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Theorem sge0cl 46401
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 6905 . . . . 5 (𝐹 = ∅ → (Σ^𝐹) = (Σ^‘∅))
2 sge00 46396 . . . . . 6 ^‘∅) = 0
32a1i 11 . . . . 5 (𝐹 = ∅ → (Σ^‘∅) = 0)
41, 3eqtrd 2776 . . . 4 (𝐹 = ∅ → (Σ^𝐹) = 0)
5 0e0iccpnf 13500 . . . . 5 0 ∈ (0[,]+∞)
65a1i 11 . . . 4 (𝐹 = ∅ → 0 ∈ (0[,]+∞))
74, 6eqeltrd 2840 . . 3 (𝐹 = ∅ → (Σ^𝐹) ∈ (0[,]+∞))
87adantl 481 . 2 ((𝜑𝐹 = ∅) → (Σ^𝐹) ∈ (0[,]+∞))
9 sge0cl.x . . . . . . 7 (𝜑𝑋𝑉)
109adantr 480 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋𝑉)
11 sge0cl.f . . . . . . 7 (𝜑𝐹:𝑋⟶(0[,]+∞))
1211adantr 480 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
13 simpr 484 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
1410, 12, 13sge0pnfval 46393 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = +∞)
15 pnfel0pnf 45546 . . . . . 6 +∞ ∈ (0[,]+∞)
1615a1i 11 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ (0[,]+∞))
1714, 16eqeltrd 2840 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ (0[,]+∞))
1817adantlr 715 . . 3 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ (0[,]+∞))
19 simpll 766 . . . 4 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 𝜑)
20 neqne 2947 . . . . 5 𝐹 = ∅ → 𝐹 ≠ ∅)
2120ad2antlr 727 . . . 4 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹 ≠ ∅)
22 simpr 484 . . . 4 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
23 0xr 11309 . . . . . 6 0 ∈ ℝ*
2423a1i 11 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 0 ∈ ℝ*)
25 pnfxr 11316 . . . . . 6 +∞ ∈ ℝ*
2625a1i 11 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → +∞ ∈ ℝ*)
279adantr 480 . . . . . . . 8 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋𝑉)
2811adantr 480 . . . . . . . . 9 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
29 simpr 484 . . . . . . . . 9 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
3028, 29fge0iccico 46390 . . . . . . . 8 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
3127, 30sge0reval 46392 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
32 elinel2 4201 . . . . . . . . . . . . 13 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
3332adantl 481 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
3411ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝐹:𝑋⟶(0[,]+∞))
35 elinel1 4200 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
36 elpwi 4606 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
3735, 36syl 17 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥𝑋)
3837adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑋)
3938adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑥𝑋)
40 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑥)
4139, 40sseldd 3983 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
4234, 41ffvelcdmd 7104 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,]+∞))
4342adantllr 719 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,]+∞))
44 nne 2943 . . . . . . . . . . . . . . . . . 18 (¬ (𝐹𝑦) ≠ +∞ ↔ (𝐹𝑦) = +∞)
4544biimpi 216 . . . . . . . . . . . . . . . . 17 (¬ (𝐹𝑦) ≠ +∞ → (𝐹𝑦) = +∞)
4645eqcomd 2742 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑦) ≠ +∞ → +∞ = (𝐹𝑦))
4746adantl 481 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ ¬ (𝐹𝑦) ≠ +∞) → +∞ = (𝐹𝑦))
4811ffund 6739 . . . . . . . . . . . . . . . . . 18 (𝜑 → Fun 𝐹)
49483ad2ant1 1133 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → Fun 𝐹)
50413impa 1109 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑋)
5111fdmd 6745 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom 𝐹 = 𝑋)
5251eqcomd 2742 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑋 = dom 𝐹)
53523ad2ant1 1133 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑋 = dom 𝐹)
5450, 53eleqtrd 2842 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑦 ∈ dom 𝐹)
55 fvelrn 7095 . . . . . . . . . . . . . . . . 17 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ ran 𝐹)
5649, 54, 55syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ran 𝐹)
5756ad5ant134 1368 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ ¬ (𝐹𝑦) ≠ +∞) → (𝐹𝑦) ∈ ran 𝐹)
