Theorem sge0cl 43020

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 6645	. . . . 5 ⊢ (𝐹 = ∅ → (Σ^‘𝐹) = (Σ^‘∅))
2		sge00 43015	. . . . . 6 ⊢ (Σ^‘∅) = 0
3	2	a1i 11	. . . . 5 ⊢ (𝐹 = ∅ → (Σ^‘∅) = 0)
4	1, 3	eqtrd 2833	. . . 4 ⊢ (𝐹 = ∅ → (Σ^‘𝐹) = 0)
5		0e0iccpnf 12837	. . . . 5 ⊢ 0 ∈ (0[,]+∞)
6	5	a1i 11	. . . 4 ⊢ (𝐹 = ∅ → 0 ∈ (0[,]+∞))
7	4, 6	eqeltrd 2890	. . 3 ⊢ (𝐹 = ∅ → (Σ^‘𝐹) ∈ (0[,]+∞))
8	7	adantl 485	. 2 ⊢ ((𝜑 ∧ 𝐹 = ∅) → (Σ^‘𝐹) ∈ (0[,]+∞))
9		sge0cl.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
10	9	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
11		sge0cl.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))
12	11	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
13		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
14	10, 12, 13	sge0pnfval 43012	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = +∞)
15		pnfel0pnf 42165	. . . . . 6 ⊢ +∞ ∈ (0[,]+∞)
16	15	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ (0[,]+∞))
17	14, 16	eqeltrd 2890	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ (0[,]+∞))
18	17	adantlr 714	. . 3 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ (0[,]+∞))
19		simpll 766	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 𝜑)
20		neqne 2995	. . . . 5 ⊢ (¬ 𝐹 = ∅ → 𝐹 ≠ ∅)
21	20	ad2antlr 726	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹 ≠ ∅)
22		simpr 488	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
23		0xr 10677	. . . . . 6 ⊢ 0 ∈ ℝ*
24	23	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 0 ∈ ℝ*)
25		pnfxr 10684	. . . . . 6 ⊢ +∞ ∈ ℝ*
26	25	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → +∞ ∈ ℝ*)
27	9	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
28	11	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
29		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
30	28, 29	fge0iccico 43009	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
31	27, 30	sge0reval 43011	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
32		elinel2 4123	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
33	32	adantl 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
34	11	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝐹:𝑋⟶(0[,]+∞))
35		elinel1 4122	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
36		elpwi 4506	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
37	35, 36	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ⊆ 𝑋)
38	37	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ⊆ 𝑋)
39	38	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑥 ⊆ 𝑋)
40		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
41	39, 40	sseldd 3916	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
42	34, 41	ffvelrnd 6829	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,]+∞))
43	42	adantllr 718	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,]+∞))
44		nne 2991	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (𝐹‘𝑦) ≠ +∞ ↔ (𝐹‘𝑦) = +∞)
45	44	biimpi 219	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝐹‘𝑦) ≠ +∞ → (𝐹‘𝑦) = +∞)
46	45	eqcomd 2804	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝐹‘𝑦) ≠ +∞ → +∞ = (𝐹‘𝑦))
47	46	adantl 485	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ ¬ (𝐹‘𝑦) ≠ +∞) → +∞ = (𝐹‘𝑦))
48	11	ffund 6491	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Fun 𝐹)
49	48	3ad2ant1 1130	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → Fun 𝐹)
50	41	3impa 1107	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
51	11	fdmd 6497	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom 𝐹 = 𝑋)
52	51	eqcomd 2804	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑋 = dom 𝐹)
53	52	3ad2ant1 1130	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑋 = dom 𝐹)
54	50, 53	eleqtrd 2892	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝐹)
55		fvelrn 6821	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ ran 𝐹)
56	49, 54, 55	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ran 𝐹)
57	56	ad5ant134 1364	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ ¬ (𝐹‘𝑦) ≠ +∞) → (𝐹‘𝑦) ∈ ran 𝐹)
58	47, 57	eqeltrd 2890	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ ¬ (𝐹‘𝑦) ≠ +∞) → +∞ ∈ ran 𝐹)
