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Theorem sge0cl 39931
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 (𝜑𝑋𝑉)
sge0cl.f (𝜑𝐹: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.
StepHypRef Expression
1 fveq2 6153 . . . . 5 (𝐹 = ∅ → (Σ^𝐹) = (Σ^‘∅))
2 sge00 39926 . . . . . 6 ^‘∅) = 0
32a1i 11 . . . . 5 (𝐹 = ∅ → (Σ^‘∅) = 0)
41, 3eqtrd 2655 . . . 4 (𝐹 = ∅ → (Σ^𝐹) = 0)
5 0e0iccpnf 12233 . . . . 5 0 ∈ (0[,]+∞)
65a1i 11 . . . 4 (𝐹 = ∅ → 0 ∈ (0[,]+∞))
74, 6eqeltrd 2698 . . 3 (𝐹 = ∅ → (Σ^𝐹) ∈ (0[,]+∞))
87adantl 482 . 2 ((𝜑𝐹 = ∅) → (Σ^𝐹) ∈ (0[,]+∞))
9 sge0cl.x . . . . . . 7 (𝜑𝑋𝑉)
109adantr 481 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋𝑉)
11 sge0cl.f . . . . . . 7 (𝜑𝐹:𝑋⟶(0[,]+∞))
1211adantr 481 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
13 simpr 477 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
1410, 12, 13sge0pnfval 39923 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = +∞)
15 pnfel0pnf 39196 . . . . . 6 +∞ ∈ (0[,]+∞)
1615a1i 11 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ (0[,]+∞))
1714, 16eqeltrd 2698 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ (0[,]+∞))
1817adantlr 750 . . 3 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ (0[,]+∞))
19 simpll 789 . . . 4 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 𝜑)
20 neqne 2798 . . . . 5 𝐹 = ∅ → 𝐹 ≠ ∅)
2120ad2antlr 762 . . . 4 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹 ≠ ∅)
22 simpr 477 . . . 4 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
23 0xr 10038 . . . . . 6 0 ∈ ℝ*
2423a1i 11 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 0 ∈ ℝ*)
25 pnfxr 10044 . . . . . 6 +∞ ∈ ℝ*
2625a1i 11 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → +∞ ∈ ℝ*)
279adantr 481 . . . . . . . 8 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋𝑉)
2811adantr 481 . . . . . . . . 9 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
29 simpr 477 . . . . . . . . 9 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
3028, 29fge0iccico 39920 . . . . . . . 8 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
3127, 30sge0reval 39922 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
32 elinel2 3783 . . . . . . . . . . . . 13 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
3332adantl 482 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
3411ad2antrr 761 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝐹:𝑋⟶(0[,]+∞))
35 elinel1 3782 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
36 elpwi 4145 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
3735, 36syl 17 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥𝑋)
3837adantl 482 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑋)
3938adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑥𝑋)
40 simpr 477 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑥)
4139, 40sseldd 3588 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
4234, 41ffvelrnd 6321 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,]+∞))
4342adantllr 754 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,]+∞))
44 nne 2794 . . . . . . . . . . . . . . . . . 18 (¬ (𝐹𝑦) ≠ +∞ ↔ (𝐹𝑦) = +∞)
4544biimpi 206 . . . . . . . . . . . . . . . . 17 (¬ (𝐹𝑦) ≠ +∞ → (𝐹𝑦) = +∞)
4645eqcomd 2627 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑦) ≠ +∞ → +∞ = (𝐹𝑦))
4746adantl 482 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ ¬ (𝐹𝑦) ≠ +∞) → +∞ = (𝐹𝑦))