5847, 57eqeltrd 2840 . . . . . . . . . . . . . 14 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ ¬ (𝐹𝑦) ≠ +∞) → +∞ ∈ ran 𝐹)
5929ad3antrrr 730 . . . . . . . . . . . . . 14 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ ¬ (𝐹𝑦) ≠ +∞) → ¬ +∞ ∈ ran 𝐹)
6058, 59condan 817 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ≠ +∞)
61 ge0xrre 45549 . . . . . . . . . . . . 13 (((𝐹𝑦) ∈ (0[,]+∞) ∧ (𝐹𝑦) ≠ +∞) → (𝐹𝑦) ∈ ℝ)
6243, 60, 61syl2anc 584 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ)
6333, 62fsumrecl 15771 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) ∈ ℝ)
6463ralrimiva 3145 . . . . . . . . . 10 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑦𝑥 (𝐹𝑦) ∈ ℝ)
65 eqid 2736 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
6665rnmptss 7142 . . . . . . . . . 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 11306 . . . . . . . . . 10 ℝ ⊆ ℝ*
6968a1i 11 . . . . . . . . 9 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ℝ ⊆ ℝ*)
7067, 69sstrd 3993 . . . . . . . 8 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ*)
71 supxrcl 13358 . . . . . . . 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 2840 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ ℝ*)
7473adantlr 715 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ ℝ*)
7552adantr 480 . . . . . . . . 9 ((𝜑𝐹 ≠ ∅) → 𝑋 = dom 𝐹)
76 neneq 2945 . . . . . . . . . . . 12 (𝐹 ≠ ∅ → ¬ 𝐹 = ∅)
7776adantl 481 . . . . . . . . . . 11 ((𝜑𝐹 ≠ ∅) → ¬ 𝐹 = ∅)
78 frel 6740 . . . . . . . . . . . . . 14 (𝐹:𝑋⟶(0[,]+∞) → Rel 𝐹)
7911, 78syl 17 . . . . . . . . . . . . 13 (𝜑 → Rel 𝐹)
8079adantr 480 . . . . . . . . . . . 12 ((𝜑𝐹 ≠ ∅) → Rel 𝐹)
81 reldm0 5937 . . . . . . . . . . . 12 (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
8280, 81syl 17 . . . . . . . . . . 11 ((𝜑𝐹 ≠ ∅) → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
8377, 82mtbid 324 . . . . . . . . . 10 ((𝜑𝐹 ≠ ∅) → ¬ dom 𝐹 = ∅)
8483neqned 2946 . . . . . . . . 9 ((𝜑𝐹 ≠ ∅) → dom 𝐹 ≠ ∅)
8575, 84eqnetrd 3007 . . . . . . . 8 ((𝜑𝐹 ≠ ∅) → 𝑋 ≠ ∅)
86 n0 4352 . . . . . . . 8 (𝑋 ≠ ∅ ↔ ∃𝑧 𝑧𝑋)
8785, 86sylib 218 . . . . . . 7 ((𝜑𝐹 ≠ ∅) → ∃𝑧 𝑧𝑋)
8887adantr 480 . . . . . 6 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → ∃𝑧 𝑧𝑋)
8923a1i 11 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → 0 ∈ ℝ*)
9011ffvelcdmda 7103 . . . . . . . . . . . . 13 ((𝜑𝑧𝑋) → (𝐹𝑧) ∈ (0[,]+∞))
9190adantlr 715 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ (0[,]+∞))
92 nne 2943 . . . . . . . . . . . . . . . . 17 (¬ (𝐹𝑧) ≠ +∞ ↔ (𝐹𝑧) = +∞)
9392biimpi 216 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑧) ≠ +∞ → (𝐹𝑧) = +∞)
9493eqcomd 2742 . . . . . . . . . . . . . . 15 (¬ (𝐹𝑧) ≠ +∞ → +∞ = (𝐹𝑧))
9594adantl 481 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) ∧ ¬ (𝐹𝑧) ≠ +∞) → +∞ = (𝐹𝑧))
9611adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑋) → 𝐹:𝑋⟶(0[,]+∞))
9796ffund 6739 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑋) → Fun 𝐹)
98 simpr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑋) → 𝑧𝑋)
9952adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑋) → 𝑋 = dom 𝐹)
10098, 99eleqtrd 2842 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑋) → 𝑧 ∈ dom 𝐹)
101 fvelrn 7095 . . . . . . . . . . . . . . . . 17 ((Fun 𝐹𝑧 ∈ dom 𝐹) → (𝐹𝑧) ∈ ran 𝐹)
10297, 100, 101syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑋) → (𝐹𝑧) ∈ ran 𝐹)
103102adantlr 715 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ran 𝐹)
104103adantr 480 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) ∧ ¬ (𝐹𝑧) ≠ +∞) → (𝐹𝑧) ∈ ran 𝐹)
10595, 104eqeltrd 2840 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) ∧ ¬ (𝐹𝑧) ≠ +∞) → +∞ ∈ ran 𝐹)
10629ad2antrr 726 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) ∧ ¬ (𝐹𝑧) ≠ +∞) → ¬ +∞ ∈ ran 𝐹)
107105, 106condan 817 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ≠ +∞)
108 ge0xrre 45549 . . . . . . . . . . . 12 (((𝐹𝑧) ∈ (0[,]+∞) ∧ (𝐹𝑧) ≠ +∞) → (𝐹𝑧) ∈ ℝ)
10991, 107, 108syl2anc 584 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℝ)
110109rexrd 11312 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℝ*)