59	29	ad3antrrr 729	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ ¬ (𝐹‘𝑦) ≠ +∞) → ¬ +∞ ∈ ran 𝐹)
60	58, 59	condan 817	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ≠ +∞)
61		ge0xrre 42168	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑦) ∈ (0[,]+∞) ∧ (𝐹‘𝑦) ≠ +∞) → (𝐹‘𝑦) ∈ ℝ)
62	43, 60, 61	syl2anc 587	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℝ)
63	33, 62	fsumrecl 15083	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ ℝ)
64	63	ralrimiva 3149	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ ℝ)
65		eqid 2798	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
66	65	rnmptss 6863	. . . . . . . . . 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 10674	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ*
69	68	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ℝ ⊆ ℝ*)
70	67, 69	sstrd 3925	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ*)
71		supxrcl 12696	. . . . . . . 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 2890	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ ℝ*)
74	73	adantlr 714	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ ℝ*)
75	52	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → 𝑋 = dom 𝐹)
76		neneq 2993	. . . . . . . . . . . 12 ⊢ (𝐹 ≠ ∅ → ¬ 𝐹 = ∅)
77	76	adantl 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → ¬ 𝐹 = ∅)
78		frel 6492	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋⟶(0[,]+∞) → Rel 𝐹)
79	11, 78	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → Rel 𝐹)
80	79	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → Rel 𝐹)
81		reldm0 5762	. . . . . . . . . . . 12 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
82	80, 81	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
83	77, 82	mtbid 327	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → ¬ dom 𝐹 = ∅)
84	83	neqned 2994	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → dom 𝐹 ≠ ∅)
85	75, 84	eqnetrd 3054	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → 𝑋 ≠ ∅)
86		n0 4260	. . . . . . . 8 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝑋)
87	85, 86	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → ∃𝑧 𝑧 ∈ 𝑋)
88	87	adantr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → ∃𝑧 𝑧 ∈ 𝑋)
89	23	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → 0 ∈ ℝ*)
90	11	ffvelrnda 6828	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ (0[,]+∞))
91	90	adantlr 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ (0[,]+∞))
92		nne 2991	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝐹‘𝑧) ≠ +∞ ↔ (𝐹‘𝑧) = +∞)
93	92	biimpi 219	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝐹‘𝑧) ≠ +∞ → (𝐹‘𝑧) = +∞)
94	93	eqcomd 2804	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝐹‘𝑧) ≠ +∞ → +∞ = (𝐹‘𝑧))
95	94	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) ∧ ¬ (𝐹‘𝑧) ≠ +∞) → +∞ = (𝐹‘𝑧))
96	11	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐹:𝑋⟶(0[,]+∞))
97	96	ffund 6491	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → Fun 𝐹)
98		simpr 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
99	52	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑋 = dom 𝐹)
100	98, 99	eleqtrd 2892	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ dom 𝐹)
101		fvelrn 6821	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ ran 𝐹)
102	97, 100, 101	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ran 𝐹)
103	102	adantlr 714	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ran 𝐹)
104	103	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) ∧ ¬ (𝐹‘𝑧) ≠ +∞) → (𝐹‘𝑧) ∈ ran 𝐹)
105	95, 104	eqeltrd 2890	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) ∧ ¬ (𝐹‘𝑧) ≠ +∞) → +∞ ∈ ran 𝐹)
106	29	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) ∧ ¬ (𝐹‘𝑧) ≠ +∞) → ¬ +∞ ∈ ran 𝐹)
107	105, 106	condan 817	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ≠ +∞)
108		ge0xrre 42168	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑧) ∈ (0[,]+∞) ∧ (𝐹‘𝑧) ≠ +∞) → (𝐹‘𝑧) ∈ ℝ)
109	91, 107, 108	syl2anc 587	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℝ)
110	109	rexrd 10680	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℝ*)
