Theorem sge0cl 47664

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 6916	. . . . 5 ⊢ (𝐹 = ∅ → (Σ^‘𝐹) = (Σ^‘∅))
2		sge00 47659	. . . . . 6 ⊢ (Σ^‘∅) = 0
3	2	a1i 11	. . . . 5 ⊢ (𝐹 = ∅ → (Σ^‘∅) = 0)
4	1, 3	eqtrd 2798	. . . 4 ⊢ (𝐹 = ∅ → (Σ^‘𝐹) = 0)
5		0e0iccpnf 13630	. . . . 5 ⊢ 0 ∈ (0[,]+∞)
6	5	a1i 11	. . . 4 ⊢ (𝐹 = ∅ → 0 ∈ (0[,]+∞))
7	4, 6	eqeltrd 2864	. . 3 ⊢ (𝐹 = ∅ → (Σ^‘𝐹) ∈ (0[,]+∞))
8	7	adantl 487	. 2 ⊢ ((𝜑 ∧ 𝐹 = ∅) → (Σ^‘𝐹) ∈ (0[,]+∞))
9		sge0cl.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
10	9	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
11		sge0cl.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))
12	11	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
13		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
14	10, 12, 13	sge0pnfval 47656	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = +∞)
15		pnfel0pnf 46813	. . . . . 6 ⊢ +∞ ∈ (0[,]+∞)
16	15	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ (0[,]+∞))
17	14, 16	eqeltrd 2864	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ (0[,]+∞))
18	17	adantlr 728	. . 3 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ (0[,]+∞))
19		simpll 779	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 𝜑)
20		neqne 2967	. . . . 5 ⊢ (¬ 𝐹 = ∅ → 𝐹 ≠ ∅)
21	20	ad2antlr 740	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹 ≠ ∅)
22		simpr 490	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
23		0xr 11395	. . . . . 6 ⊢ 0 ∈ ℝ*
24	23	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 0 ∈ ℝ*)
25		pnfxr 11402	. . . . . 6 ⊢ +∞ ∈ ℝ*
26	25	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → +∞ ∈ ℝ*)
27	9	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
28	11	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
29		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
30	28, 29	fge0iccico 47653	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
31	27, 30	sge0reval 47655	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
32		elinel2 4102	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
33	32	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
34	11	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝐹:𝑋⟶(0[,]+∞))
35		elinel1 4101	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
36		elpwi 4567	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
37	35, 36	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ⊆ 𝑋)
38	37	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ⊆ 𝑋)
39	38	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑥 ⊆ 𝑋)
40		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
41	39, 40	sseldd 3914	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
42	34, 41	ffvelcdmd 7117	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,]+∞))
43	42	adantllr 732	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,]+∞))
44		nne 2963	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (𝐹‘𝑦) ≠ +∞ ↔ (𝐹‘𝑦) = +∞)
45	44	biimpi 219	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝐹‘𝑦) ≠ +∞ → (𝐹‘𝑦) = +∞)
46	45	eqcomd 2769	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝐹‘𝑦) ≠ +∞ → +∞ = (𝐹‘𝑦))
47	46	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ ¬ (𝐹‘𝑦) ≠ +∞) → +∞ = (𝐹‘𝑦))
48	11	ffund 6744	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Fun 𝐹)
49	48	3ad2ant1 1151	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → Fun 𝐹)
50	41	3impa 1127	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
51	11	fdmd 6750	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom 𝐹 = 𝑋)
52	51	eqcomd 2769	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑋 = dom 𝐹)
53	52	3ad2ant1 1151	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑋 = dom 𝐹)
54	50, 53	eleqtrd 2866	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝐹)
55		fvelrn 7108	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ ran 𝐹)
56	49, 54, 55	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ran 𝐹)
57	56	ad5ant134 1393	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ ¬ (𝐹‘𝑦) ≠ +∞) → (𝐹‘𝑦) ∈ ran 𝐹)
58	47, 57	eqeltrd 2864	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ ¬ (𝐹‘𝑦) ≠ +∞) → +∞ ∈ ran 𝐹)
59	29	ad3antrrr 743	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ ¬ (𝐹‘𝑦) ≠ +∞) → ¬ +∞ ∈ ran 𝐹)