48 ffun 6010 . . . . . . . . . . . . . . . . . . 19 (𝐹:𝑋⟶(0[,]+∞) → Fun 𝐹)
4911, 48syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → Fun 𝐹)
50493ad2ant1 1080 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → Fun 𝐹)
51413impa 1256 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑋)
52 fdm 6013 . . . . . . . . . . . . . . . . . . . . 21 (𝐹:𝑋⟶(0[,]+∞) → dom 𝐹 = 𝑋)
5311, 52syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom 𝐹 = 𝑋)
5453eqcomd 2627 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑋 = dom 𝐹)
55543ad2ant1 1080 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑋 = dom 𝐹)
5651, 55eleqtrd 2700 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑦 ∈ dom 𝐹)
57 fvelrn 6313 . . . . . . . . . . . . . . . . 17 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ ran 𝐹)
5850, 56, 57syl2anc 692 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ran 𝐹)
5958ad5ant134 1310 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ ¬ (𝐹𝑦) ≠ +∞) → (𝐹𝑦) ∈ ran 𝐹)
6047, 59eqeltrd 2698 . . . . . . . . . . . . . 14 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ ¬ (𝐹𝑦) ≠ +∞) → +∞ ∈ ran 𝐹)
6129ad3antrrr 765 . . . . . . . . . . . . . 14 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ ¬ (𝐹𝑦) ≠ +∞) → ¬ +∞ ∈ ran 𝐹)
6260, 61condan 834 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ≠ +∞)
63 ge0xrre 39200 . . . . . . . . . . . . 13 (((𝐹𝑦) ∈ (0[,]+∞) ∧ (𝐹𝑦) ≠ +∞) → (𝐹𝑦) ∈ ℝ)
6443, 62, 63syl2anc 692 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ)
6533, 64fsumrecl 14406 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) ∈ ℝ)
6665ralrimiva 2961 . . . . . . . . . 10 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑦𝑥 (𝐹𝑦) ∈ ℝ)
67 eqid 2621 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
6867rnmptss 6353 . . . . . . . . . 10 (∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑦𝑥 (𝐹𝑦) ∈ ℝ → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ)
6966, 68syl 17 . . . . . . . . 9 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ)
70 ressxr 10035 . . . . . . . . . 10 ℝ ⊆ ℝ*
7170a1i 11 . . . . . . . . 9 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ℝ ⊆ ℝ*)
7269, 71sstrd 3597 . . . . . . . 8 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ*)
73 supxrcl 12096 . . . . . . . 8 (ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ* → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) ∈ ℝ*)
7472, 73syl 17 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) ∈ ℝ*)
7531, 74eqeltrd 2698 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ ℝ*)
7675adantlr 750 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ ℝ*)
7754adantr 481 . . . . . . . . 9 ((𝜑𝐹 ≠ ∅) → 𝑋 = dom 𝐹)
78 neneq 2796 . . . . . . . . . . . 12 (𝐹 ≠ ∅ → ¬ 𝐹 = ∅)
7978adantl 482 . . . . . . . . . . 11 ((𝜑𝐹 ≠ ∅) → ¬ 𝐹 = ∅)
80 frel 6012 . . . . . . . . . . . . . 14 (𝐹:𝑋⟶(0[,]+∞) → Rel 𝐹)
8111, 80syl 17 . . . . . . . . . . . . 13 (𝜑 → Rel 𝐹)
8281adantr 481 . . . . . . . . . . . 12 ((𝜑𝐹 ≠ ∅) → Rel 𝐹)
83 reldm0 5308 . . . . . . . . . . . 12 (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
8482, 83syl 17 . . . . . . . . . . 11 ((𝜑𝐹 ≠ ∅) → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
8579, 84mtbid 314 . . . . . . . . . 10 ((𝜑𝐹 ≠ ∅) → ¬ dom 𝐹 = ∅)
8685neqned 2797 . . . . . . . . 9 ((𝜑𝐹 ≠ ∅) → dom 𝐹 ≠ ∅)
8777, 86eqnetrd 2857 . . . . . . . 8 ((𝜑𝐹 ≠ ∅) → 𝑋 ≠ ∅)
88 n0 3912 . . . . . . . 8 (𝑋 ≠ ∅ ↔ ∃𝑧 𝑧𝑋)
8987, 88sylib 208 . . . . . . 7 ((𝜑𝐹 ≠ ∅) → ∃𝑧 𝑧𝑋)
