . . . 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 2837	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ ℝ*)
74	73	adantlr 716	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ ℝ*)
75	52	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → 𝑋 = dom 𝐹)
76		neneq 2939	. . . . . . . . . . . 12 ⊢ (𝐹 ≠ ∅ → ¬ 𝐹 = ∅)
77	76	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → ¬ 𝐹 = ∅)
78		frel 6668	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋⟶(0[,]+∞) → Rel 𝐹)
79	11, 78	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → Rel 𝐹)
80	79	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → Rel 𝐹)
81		reldm0 5878	. . . . . . . . . . . 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 2940	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → dom 𝐹 ≠ ∅)
85	75, 84	eqnetrd 3000	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → 𝑋 ≠ ∅)
86		n0 4294	. . . . . . . 8 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝑋)
87	85, 86	sylib 218	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → ∃𝑧 𝑧 ∈ 𝑋)
88	87	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → ∃𝑧 𝑧 ∈ 𝑋)
89	23	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → 0 ∈ ℝ*)
90	11	ffvelcdmda 7031	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ (0[,]+∞))
91	90	adantlr 716	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ (0[,]+∞))
92		nne 2937	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝐹‘𝑧) ≠ +∞ ↔ (𝐹‘𝑧) = +∞)
93	92	biimpi 216	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝐹‘𝑧) ≠ +∞ → (𝐹‘𝑧) = +∞)
94	93	eqcomd 2743	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝐹‘𝑧) ≠ +∞ → +∞ = (𝐹‘𝑧))
95	94	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) ∧ ¬ (𝐹‘𝑧) ≠ +∞) → +∞ = (𝐹‘𝑧))
96	11	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐹:𝑋⟶(0[,]+∞))
97	96	ffund 6667	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → Fun 𝐹)
98		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
99	52	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑋 = dom 𝐹)
100	98, 99	eleqtrd 2839	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ dom 𝐹)
101		fvelrn 7023	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ ran 𝐹)
102	97, 100, 101	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ran 𝐹)
103	102	adantlr 716	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ran 𝐹)
104	103	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) ∧ ¬ (𝐹‘𝑧) ≠ +∞) → (𝐹‘𝑧) ∈ ran 𝐹)
105	95, 104	eqeltrd 2837	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) ∧ ¬ (𝐹‘𝑧) ≠ +∞) → +∞ ∈ ran 𝐹)
106	29	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) ∧ ¬ (𝐹‘𝑧) ≠ +∞) → ¬ +∞ ∈ ran 𝐹)
107	105, 106	condan 818	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ≠ +∞)
108		ge0xrre 45982	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑧) ∈ (0[,]+∞) ∧ (𝐹‘𝑧) ≠ +∞) → (𝐹‘𝑧) ∈ ℝ)
109	91, 107, 108	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℝ)
110	109	rexrd 11189	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℝ*)
111	73	adantr 480	. . . . . . . . . 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 13349	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹‘𝑧) ∈ (0[,]+∞)) → 0 ≤ (𝐹‘𝑧))
115	112, 113, 90, 114	syl3anc 1374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 0 ≤ (𝐹‘𝑧))
116	115	adantlr 716	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → 0 ≤ (𝐹‘𝑧))
117	70	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ*)
118		snelpwi 5392	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑋 → {𝑧} ∈ 𝒫 𝑋)
119		snfi 8984	. . . . . . . . . . . . . . . . 17 ⊢ {𝑧} ∈ Fin
120	119	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑋 → {𝑧} ∈ Fin)
121	118, 120	elind 4141	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑋 → {𝑧} ∈ (𝒫 𝑋 ∩ Fin))
122	121	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → {𝑧} ∈ (𝒫 𝑋 ∩ Fin))
123		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
124	109	recnd 11167	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℂ)
125		fveq2 6835	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
126	125	sumsn 15702	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ ℂ) → Σ𝑦 ∈ {𝑧} (𝐹‘𝑦) = (𝐹‘𝑧))
127	123, 124, 126	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → Σ𝑦 ∈ {𝑧} (𝐹‘𝑦) = (𝐹‘𝑧))
128	127	eqcomd 2743	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) = Σ𝑦 ∈ {𝑧} (𝐹‘𝑦))
129		sumeq1 15645	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = {𝑧} → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) = Σ𝑦 ∈ {𝑧} (𝐹‘𝑦))
130	129	rspceeqv 3588	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹‘𝑧) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
132	65	elrnmpt 5908	. . . . . . . . . . . . . 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 257	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))
135		supxrub 13270	. . . . . . . . . . . 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 585	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ≤ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
137	31	eqcomd 2743	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ) = (Σ^‘𝐹))
138	137	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ) = (Σ^‘𝐹))
139	136, 138	breqtrd 5112	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ≤ (Σ^‘𝐹))
140	89, 110, 111, 116, 139	xrletrd 13107	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → 0 ≤ (Σ^‘𝐹))
141	140	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑧 ∈ 𝑋 → 0 ≤ (Σ^‘𝐹)))
142	141	adantlr 716	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (𝑧 ∈ 𝑋 → 0 ≤ (Σ^‘𝐹)))
143	142	exlimdv 1935	. . . . . 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 13075	. . . . . . 7 ⊢ ((Σ^‘𝐹) ∈ ℝ* → (Σ^‘𝐹) ≤ +∞)
146	73, 145	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ≤ +∞)
147	146	adantlr 716	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ≤ +∞)
148	24, 26, 74, 144, 147	eliccxrd 45978	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ (0[,]+∞))
149	19, 21, 22, 148	syl21anc 838	. . 3 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ (0[,]+∞))
150	18, 149	pm2.61dan 813	. 2 ⊢ ((𝜑 ∧ ¬ 𝐹 = ∅) → (Σ^‘𝐹) ∈ (0[,]+∞))
151	8, 150	pm2.61dan 813	1 ⊢ (𝜑 → (Σ^‘𝐹) ∈ (0[,]+∞))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  {csn 4568   class class class wbr 5086   ↦ cmpt 5167  dom cdm 5625  ran crn 5626  Rel wrel 5630  Fun wfun 6487  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361  Fincfn 8887  supcsup 9347  ℂcc 11030  ℝcr 11031  0cc0 11032  +∞cpnf 11170  ℝ^*cxr 11172   < clt 11173   ≤ cle 11174  [,]cicc 13295  Σcsu 15642  Σ^{^}csumge0 46811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-inf2 9556  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-pre-sup 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-sup 9349  df-oi 9419  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-div 11802  df-nn 12169  df-2 12238  df-3 12239  df-n0 12432  df-z 12519  df-uz 12783  df-rp 12937  df-ico 13298  df-icc 13299  df-fz 13456  df-fzo 13603  df-seq 13958  df-exp 14018  df-hash 14287  df-cj 15055  df-re 15056  df-im 15057  df-sqrt 15191  df-abs 15192  df-clim 15444  df-sum 15643  df-sumge0 46812

This theorem is referenced by:  sge0ge0  46833  sge0xrcl  46834  sge0split  46858  sge0iunmptlemre  46864  sge0iunmpt  46867  sge0nemnf  46869  sge0clmpt  46874  sge0isum  46876  psmeasure  46920  ovnsupge0  47006  ovnsubaddlem1  47019  sge0hsphoire  47038  hoidmvlelem1  47044  hspmbllem2  47076