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Theorem sge0cl 42548
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 6669 . . . . 5 (𝐹 = ∅ → (Σ^𝐹) = (Σ^‘∅))
2 sge00 42543 . . . . . 6 ^‘∅) = 0
32a1i 11 . . . . 5 (𝐹 = ∅ → (Σ^‘∅) = 0)
41, 3eqtrd 2861 . . . 4 (𝐹 = ∅ → (Σ^𝐹) = 0)
5 0e0iccpnf 12842 . . . . 5 0 ∈ (0[,]+∞)
65a1i 11 . . . 4 (𝐹 = ∅ → 0 ∈ (0[,]+∞))
74, 6eqeltrd 2918 . . 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 485 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
1410, 12, 13sge0pnfval 42540 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = +∞)
15 pnfel0pnf 41688 . . . . . 6 +∞ ∈ (0[,]+∞)
1615a1i 11 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ (0[,]+∞))
1714, 16eqeltrd 2918 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ (0[,]+∞))
1817adantlr 711 . . 3 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ (0[,]+∞))
19 simpll 763 . . . 4 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 𝜑)
20 neqne 3029 . . . . 5 𝐹 = ∅ → 𝐹 ≠ ∅)
2120ad2antlr 723 . . . 4 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹 ≠ ∅)
22 simpr 485 . . . 4 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
23 0xr 10682 . . . . . 6 0 ∈ ℝ*
2423a1i 11 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 0 ∈ ℝ*)
25 pnfxr 10689 . . . . . 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 485 . . . . . . . . 9 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
3028, 29fge0iccico 42537 . . . . . . . 8 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
3127, 30sge0reval 42539 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
32 elinel2 4177 . . . . . . . . . . . . 13 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
3332adantl 482 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
3411ad2antrr 722 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝐹:𝑋⟶(0[,]+∞))
35 elinel1 4176 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
36 elpwi 4554 . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑥)
4139, 40sseldd 3972 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
4234, 41ffvelrnd 6850 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,]+∞))
4342adantllr 715 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,]+∞))
44 nne 3025 . . . . . . . . . . . . . . . . . 18 (¬ (𝐹𝑦) ≠ +∞ ↔ (𝐹𝑦) = +∞)
4544biimpi 217 . . . . . . . . . . . . . . . . 17 (¬ (𝐹𝑦) ≠ +∞ → (𝐹𝑦) = +∞)
4645eqcomd 2832 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑦) ≠ +∞ → +∞ = (𝐹𝑦))
4746adantl 482 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ ¬ (𝐹𝑦) ≠ +∞) → +∞ = (𝐹𝑦))
4811ffund 6517 . . . . . . . . . . . . . . . . . 18 (𝜑 → Fun 𝐹)
49483ad2ant1 1127 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → Fun 𝐹)
50413impa 1104 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑋)
5111fdmd 6522 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom 𝐹 = 𝑋)
5251eqcomd 2832 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑋 = dom 𝐹)
53523ad2ant1 1127 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑋 = dom 𝐹)
5450, 53eleqtrd 2920 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑦 ∈ dom 𝐹)
55 fvelrn 6842 . . . . . . . . . . . . . . . . 17 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ ran 𝐹)
5649, 54, 55syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ran 𝐹)
5756ad5ant134 1361 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ ¬ (𝐹𝑦) ≠ +∞) → (𝐹𝑦) ∈ ran 𝐹)
5847, 57eqeltrd 2918 . . . . . . . . . . . . . 14 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ ¬ (𝐹𝑦) ≠ +∞) → +∞ ∈ ran 𝐹)
5929ad3antrrr 726 . . . . . . . . . . . . . 14 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ ¬ (𝐹𝑦) ≠ +∞) → ¬ +∞ ∈ ran 𝐹)
6058, 59condan 814 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ≠ +∞)
61 ge0xrre 41691 . . . . . . . . . . . . 13 (((𝐹𝑦) ∈ (0[,]+∞) ∧ (𝐹𝑦) ≠ +∞) → (𝐹𝑦) ∈ ℝ)
6243, 60, 61syl2anc 584 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ)
6333, 62fsumrecl 15086 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) ∈ ℝ)
6463ralrimiva 3187 . . . . . . . . . 10 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑦𝑥 (𝐹𝑦) ∈ ℝ)
65 eqid 2826 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