9089adantr 481 . . . . . 6 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → ∃𝑧 𝑧𝑋)
9123a1i 11 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → 0 ∈ ℝ*)
9211ffvelrnda 6320 . . . . . . . . . . . . 13 ((𝜑𝑧𝑋) → (𝐹𝑧) ∈ (0[,]+∞))
9392adantlr 750 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ (0[,]+∞))
94 nne 2794 . . . . . . . . . . . . . . . . 17 (¬ (𝐹𝑧) ≠ +∞ ↔ (𝐹𝑧) = +∞)
9594biimpi 206 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑧) ≠ +∞ → (𝐹𝑧) = +∞)
9695eqcomd 2627 . . . . . . . . . . . . . . 15 (¬ (𝐹𝑧) ≠ +∞ → +∞ = (𝐹𝑧))
9796adantl 482 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) ∧ ¬ (𝐹𝑧) ≠ +∞) → +∞ = (𝐹𝑧))
9811adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑋) → 𝐹:𝑋⟶(0[,]+∞))
9998, 48syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑋) → Fun 𝐹)
100 simpr 477 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑋) → 𝑧𝑋)
10154adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑋) → 𝑋 = dom 𝐹)
102100, 101eleqtrd 2700 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑋) → 𝑧 ∈ dom 𝐹)
103 fvelrn 6313 . . . . . . . . . . . . . . . . 17 ((Fun 𝐹𝑧 ∈ dom 𝐹) → (𝐹𝑧) ∈ ran 𝐹)
10499, 102, 103syl2anc 692 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑋) → (𝐹𝑧) ∈ ran 𝐹)
105104adantlr 750 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ran 𝐹)
106105adantr 481 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) ∧ ¬ (𝐹𝑧) ≠ +∞) → (𝐹𝑧) ∈ ran 𝐹)
10797, 106eqeltrd 2698 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) ∧ ¬ (𝐹𝑧) ≠ +∞) → +∞ ∈ ran 𝐹)
10829ad2antrr 761 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) ∧ ¬ (𝐹𝑧) ≠ +∞) → ¬ +∞ ∈ ran 𝐹)
109107, 108condan 834 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ≠ +∞)
110 ge0xrre 39200 . . . . . . . . . . . 12 (((𝐹𝑧) ∈ (0[,]+∞) ∧ (𝐹𝑧) ≠ +∞) → (𝐹𝑧) ∈ ℝ)
11193, 109, 110syl2anc 692 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℝ)
112111rexrd 10041 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℝ*)
11375adantr 481 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (Σ^𝐹) ∈ ℝ*)
11423a1i 11 . . . . . . . . . . . 12 ((𝜑𝑧𝑋) → 0 ∈ ℝ*)
11525a1i 11 . . . . . . . . . . . 12 ((𝜑𝑧𝑋) → +∞ ∈ ℝ*)
116 iccgelb 12180 . . . . . . . . . . . 12 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹𝑧) ∈ (0[,]+∞)) → 0 ≤ (𝐹𝑧))
117114, 115, 92, 116syl3anc 1323 . . . . . . . . . . 11 ((𝜑𝑧𝑋) → 0 ≤ (𝐹𝑧))
118117adantlr 750 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → 0 ≤ (𝐹𝑧))
11972adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ*)
120 snelpwi 4878 . . . . . . . . . . . . . . . 16 (𝑧𝑋 → {𝑧} ∈ 𝒫 𝑋)
121 snfi 7990 . . . . . . . . . . . . . . . . 17 {𝑧} ∈ Fin
122121a1i 11 . . . . . . . . . . . . . . . 16 (𝑧𝑋 → {𝑧} ∈ Fin)
123120, 122elind 3781 . . . . . . . . . . . . . . 15 (𝑧𝑋 → {𝑧} ∈ (𝒫 𝑋 ∩ Fin))
124123adantl 482 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → {𝑧} ∈ (𝒫 𝑋 ∩ Fin))
125 simpr 477 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → 𝑧𝑋)
126111recnd 10020 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℂ)
127 fveq2 6153 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
128127sumsn 14416 . . . . . . . . . . . . . . . 16 ((𝑧𝑋 ∧ (𝐹𝑧) ∈ ℂ) → Σ𝑦 ∈ {𝑧} (𝐹𝑦) = (𝐹𝑧))
129125, 126, 128syl2anc 692 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → Σ𝑦 ∈ {𝑧} (𝐹𝑦) = (𝐹𝑧))
130129eqcomd 2627 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) = Σ𝑦 ∈ {𝑧} (𝐹𝑦))
131 sumeq1 14361 . . . . . . . . . . . . . . . 16 (𝑥 = {𝑧} → Σ𝑦𝑥 (𝐹𝑦) = Σ𝑦 ∈ {𝑧} (𝐹𝑦))