11173adantr 480 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (Σ^𝐹) ∈ ℝ*)
11223a1i 11 . . . . . . . . . . . 12 ((𝜑𝑧𝑋) → 0 ∈ ℝ*)
11325a1i 11 . . . . . . . . . . . 12 ((𝜑𝑧𝑋) → +∞ ∈ ℝ*)
114 iccgelb 13444 . . . . . . . . . . . 12 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹𝑧) ∈ (0[,]+∞)) → 0 ≤ (𝐹𝑧))
115112, 113, 90, 114syl3anc 1372 . . . . . . . . . . 11 ((𝜑𝑧𝑋) → 0 ≤ (𝐹𝑧))
116115adantlr 715 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → 0 ≤ (𝐹𝑧))
11770adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ*)
118 snelpwi 5447 . . . . . . . . . . . . . . . 16 (𝑧𝑋 → {𝑧} ∈ 𝒫 𝑋)
119 snfi 9084 . . . . . . . . . . . . . . . . 17 {𝑧} ∈ Fin
120119a1i 11 . . . . . . . . . . . . . . . 16 (𝑧𝑋 → {𝑧} ∈ Fin)
121118, 120elind 4199 . . . . . . . . . . . . . . 15 (𝑧𝑋 → {𝑧} ∈ (𝒫 𝑋 ∩ Fin))
122121adantl 481 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → {𝑧} ∈ (𝒫 𝑋 ∩ Fin))
123 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → 𝑧𝑋)
124109recnd 11290 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℂ)
125 fveq2 6905 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
126125sumsn 15783 . . . . . . . . . . . . . . . 16 ((𝑧𝑋 ∧ (𝐹𝑧) ∈ ℂ) → Σ𝑦 ∈ {𝑧} (𝐹𝑦) = (𝐹𝑧))
127123, 124, 126syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → Σ𝑦 ∈ {𝑧} (𝐹𝑦) = (𝐹𝑧))
128127eqcomd 2742 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) = Σ𝑦 ∈ {𝑧} (𝐹𝑦))
129 sumeq1 15726 . . . . . . . . . . . . . . 15 (𝑥 = {𝑧} → Σ𝑦𝑥 (𝐹𝑦) = Σ𝑦 ∈ {𝑧} (𝐹𝑦))
130129rspceeqv 3644 . . . . . . . . . . . . . 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 5968 . . . . . . . . . . . . . 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 257 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
135 supxrub 13367 . . . . . . . . . . . 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 2742 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) = (Σ^𝐹))
138137adantr 480 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) = (Σ^𝐹))
139136, 138breqtrd 5168 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ≤ (Σ^𝐹))
14089, 110, 111, 116, 139xrletrd 13205 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → 0 ≤ (Σ^𝐹))
141140ex 412 . . . . . . . 8 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑧𝑋 → 0 ≤ (Σ^𝐹)))
142141adantlr 715 . . . . . . 7 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (𝑧𝑋 → 0 ≤ (Σ^𝐹)))
143142exlimdv 1932 . . . . . 6 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (∃𝑧 𝑧𝑋 → 0 ≤ (Σ^𝐹)))
14488, 143mpd 15 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 0 ≤ (Σ^𝐹))
145 pnfge 13173 . . . . . . 7 ((Σ^𝐹) ∈ ℝ* → (Σ^𝐹) ≤ +∞)
14673, 145syl 17 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ≤ +∞)
147146adantlr 715 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ≤ +∞)
14824, 26, 74, 144, 147eliccxrd 45545 . . . 4 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ (0[,]+∞))
14919, 21, 22, 148syl21anc 837 . . 3 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ (0[,]+∞))
15018, 149pm2.61dan 812 . 2 ((𝜑 ∧ ¬ 𝐹 = ∅) → (Σ^𝐹) ∈ (0[,]+∞))
1518, 150pm2.61dan 812 1 (𝜑 → (Σ^𝐹) ∈ (0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wex 1778  wcel 2107  wne 2939  wral 3060  wrex 3069  cin 3949  wss 3950  c0 4332  𝒫 cpw 4599  {csn 4625   class class class wbr 5142  cmpt 5224  dom cdm 5684  ran crn 5685  Rel wrel 5689  Fun wfun 6554  wf 6556  cfv 6560  (class class class)co 7432  Fincfn 8986  supcsup 9481  cc 11154  cr 11155  0cc0 11156  +∞cpnf 11293  *cxr 11295   < clt 11296  cle 11297  [,]cicc 13391  Σcsu 15723  Σ^csumge0 46382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-z 12616  df-uz 12880  df-rp 13036  df-ico 13394  df-icc 13395  df-fz 13549  df-fzo 13696  df-seq 14044  df-exp 14104  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-clim 15525  df-sum 15724  df-sumge0 46383
This theorem is referenced by:  sge0ge0  46404  sge0xrcl  46405  sge0split  46429  sge0iunmptlemre  46435  sge0iunmpt  46438  sge0nemnf  46440  sge0clmpt  46445  sge0isum  46447  psmeasure  46491  ovnsupge0  46577  ovnsubaddlem1  46590  sge0hsphoire  46609  hoidmvlelem1  46615  hspmbllem2  46647
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