60	58, 59	condan 830	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ≠ +∞)
61		ge0xrre 46816	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑦) ∈ (0[,]+∞) ∧ (𝐹‘𝑦) ≠ +∞) → (𝐹‘𝑦) ∈ ℝ)
62	43, 60, 61	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℝ)
63	33, 62	fsumrecl 15944	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ ℝ)
64	63	ralrimiva 3158	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ ℝ)
65		eqid 2763	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
66	65	rnmptss 7155	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ ℝ → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ)
67	64, 66	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ)
68		ressxr 11392	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ*
69	68	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ℝ ⊆ ℝ*)
70	67, 69	sstrd 3923	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ*)
71		supxrcl 13485	. . . . . . . 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 18	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ) ∈ ℝ*)
73	31, 72	eqeltrd 2864	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ ℝ*)
74	73	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ ℝ*)
75	52	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → 𝑋 = dom 𝐹)
76		neneq 2965	. . . . . . . . . . . 12 ⊢ (𝐹 ≠ ∅ → ¬ 𝐹 = ∅)
77	76	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → ¬ 𝐹 = ∅)
78		frel 6745	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋⟶(0[,]+∞) → Rel 𝐹)
79	11, 78	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → Rel 𝐹)
80	79	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → Rel 𝐹)
81		reldm0 5922	. . . . . . . . . . . 12 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
82	80, 81	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
83	77, 82	mtbid 327	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → ¬ dom 𝐹 = ∅)
84	83	neqned 2966	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → dom 𝐹 ≠ ∅)
85	75, 84	eqnetrd 3026	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → 𝑋 ≠ ∅)
86		n0 4303	. . . . . . . 8 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝑋)
87	85, 86	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → ∃𝑧 𝑧 ∈ 𝑋)
88	87	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → ∃𝑧 𝑧 ∈ 𝑋)
89	23	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → 0 ∈ ℝ*)
90	11	ffvelcdmda 7116	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ (0[,]+∞))
91	90	adantlr 728	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ (0[,]+∞))
92		nne 2963	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝐹‘𝑧) ≠ +∞ ↔ (𝐹‘𝑧) = +∞)
93	92	biimpi 219	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝐹‘𝑧) ≠ +∞ → (𝐹‘𝑧) = +∞)
94	93	eqcomd 2769	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝐹‘𝑧) ≠ +∞ → +∞ = (𝐹‘𝑧))
95	94	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) ∧ ¬ (𝐹‘𝑧) ≠ +∞) → +∞ = (𝐹‘𝑧))
96	11	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐹:𝑋⟶(0[,]+∞))
97	96	ffund 6744	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → Fun 𝐹)
98		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
99	52	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑋 = dom 𝐹)
100	98, 99	eleqtrd 2866	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ dom 𝐹)
101		fvelrn 7108	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ ran 𝐹)
102	97, 100, 101	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ran 𝐹)
103	102	adantlr 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ran 𝐹)
104	103	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) ∧ ¬ (𝐹‘𝑧) ≠ +∞) → (𝐹‘𝑧) ∈ ran 𝐹)
105	95, 104	eqeltrd 2864	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) ∧ ¬ (𝐹‘𝑧) ≠ +∞) → +∞ ∈ ran 𝐹)
106	29	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) ∧ ¬ (𝐹‘𝑧) ≠ +∞) → ¬ +∞ ∈ ran 𝐹)
107	105, 106	condan 830	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ≠ +∞)
108		ge0xrre 46816	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑧) ∈ (0[,]+∞) ∧ (𝐹‘𝑧) ≠ +∞) → (𝐹‘𝑧) ∈ ℝ)
109	91, 107, 108	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℝ)
110	109	rexrd 11398	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℝ*)