132131eqeq2d 2631 . . . . . . . . . . . . . . 15 (𝑥 = {𝑧} → ((𝐹𝑧) = Σ𝑦𝑥 (𝐹𝑦) ↔ (𝐹𝑧) = Σ𝑦 ∈ {𝑧} (𝐹𝑦)))
133132rspcev 3298 . . . . . . . . . . . . . 14 (({𝑧} ∈ (𝒫 𝑋 ∩ Fin) ∧ (𝐹𝑧) = Σ𝑦 ∈ {𝑧} (𝐹𝑦)) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑧) = Σ𝑦𝑥 (𝐹𝑦))
134124, 130, 133syl2anc 692 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑧) = Σ𝑦𝑥 (𝐹𝑦))
13567elrnmpt 5337 . . . . . . . . . . . . . 14 ((𝐹𝑧) ∈ (0[,]+∞) → ((𝐹𝑧) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑧) = Σ𝑦𝑥 (𝐹𝑦)))
13693, 135syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑧) = Σ𝑦𝑥 (𝐹𝑦)))
137134, 136mpbird 247 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
138 supxrub 12105 . . . . . . . . . . . 12 ((ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ* ∧ (𝐹𝑧) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → (𝐹𝑧) ≤ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
139119, 137, 138syl2anc 692 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ≤ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
14031eqcomd 2627 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) = (Σ^𝐹))
141140adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) = (Σ^𝐹))
142139, 141breqtrd 4644 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ≤ (Σ^𝐹))
14391, 112, 113, 118, 142xrletrd 11945 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → 0 ≤ (Σ^𝐹))
144143ex 450 . . . . . . . 8 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑧𝑋 → 0 ≤ (Σ^𝐹)))
145144adantlr 750 . . . . . . 7 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (𝑧𝑋 → 0 ≤ (Σ^𝐹)))
146145exlimdv 1858 . . . . . 6 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (∃𝑧 𝑧𝑋 → 0 ≤ (Σ^𝐹)))
14790, 146mpd 15 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 0 ≤ (Σ^𝐹))
148 pnfge 11916 . . . . . . 7 ((Σ^𝐹) ∈ ℝ* → (Σ^𝐹) ≤ +∞)
14975, 148syl 17 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ≤ +∞)
150149adantlr 750 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ≤ +∞)
15124, 26, 76, 147, 150eliccxrd 39195 . . . 4 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ (0[,]+∞))
15219, 21, 22, 151syl21anc 1322 . . 3 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ (0[,]+∞))
15318, 152pm2.61dan 831 . 2 ((𝜑 ∧ ¬ 𝐹 = ∅) → (Σ^𝐹) ∈ (0[,]+∞))
1548, 153pm2.61dan 831 1 (𝜑 → (Σ^𝐹) ∈ (0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135  {csn 4153   class class class wbr 4618  cmpt 4678  dom cdm 5079  ran crn 5080  Rel wrel 5084  Fun wfun 5846  wf 5848  cfv 5852  (class class class)co 6610  Fincfn 7907  supcsup 8298  cc 9886  cr 9887  0cc0 9888  +∞cpnf 10023  *cxr 10025   < clt 10026  cle 10027  [,]cicc 12128  Σcsu 14358  Σ^csumge0 39912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-sup 8300  df-oi 8367  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-n0 11245  df-z 11330  df-uz 11640  df-rp 11785  df-ico 12131  df-icc 12132  df-fz 12277  df-fzo 12415  df-seq 12750  df-exp 12809  df-hash 13066  df-cj 13781  df-re 13782  df-im 13783  df-sqrt 13917  df-abs 13918  df-clim 14161  df-sum 14359  df-sumge0 39913
This theorem is referenced by:  sge0ge0  39934  sge0xrcl  39935  sge0split  39959  sge0iunmptlemre  39965  sge0iunmpt  39968  sge0nemnf  39970  sge0clmpt  39975  sge0isum  39977  psmeasure  40021  ovnsupge0  40104  ovnsubaddlem1  40117  sge0hsphoire  40136  hoidmvlelem1  40142  hspmbllem2  40174
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