111	73	adantr 484	. . . . . . . . . 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 12781	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹‘𝑧) ∈ (0[,]+∞)) → 0 ≤ (𝐹‘𝑧))
115	112, 113, 90, 114	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 0 ≤ (𝐹‘𝑧))
116	115	adantlr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → 0 ≤ (𝐹‘𝑧))
117	70	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ*)
118		snelpwi 5302	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑋 → {𝑧} ∈ 𝒫 𝑋)
119		snfi 8577	. . . . . . . . . . . . . . . . 17 ⊢ {𝑧} ∈ Fin
120	119	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑋 → {𝑧} ∈ Fin)
121	118, 120	elind 4121	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑋 → {𝑧} ∈ (𝒫 𝑋 ∩ Fin))
122	121	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → {𝑧} ∈ (𝒫 𝑋 ∩ Fin))
123		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
124	109	recnd 10658	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℂ)
125		fveq2 6645	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
126	125	sumsn 15093	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ ℂ) → Σ𝑦 ∈ {𝑧} (𝐹‘𝑦) = (𝐹‘𝑧))
127	123, 124, 126	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → Σ𝑦 ∈ {𝑧} (𝐹‘𝑦) = (𝐹‘𝑧))
128	127	eqcomd 2804	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) = Σ𝑦 ∈ {𝑧} (𝐹‘𝑦))
129		sumeq1 15037	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = {𝑧} → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) = Σ𝑦 ∈ {𝑧} (𝐹‘𝑦))
130	129	rspceeqv 3586	. . . . . . . . . . . . . 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 587	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹‘𝑧) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
132	65	elrnmpt 5792	. . . . . . . . . . . . . 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 260	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))
135		supxrub 12705	. . . . . . . . . . . 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 587	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ≤ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
137	31	eqcomd 2804	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ) = (Σ^‘𝐹))
138	137	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ) = (Σ^‘𝐹))
139	136, 138	breqtrd 5056	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ≤ (Σ^‘𝐹))
140	89, 110, 111, 116, 139	xrletrd 12543	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → 0 ≤ (Σ^‘𝐹))
141	140	ex 416	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑧 ∈ 𝑋 → 0 ≤ (Σ^‘𝐹)))
142	141	adantlr 714	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (𝑧 ∈ 𝑋 → 0 ≤ (Σ^‘𝐹)))
143	142	exlimdv 1934	. . . . . 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 12513	. . . . . . 7 ⊢ ((Σ^‘𝐹) ∈ ℝ* → (Σ^‘𝐹) ≤ +∞)
146	73, 145	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ≤ +∞)
147	146	adantlr 714	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ≤ +∞)
148	24, 26, 74, 144, 147	eliccxrd 42164	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ (0[,]+∞))
149	19, 21, 22, 148	syl21anc 836	. . 3 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ (0[,]+∞))
150	18, 149	pm2.61dan 812	. 2 ⊢ ((𝜑 ∧ ¬ 𝐹 = ∅) → (Σ^‘𝐹) ∈ (0[,]+∞))
151	8, 150	pm2.61dan 812	1 ⊢ (𝜑 → (Σ^‘𝐹) ∈ (0[,]+∞))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  {csn 4525   class class class wbr 5030   ↦ cmpt 5110  dom cdm 5519  ran crn 5520  Rel wrel 5524  Fun wfun 6318  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  Fincfn 8492  supcsup 8888  ℂcc 10524  ℝcr 10525  0cc0 10526  +∞cpnf 10661  ℝ^*cxr 10663   < clt 10664   ≤ cle 10665  [,]cicc 12729  Σcsu 15034  Σ^{^}csumge0 43001

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-ico 12732  df-icc 12733  df-fz 12886  df-fzo 13029  df-seq 13365  df-exp 13426  df-hash 13687  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-sum 15035  df-sumge0 43002

This theorem is referenced by:  sge0ge0  43023  sge0xrcl  43024  sge0split  43048  sge0iunmptlemre  43054  sge0iunmpt  43057  sge0nemnf  43059  sge0clmpt  43064  sge0isum  43066  psmeasure  43110  ovnsupge0  43196  ovnsubaddlem1  43209  sge0hsphoire  43228  hoidmvlelem1  43234  hspmbllem2  43266