111	73	adantr 486	. . . . . . . . . 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 13573	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹‘𝑧) ∈ (0[,]+∞)) → 0 ≤ (𝐹‘𝑧))
115	112, 113, 90, 114	syl3anc 1399	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 0 ≤ (𝐹‘𝑧))
116	115	adantlr 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → 0 ≤ (𝐹‘𝑧))
117	70	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ*)
118		snelpwi 5419	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑋 → {𝑧} ∈ 𝒫 𝑋)
119		snfi 9109	. . . . . . . . . . . . . . . . 17 ⊢ {𝑧} ∈ Fin
120	119	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑋 → {𝑧} ∈ Fin)
121	118, 120	elind 4100	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑋 → {𝑧} ∈ (𝒫 𝑋 ∩ Fin))
122	121	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → {𝑧} ∈ (𝒫 𝑋 ∩ Fin))
123		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
124	109	recnd 11376	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℂ)
125		fveq2 6916	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
126	125	sumsn 15956	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ ℂ) → Σ𝑦 ∈ {𝑧} (𝐹‘𝑦) = (𝐹‘𝑧))
127	123, 124, 126	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → Σ𝑦 ∈ {𝑧} (𝐹‘𝑦) = (𝐹‘𝑧))
128	127	eqcomd 2769	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) = Σ𝑦 ∈ {𝑧} (𝐹‘𝑦))
129		sumeq1 15900	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = {𝑧} → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) = Σ𝑦 ∈ {𝑧} (𝐹‘𝑦))
130	129	rspceeqv 3602	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹‘𝑧) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
132	65	elrnmpt 5952	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 13494	. . . . . . . . . . . 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 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ≤ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
137	31	eqcomd 2769	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ) = (Σ^‘𝐹))
138	137	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ) = (Σ^‘𝐹))
139	136, 138	breqtrd 5137	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ≤ (Σ^‘𝐹))
140	89, 110, 111, 116, 139	xrletrd 13331	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧 ∈ 𝑋) → 0 ≤ (Σ^‘𝐹))
141	140	ex 418	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑧 ∈ 𝑋 → 0 ≤ (Σ^‘𝐹)))
142	141	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (𝑧 ∈ 𝑋 → 0 ≤ (Σ^‘𝐹)))
143	142	exlimdv 1968	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (∃𝑧 𝑧 ∈ 𝑋 → 0 ≤ (Σ^‘𝐹)))
144	88, 143	mpd 16	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 0 ≤ (Σ^‘𝐹))
145		pnfge 13299	. . . . . . 7 ⊢ ((Σ^‘𝐹) ∈ ℝ* → (Σ^‘𝐹) ≤ +∞)
146	73, 145	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ≤ +∞)
147	146	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ≤ +∞)
148	24, 26, 74, 144, 147	eliccxrd 46812	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ (0[,]+∞))
149	19, 21, 22, 148	syl21anc 851	. . 3 ⊢ (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) ∈ (0[,]+∞))
150	18, 149	pm2.61dan 825	. 2 ⊢ ((𝜑 ∧ ¬ 𝐹 = ∅) → (Σ^‘𝐹) ∈ (0[,]+∞))
151	8, 150	pm2.61dan 825	1 ⊢ (𝜑 → (Σ^‘𝐹) ∈ (0[,]+∞))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  𝒫 cpw 4560  {csn 4588   class class class wbr 5108   ↦ cmpt 5192  dom cdm 5658  ran crn 5659  Rel wrel 5663  Fun wfun 6560  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  Fincfn 9011  supcsup 9469  ℂcc 11237  ℝcr 11238  0cc0 11239  +∞cpnf 11379  ℝ^*cxr 11381   < clt 11382   ≤ cle 11383  [,]cicc 13519  Σcsu 15897  Σ^{^}csumge0 47645

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-sup 9471  df-oi 9541  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-n0 12647  df-z 12734  df-uz 13006  df-rp 13161  df-ico 13522  df-icc 13523  df-fz 13680  df-fzo 13829  df-seq 14185  df-exp 14245  df-hash 14516  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-abs 15446  df-clim 15699  df-sum 15898  df-sumge0 47646

This theorem is used by:  sge0ge0  47667  sge0xrcl  47668  sge0split  47692  sge0iunmptlemre  47698  sge0iunmpt  47701  sge0nemnf  47703  sge0clmpt  47708  sge0isum  47710  psmeasure  47754  ovnsupge0  47840  ovnsubaddlem1  47853  sge0hsphoire  47872  hoidmvlelem1  47878  hspmbllem2  47910