6665rnmptss 6884 . . . . . . . . . 10 (∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑦𝑥 (𝐹𝑦) ∈ ℝ → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ)
6764, 66syl 17 . . . . . . . . 9 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ)
68 ressxr 10679 . . . . . . . . . 10 ℝ ⊆ ℝ*
6968a1i 11 . . . . . . . . 9 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ℝ ⊆ ℝ*)
7067, 69sstrd 3981 . . . . . . . 8 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ*)
71 supxrcl 12703 . . . . . . . 8 (ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ* → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) ∈ ℝ*)
7270, 71syl 17 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) ∈ ℝ*)
7331, 72eqeltrd 2918 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ ℝ*)
7473adantlr 711 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ ℝ*)
7552adantr 481 . . . . . . . . 9 ((𝜑𝐹 ≠ ∅) → 𝑋 = dom 𝐹)
76 neneq 3027 . . . . . . . . . . . 12 (𝐹 ≠ ∅ → ¬ 𝐹 = ∅)
7776adantl 482 . . . . . . . . . . 11 ((𝜑𝐹 ≠ ∅) → ¬ 𝐹 = ∅)
78 frel 6518 . . . . . . . . . . . . . 14 (𝐹:𝑋⟶(0[,]+∞) → Rel 𝐹)
7911, 78syl 17 . . . . . . . . . . . . 13 (𝜑 → Rel 𝐹)
8079adantr 481 . . . . . . . . . . . 12 ((𝜑𝐹 ≠ ∅) → Rel 𝐹)
81 reldm0 5797 . . . . . . . . . . . 12 (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
8280, 81syl 17 . . . . . . . . . . 11 ((𝜑𝐹 ≠ ∅) → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
8377, 82mtbid 325 . . . . . . . . . 10 ((𝜑𝐹 ≠ ∅) → ¬ dom 𝐹 = ∅)
8483neqned 3028 . . . . . . . . 9 ((𝜑𝐹 ≠ ∅) → dom 𝐹 ≠ ∅)
8575, 84eqnetrd 3088 . . . . . . . 8 ((𝜑𝐹 ≠ ∅) → 𝑋 ≠ ∅)
86 n0 4314 . . . . . . . 8 (𝑋 ≠ ∅ ↔ ∃𝑧 𝑧𝑋)
8785, 86sylib 219 . . . . . . 7 ((𝜑𝐹 ≠ ∅) → ∃𝑧 𝑧𝑋)
8887adantr 481 . . . . . 6 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → ∃𝑧 𝑧𝑋)
8923a1i 11 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → 0 ∈ ℝ*)
9011ffvelrnda 6849 . . . . . . . . . . . . 13 ((𝜑𝑧𝑋) → (𝐹𝑧) ∈ (0[,]+∞))
9190adantlr 711 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ (0[,]+∞))
92 nne 3025 . . . . . . . . . . . . . . . . 17 (¬ (𝐹𝑧) ≠ +∞ ↔ (𝐹𝑧) = +∞)
9392biimpi 217 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑧) ≠ +∞ → (𝐹𝑧) = +∞)
9493eqcomd 2832 . . . . . . . . . . . . . . 15 (¬ (𝐹𝑧) ≠ +∞ → +∞ = (𝐹𝑧))
9594adantl 482 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) ∧ ¬ (𝐹𝑧) ≠ +∞) → +∞ = (𝐹𝑧))
9611adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑋) → 𝐹:𝑋⟶(0[,]+∞))
9796ffund 6517 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑋) → Fun 𝐹)
98 simpr 485 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑋) → 𝑧𝑋)
9952adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑋) → 𝑋 = dom 𝐹)
10098, 99eleqtrd 2920 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑋) → 𝑧 ∈ dom 𝐹)
101 fvelrn 6842 . . . . . . . . . . . . . . . . 17 ((Fun 𝐹𝑧 ∈ dom 𝐹) → (𝐹𝑧) ∈ ran 𝐹)
10297, 100, 101syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑋) → (𝐹𝑧) ∈ ran 𝐹)
103102adantlr 711 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ran 𝐹)
104103adantr 481 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) ∧ ¬ (𝐹𝑧) ≠ +∞) → (𝐹𝑧) ∈ ran 𝐹)
10595, 104eqeltrd 2918 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) ∧ ¬ (𝐹𝑧) ≠ +∞) → +∞ ∈ ran 𝐹)
10629ad2antrr 722 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) ∧ ¬ (𝐹𝑧) ≠ +∞) → ¬ +∞ ∈ ran 𝐹)
107105, 106condan 814 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ≠ +∞)
108 ge0xrre 41691 . . . . . . . . . . . 12 (((𝐹𝑧) ∈ (0[,]+∞) ∧ (𝐹𝑧) ≠ +∞) → (𝐹𝑧) ∈ ℝ)
10991, 107, 108syl2anc 584 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℝ)
110109rexrd 10685 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℝ*)
11173adantr 481 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (Σ^𝐹) ∈ ℝ*)
11223a1i 11 . . . . . . . . . . . 12 ((𝜑𝑧𝑋) → 0 ∈ ℝ*)
11325a1i 11 . . . . . . . . . . . 12 ((𝜑𝑧𝑋) → +∞ ∈ ℝ*)
114 iccgelb 12788 . . . . . . . . . . . 12 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹𝑧) ∈ (0[,]+∞)) → 0 ≤ (𝐹𝑧))
115112, 113, 90, 114syl3anc 1365 . . . . . . . . . . 11 ((𝜑𝑧𝑋) → 0 ≤ (𝐹𝑧))
116115adantlr 711 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → 0 ≤ (𝐹𝑧))
11770adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ*)
118 snelpwi 5333 . . . . . . . . . . . . . . . 16 (𝑧𝑋 → {𝑧} ∈ 𝒫 𝑋)
119 snfi 8588 . . . . . . . . . . . . . . . . 17 {𝑧} ∈ Fin
120119a1i 11 . . . . . . . . . . . . . . . 16 (𝑧𝑋 → {𝑧} ∈ Fin)
121118, 120elind 4175 . . . . . . . . . . . . . . 15 (𝑧𝑋 → {𝑧} ∈ (𝒫 𝑋 ∩ Fin))
122121adantl 482 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → {𝑧} ∈ (𝒫 𝑋 ∩ Fin))
123 simpr 485 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → 𝑧𝑋)
124109recnd 10663 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℂ)
125 fveq2 6669 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
126125sumsn 15096 . . . . . . . . . . . . . . . 16 ((𝑧𝑋 ∧ (𝐹𝑧) ∈ ℂ) → Σ𝑦 ∈ {𝑧} (𝐹𝑦) = (𝐹𝑧))
127123, 124, 126syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → Σ𝑦 ∈ {𝑧} (𝐹𝑦) = (𝐹𝑧))
128127eqcomd 2832 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) = Σ𝑦 ∈ {𝑧} (𝐹𝑦))
129 sumeq1 15040 . . . . . . . . . . . . . . 15 (𝑥 = {𝑧} → Σ𝑦𝑥 (𝐹𝑦) = Σ𝑦 ∈ {𝑧} (𝐹𝑦))
130129rspceeqv 3642 . . . . . . . . . . . . . 14 (({𝑧} ∈ (𝒫 𝑋 ∩ Fin) ∧ (𝐹𝑧) = Σ𝑦 ∈ {𝑧} (𝐹𝑦)) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑧) = Σ𝑦𝑥 (𝐹𝑦))
131122, 128, 130syl2anc 584 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑧) = Σ𝑦𝑥 (𝐹𝑦))
13265elrnmpt 5827 . . . . . . . . . . . . . 14 ((𝐹𝑧) ∈ (0[,]+∞) → ((𝐹𝑧) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑧) = Σ𝑦𝑥 (𝐹𝑦)))
13391, 132syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑧) = Σ𝑦𝑥 (𝐹𝑦)))
134131, 133mpbird 258 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
135 supxrub 12712 . . . . . . . . . . . 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*, < ))
136117, 134, 135syl2anc 584 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ≤ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
13731eqcomd 2832 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) = (Σ^𝐹))
138137adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) = (Σ^𝐹))
139136, 138breqtrd 5089 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ≤ (Σ^𝐹))
14089, 110, 111, 116, 139xrletrd 12550 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → 0 ≤ (Σ^𝐹))
141140ex 413 . . . . . . . 8 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑧𝑋 → 0 ≤ (Σ^𝐹)))
142141adantlr 711 . . . . . . 7 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (𝑧𝑋 → 0 ≤ (Σ^𝐹)))
143142exlimdv 1927 . . . . . 6 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (∃𝑧 𝑧𝑋 → 0 ≤ (Σ^𝐹)))
14488, 143mpd 15 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 0 ≤ (Σ^𝐹))
145 pnfge 12520 . . . . . . 7 ((Σ^𝐹) ∈ ℝ* → (Σ^𝐹) ≤ +∞)
14673, 145syl 17 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ≤ +∞)
147146adantlr 711 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ≤ +∞)
14824, 26, 74, 144, 147eliccxrd 41687 . . . 4 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ (0[,]+∞))
14919, 21, 22, 148syl21anc 835 . . 3 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ (0[,]+∞))
15018, 149pm2.61dan 809 . 2 ((𝜑 ∧ ¬ 𝐹 = ∅) → (Σ^𝐹) ∈ (0[,]+∞))
1518, 150pm2.61dan 809 1 (𝜑 → (Σ^𝐹) ∈ (0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2107  wne 3021  wral 3143  wrex 3144  cin 3939  wss 3940  c0 4295  𝒫 cpw 4542  {csn 4564   class class class wbr 5063  cmpt 5143  dom cdm 5554  ran crn 5555  Rel wrel 5559  Fun wfun 6348  wf 6350  cfv 6354  (class class class)co 7150  Fincfn 8503  supcsup 8898  cc 10529  cr 10530  0cc0 10531  +∞cpnf 10666  *cxr 10668   < clt 10669  cle 10670  [,]cicc 12736  Σcsu 15037  Σ^csumge0 42529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-inf2 9098  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-oi 8968  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-rp 12385  df-ico 12739  df-icc 12740  df-fz 12888  df-fzo 13029  df-seq 13365  df-exp 13425  df-hash 13686  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-sum 15038  df-sumge0 42530
This theorem is referenced by:  sge0ge0  42551  sge0xrcl  42552  sge0split  42576  sge0iunmptlemre  42582  sge0iunmpt  42585  sge0nemnf  42587  sge0clmpt  42592  sge0isum  42594  psmeasure  42638  ovnsupge0  42724  ovnsubaddlem1  42737  sge0hsphoire  42756  hoidmvlelem1  42762  hspmbllem2  42794
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